Vijaya Sree

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Go Math Grade 7 Answer Key Chapter 5 Percent Increase and Decrease

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Chapter 5 – Percent Increase and Decrease

• Percent Increase and Decrease – Guided Practice – Page No. 144
• Percent Increase and Decrease – Independent Practice – Page No. 145
• Percent Increase and Decrease – Page No. 146

Chapter 5 – Rewriting Percent Expressions

• Rewriting Percent Expressions – Guided Practice – Page No. 150
• Rewriting Percent Expressions – Independent Practice – Page No. 151
• Rewriting Percent Expressions – Page No. 152

Chapter 5 – Applications of Percent

• Applications of Percent – Guided Practice – Page No. 156
• Applications of Percent – Independent Practice – Page No. 157
• Applications of Percent – Page No. 158

Chapter 5

• MODULE QUIZ – 5.1 Percent Increase and Decrease – Page No. 159
• Selected Response – Page No. 160
• Module 5 – Page No. 162
• Unit 2 Performance Tasks – Page No. 163
• Unit 2 Performance Tasks (con’td) – Page No. 164
• Selected Response – Page No. 165
• Page No. 166

Percent Increase and Decrease – Guided Practice – Page No. 144

Find each percent increase. Round to the nearest percent.

Question 1.
From $5 to $8
______ %

Answer: 60%

Explanation:
Percent Charge = Amount of Change/Original Amount
Original amount = 5
Final amount = 8
8 – 5 = 3
Percent change = 3/5 = 0.6 = 60%

Question 2.
From 20 students to 30 students
______ %

Answer: 50%

Explanation:
Percent Charge = Amount of Change/Original Amount
Original amount = 20
Final amount = 30
We find the amount of change
30 – 20 = 10
We determine the percent of the increase
Percent change = 10/20 = 0.5 = 50%

Question 5.
From 13 friends to 14 friends
______ %

Answer: 8%

Explanation:
Percent Charge = Amount of Change/Original Amount
Original amount = 13
Final amount = 14
We find the amount of change
14 – 13 = 1
We determine the percent of increase and round it to the nearest percent
Percent Change = 1/13 ≈ 0.08 = 8%

Question 6.
From 5 miles to 16 miles
______ %

Answer: 220%

Explanation:
Percent Charge = Amount of Change/Original Amount
Original amount = 5
Final amount = 16
We find the amount of change
16 – 5 = 11
We determine the percent of increase and round it to the nearest percent
Percent Change = 11/5 = 2.2 = 220%

Question 7.
Nathan usually drinks 36 ounces of water per day. He read that he should drink 64 ounces of water per day. If he starts drinking 64 ounces, what is the percent increase? Round to the nearest percent.
______ %

Answer: 78%

Explanation:
Given,
Nathan usually drinks 36 ounces of water per day. He read that he should drink 64 ounces of water per day.
Original Amount: 36
Final Amount: 64
Percent Charge = Amount of Change/Original Amount
We find the amount of change
64 – 36 = 28
We determine the percent of increase and round it to the nearest percent
Percent Change = 28/36 ≈ 0.78 = 78%
Thus the nearest percent is 78%

Find each percent decrease. Round to the nearest percent.

Question 8.
From $80 to $64
______ %

Answer: 20%

Explanation:
Percent Charge = Amount of Change/Original Amount
Original amount = 80
Final amount = 64
We find the amount of change
Amount of change = Greater value – Lesser value
= 80 – 64 = 16
We determine the percent of increase and round it to the nearest percent
Percent Change = 16/80 = 0.20 = 20%
Thus the nearest percent is 20%

Question 11.
From 145 pounds to 132 pounds
______ %

Answer: 9%

Explanation:
Percent Charge = Amount of Change/Original Amount
Original amount = 145
Final amount = 132
We find the amount of change
Amount of change = Greater value – Lesser value
145 – 132 = 13
We determine the percent of increase and round it to the nearest percent
Percent Change = 13/145 ≈ 0.09 = 9%
The nearest percent is 9%

Question 12.
From 64 photos to 21 photos
______ %

Answer: 67%

Explanation:
Percent Charge = Amount of Change/Original Amount
Original amount = 64
Final amount = 21
We find the amount of change
Amount of change = Greater value – Lesser value
64 – 21 = 43
We determine the percent of increase and round it to the nearest percent
Percent Change = 43/64 ≈ 0.67 = 67%
Therefore the nearest percent is 67%

Question 13.
From 16 bagels to 0 bagels
______ %

Answer: 100%

Explanation:
Percent Charge = Amount of Change/Original Amount
Original amount = 16
Final amount = 0
We find the amount of change
Amount of change = Greater value – Lesser value
16 – 0 = 16
We determine the percent of increase and round it to the nearest percent
Percent Change = 16/16 = 1.0% = 100%

Find the new amount given the original amount and the percent of change.

Question 15.
$9; 10% increase
$ ______

Answer: $9.90

Explanation:
Percent Charge = Amount of Change/Original Amount
Original amount = 9
Increase = 10%
We find the amount of change
0.1 × 9 = 0.90
New Amount = Original Amount + Amount of Change
9 + 0.90 = 9.90

Question 16.
48 cookies; 25% decrease
______ cookies

Answer: 36 cookies

Explanation:
Original amount = 48
Decrease = 25%
We find the amount of change
0.25 × 48 = 12
New Amount = Original Amount – Amount of Change
48 – 12 = 36
Thus the answer is 36 cookies.

Question 17.
340 pages; 20% decrease
______ pages

Answer: 272 pages

Explanation:
Original Amount: 340 pages
Decrease: 20%
We find the amount of change
0.20 × 340 = 68
New Amount = Original Amount – Amount of Change
340 – 68 = 272
The answer is 272 pages.

Question 18.
28 members; 50% increase
______ members

Answer: 42 members

Explanation:
Original Amount: 28
Increase: 50%
We find the amount of change
0.5 × 28 = 14
New amount = Original Amount + Amount of Change
28 + 14 = 42
The answer is 42 members

Question 19.
$29,000; 4% decrease
$ ______

Answer: $27,840

Explanation:
Original Amount: 29000
Decrease: 4%
We find the amount of change
0.04 × 29000 = 1160
New Amount = Original Amount – Amount of Change
29000 – 1160 = 27840
The answer is $27,840

Essential Question Check-In

Question 22.
What process do you use to find the percent change of a quantity?
Type below:
_____________

Answer: In order to find the percent change of a quantity, we determine the amount of change in the quantity and divide it by the original amount.

Percent Increase and Decrease – Independent Practice – Page No. 145

Question 23.
Complete the table.

Type below:
_____________

Answer: bike: 13%, scooter 24%, increase, tennis racket: $83, skis: $435

Explanation:
Since the new price is less than the original price, it is a percent decrease. percent decreases can be found using the equation percent decrease = (original – new)/original
Bike: 110 – 96/110 = 14/110 ≈ 13%
Scooter: 56 – 45/45 = 11/45 ≈ 24%
Use the equation percent increase = new – original/original
let x be the new price
skis: (580 – x)/580 = 0.25
580 – x = 0.25 × 580
580 – x = 145
x = 580 – 145 = 435
The new price is $435

Question 24.
Multiple Representations
The bar graph shows the number of hurricanes in the Atlantic Basin from 2006–2011.

a. Find the amount of change and the percent of decrease in the number of hurricanes from 2008 to 2009 and from 2010 to 2011. Compare the amounts of change and percents of decrease.
Type below:
_____________

Answer: 2008 to 2009 has a smaller amount of change but a larger percent of decrease.

Explanation:
2008 to 2009:
amount of change: 8 – 3 = 5
percent decrease: 5/8 = 0.625 = 62.5%
2010 to 2011:
amount of change: 12 – 7 = 5
percent decrease: 5/12 ≈ 0.416 = 41.6%
The amount of change for 2010 to 2011 was greater than the amount of change for 2008 to 2009 but 2008 to 209 had a greater percent decrease than 2010 to 2011.

Question 24.
b. Between which two years was the percent of change the greatest? What was the percent of change during that period?
_______ %

Answer: 2009 and 2010, 300%

Explanation:
Use the percent change = amount of change/original amount.
The biggest change in heights is between 2009 and 2010.
The percent change is (12-3)/3 = 9/3 = 3 = 300%

Question 25.
Represent Real-World Problems
Cheese sticks that were previously priced at “5 for $1” are now “4 for $1”. Find each percent of change and show your work.
a. Find the percent decrease in the number of cheese sticks you can buy for $1.
_______ %

Answer: 20% decrease

Explanation:
Use the percent change = amount of change/original amount.
(5 – 4)/5 = 1/5 = 0.2 = 20% decrease

Question 25.
b. Find the percent increase in the price per cheese stick.
_______ %

Answer: 25% increase

Explanation:
First, find the price per cheese stick at each price.
Use the percent change = amount of change/original amount.
1.00/5 = 0.20
1/4 = 0.25
(0.25 – 0.20)/0.20 = 0.05/0.20 = 25% increase

Percent Increase and Decrease – Page No. 146

Question 26.
Percent error calculations are used to determine how close to the true values, or how accurate, experimental values really are. The formula is similar to finding percent of change.

chemistry class, Charlie records the volume of a liquid as 13.3 milliliters. The actual volume is 13.6 milliliters. What is his percent error? Round to the nearest percent.
_______ %

Answer: 2%

Explanation:
Use the formula

|13.3 – 13.6|/13.6 = |-0.3|/13.6 ≈ 0.02 = 2%

H.O.T.

Focus on Higher Order Thinking

Question 27.
Look for a Pattern
Leroi and Sylvia both put $100 in a savings account. Leroi decides he will put in an additional $10 each week. Sylvia decides to put in an additional 10% of the amount in the account each week.
a. Who has more money after the first additional deposit? Explain.
___________

Answer: the same

Explanation:
Since 10% of 100 is 100(0.10) = 10, they both make an additional deposit of 10, so they have the same amount of money after the first additional deposit.

Question 27.
b. Who has more money after the second additional deposit? Explain.
___________

Answer: Sylvia

Explanation:
Both Lerio and Sylvia have $110 in their account after their first deposits since they both started with $100 and both deposited $10 for their first deposit.
After the second deposit, Lerio has 110 + 10 = $120.
Sylvia has 110 + 0.10(110) = 110 + 11 = $121
So she has more money after the second deposit.

Question 27.
c. How do you think the amounts in the two accounts will compare after a month? A year?
Type below:
___________

Answer: Sylvia will continue to have more money after a month and a year since 10% of the balance is going to be greater than the 10 deposit that Leroi is making.

Question 28.
Critical Thinking
Suppose an amount increases by 100%, then decreases by 100%. Find the final amount. Would the situation change if the original increase was 150%? Explain your reasoning.
Type below:
___________

Answer: If an amount increases by 100%, then it will double. If it then decreases by 100%, it will become 0.
If you increase a number by 150% and then decrease it by 150%, you will not get to 0. 150% increase of 100 is 100 + 150 = 250.
A decrease of 150% is then 250 – 1.5(250) = 250 – 375 = -125

Rewriting Percent Expressions – Guided Practice – Page No. 150

Question 1.
Dana buys dress shirts from a clothing manufacturer for s dollars each and then sells the dress shirts in her retail clothing store at a 35% markup.
a. Write the markup as a decimal.
______

Answer: To convert a percent to a decimal, move the decimal place two places to the left. Therefore, 35% as a decimal is 0.35.

Question 1.
b. Write an expression for the retail price of the dress shirt.
Type below:
___________

Answer:
To write the expression, use the formula
retail price = original place + markup
Since s is the original place, if the markup is 35% = 0.35, then the markup is 0.35s.

Question 1.
c. What is the retail price of a dress shirt that Dana purchased for $32.00?
$ ______

Answer: Plugging in s = 32 into the expression gives a retail price of 1.35 = 1.35(32) = $43.20

Question 1.
d. How much was added to the original price of the dress shirt?
$ ______

Answer: The amount added to the original price is the amount of the markup. Since the amount of the markup is 0.35s and s = 32, then the amount of the markup was 0.35s = 0.35(32) = $11.20.
You can also find the amount of markup by subtracting the retail price and the original price. Since the retail price is $43.20 and the original price is $32, then the markup amount is $43.20 – $32 = $11.20

List the markup and retail price of each item. Round to two decimal places when necessary.

Question 2.

Markup: $ ______ Retail Price: $ ______

Answer: Markup: $ 2.70 Retail Price: $ 20.70

Explanation:

Use the formula markup = price(markup%)
18(0.15) = 2.70
Use the retail price formula = price + markup
18 + 2.70 = 20.70

Question 3.

Markup: $ ______ Retail Price: $ ______

Answer:

Use the formula markup = (price)(markup %)
22.50(0.42) = 9.45
Use the retail price formula = price + markup
22.50 + 9.45 = 31.95

Question 4.

Markup: $ ______ Retail Price: $ ______

Answer:

Use the formula markup = (price)(markup %)
= 33.75(0.75) = 25.31
Use the formula retail price = price + markup
33.75 + 25.31 = 59.06

Question 5.

Markup: $ ______ Retail Price: $ ______

Answer:

Use the formula markup = (price)(markup %)
= 74.99(0.33) = 24.75
Use the formula retail price = price + markup
74.99 + 24.75 = 99.74

Question 6.

Markup: $ ______ Retail Price: $ ______

Answer:

Use the formula markup = (price)(markup %)
48.60(1.00) = 48.60
Use the formula retail price = price + markup
48.60 + 48.60 = 97.20

Question 7.

Markup: $ ______ Retail Price: $ ______

Answer:

Use the formula markup = (price)(markup %)
= 185 × 1.25 = 231.25
Use the formula retail price = price + markup
185 + 231.25 = 461.25

Find the sale price of each item. Round to two decimal places when necessary.

Question 8.
Original price: $45.00; Markdown: 22%
$ ______

Answer:
Use the formula markup = (price)(markup %)
45(0.22) = 9.90
Markdown is 9.90
Use the formula retail price = price + markup
45 – 9.90 = 35.10
The sale price is $35.10

Question 11.
Original price: $279.99, Markdown: 75%
$ ______

Answer:
Use the formula markup = (price)(markup %)
279.99 × 0.75 = 209.99
Use the formula retail price = price – markup
279.99 – 209.99 = 70

Essential Question Check-In

Rewriting Percent Expressions – Independent Practice – Page No. 151

Question 13.
A bookstore manager marks down the price of older hardcover books, which originally sell for b dollars, by 46%.
a. Write the markdown as a decimal.
______

Answer: 0.46

Explanation:
To convert a percent to decimal form, move the decimal point 2 places to the left and don’t write the percent symbol. Therefore, 46% as a decimal is 0.46.

Question 13.
b. Write an expression for the sale price of the hardcover book.
Type below:
____________

Answer: 0.54b

Explanation:
The sale price is the original price minus the discount amount. If the original price is discounted 46% and the original price is b dollars, the amount of the discount is 46% of b = 0.46b.
The sale price is then b – 0.46b = (1 – 0.46)b = 0.54b

Question 13.
c. What is the sale price of a hardcover book for which the original retail price was $29.00?
$ ______

Answer: $15.66

Explanation:
From part (b), the sale price of an item with an original price of b dollars is 0.54b. If the original price is then b = 29 dollars, the sale price is 0.54b = 0.54 × 29 = $15.66

Question 13.
d. If you buy the book in part c, how much do you save by paying the sale price?
$ ______

Answer: $13.34

Explanation:
The amount of savings is the difference between the original price and the sale price. If the original price is $29 and the sale price is $15.66, then the amount of savings is $29.00 – $15.66 = $13.34

Question 14.
Raquela’s coworker made price tags for several items that are to be marked down by 35%. Match each Regular Price to the correct Sale Price, if possible. Not all sales tags match an item.

Type below:
_____________

Answer:
35% markdown means the expression for the sales price is p – 0.35p = 0.65p. Plug in the regular prices for p to find the sale prices. Remember the directions stated not all sales tags will match a regular price so you won’t be able to match every regular price ticket with a sale price ticket.
0.65(3.29) = 2.14
0.65(4.19) = 2.72
0.65(2.79) = 1.81
0.65(3.09) = 2.01
0.65(3.77) = 2.45

Question 15.
Communicate Mathematical Ideas
For each situation, give an example that includes the original price and final price after markup or markdown.
a. A markdown that is greater than 99% but less than 100%
Type below:
_____________

Answer:
A markdown that is greater than 99% but less than 100% could be 99.5%. If the original price is $100, then the final price is 100 – 100(0.995) = 100 – 99.50 = 0.50

Question 15.
b. A markdown that is less than 1%
Type below:
_____________

Answer:
A markdown that is less then 1% could be 0.5%. If the original price is $100, then the final price would be 100 – 0.005(100) = 100 – 0.50 = 99.50

Question 15.
c. A markup that is more than 200%
Type below:
_____________

Answer:
A markup that is more than 200% could be 300%. If the original price is $100, then the final price would be 100 + 100 (3.00) = 100 + 300 = 400

Rewriting Percent Expressions – Page No. 152

Question 16.
Represent Real-World Problems
Harold works at a men’s clothing store, which marks up its retail clothing by 27%. The store purchases pants for $74.00, suit jackets for $325.00, and dress shirts for $48.00. How much will Harold charge a customer for two pairs of pants, three dress shirts, and a suit jacket?
$ __________

Answer: $783.59

Explanation:
Given,
Harold works at a men’s clothing store, which marks up its retail clothing by 27%.
The store purchases pants for $74.00, suit jackets for $325.00, and dress shirts for $48.00.
If the markup is 27%, then the expression for the retail price is p + 0.27p = 1.27p
where p is the original price.
The retail price of the pants is then 1.27(74) = 93.98.
The retail price of the suit jackets is 1.27(325) = 412.75
The retail price of the dress shirts is 1.27(48) = 60.96
The total for two pants, three dress shirts, and one suit jacket would then be 2(93.98) + 3(60.96) + 412.75
= 187.96 + 182.88 + 412.75 = 783.59

Question 17.
Analyze Relationships
Your family needs a set of 4 tires. Which of the following deals would you prefer? Explain.

Type below:
____________

Answer: I and III

Explanation:
The percent discount for buying 3 tires and getting one free is 25% since you are getting 1/4 of the tires for free and 1/4 off = 25%.
This means deal (I) and deal (III) are the same. They are greater than a 20% discount so deals (I) and (III) are preferable.

H.O.T.

Focus on Higher Order Thinking

Question 18.
Critique Reasoning
Margo purchases bulk teas from a warehouse and marks up those prices by 20% for retail sale. When teas go unsold for more than two months, Margo marks down the retail price by 20%. She says that she is breaking even, that is, she is getting the same price for the tea that she paid for it. Is she correct? Explain.
_______

Answer:
She is not correct. If she originally purchases the teas for $100 and then marks the price up 20%,
the retail price would then be 100 + 0.20(100) = 100 + 20 = 120.
The sales price would then be 120 – 0.2(120) = 120 – 24 = 96.
This is less than the purchase price so she is losing money, and not breaking even.

Question 20.
Persevere in Problem-Solving
At Danielle’s clothing boutique, if an item does not sell for eight weeks, she marks it down by 15%. If it remains unsold after that, she marks it down an additional 5% each week until she can no longer make a profit. Then she donates it to charity.

Rafael wants to buy a coat originally priced at $150, but he can’t afford more than $110. If Danielle paid $100 for the coat, during which week(s) could Rafael buy the coat within his budget? Justify your answer.
Type below:
_____________

Answer:
The expression for the markdown on the 8th week is p – 0.15p = 0.85p since it will get marked down 15% on the 8th week.
The expression for the additional markdowns is p – 0.05p = 0.95p since it will get marked down an additional 5% every week after the 8th week.
On the 8th week, it will be marked down to 0.85(150) = 127.50. This is more than Rafael can afford.
On the 9th week, it will be marked down to 0.95(127.50) = 121.13. This is still more than Rafael can afford.
On the 10th week, it will be marked down to 0.95(121.13) = 115.07. This is still more than Rafael can afford.
On the 11th week, it will be marked down to 0.95(115.07) = 109.32. Rafael can afford this price so he must wait until the 11th week.

Applications of Percent – Guided Practice – Page No. 156

Question 1.
5% of $30 =
$ _______

Answer: $1.5

Explanation:
We have to find:
5% of $30
0.50 × 30 = $1.5

Question 4.
150% of $22 =
$ _______

Answer: $33

Explanation:
We have to find:
150% of $22
1.5 × 22 = 33

Question 5.
1% of $80 =
$ _______

Answer: $0.8

Explanation:
We have to find:
1% of $80
0.01 × 80 = 0.8

Question 6.
200% of $5 =
$ _______

Answer: $10

Explanation:
We have to find:
200% of $5
2 × 5 = 10

Question 7.
Brandon buys a radio for $43.99 in a state where the sales tax is 7%.
a. How much does he pay in taxes?
$ _______

Answer: 3.08

Explanation:
We have to find the amount he pays in taxes by multiplying the cost by the sales tax percentage in decimal form remember to round to 2 decimal places.
43.99(0.07) = 3.08

Question 7.
b. What is the total Brandon pays for the radio?
$ _______

Answer: 47.07

Explanation:
To find the total Brandon pays for the radio we have to add the sales tax amount to the cost to find the total amount he pays.
43.99 + 3.08 = 47.07
Thus the total Brandon pays for the radio is $47.07.

Question 8.
Luisa’s restaurant bill comes to $75.50, and she leaves a 15% tip. What is Luisa’s total restaurant bill?
$ _______

Answer: $86.25

Explanation:
Given that,
Luisa’s restaurant bill comes to $75.50, and she leaves a 15% tip.
Use the formula for the total restaurant bill:
T = P + x. P
Where T represents the total bill, P represents Luisa’s bill and x represents percents for tip, then the total restaurant bill is:
T = 75 + 0.15 (75)
T = 75 + 11.25
T = $86.25
Therefore Lusia’s total restaurant bill is $86.25

Question 9.
Joe borrowed $2,000 from the bank at a rate of 7% simple interest per year. How much interest did he pay in 5 years?
$ _______

Answer: 700

Explanation:
Joe borrowed $2,000 from the bank at a rate of 7% simple interest per year.
We have to find the amount of interest per year
2000(0.07) = 140
Find the amount of interest for 5 years
140(5) = 700
Thus Joe pays $700 in 5 years.

Question 11.
Martin finds a shirt on sale for 10% off at a department store. The original price was $20. Martin must also pay 8.5% sales tax.
a. How much is the shirt before taxes are applied?
$ _______

Answer: 18

Explanation:
We have to find the sales price of the shirt
20 – 0.1(20) = 20 – 2 = 18
The price of the shirt before taxes are applied is $18.

Question 11.
b. How much is the shirt after taxes are applied?
$ _______

Answer: 19.53

Explanation:
We have to find the price after the sales tax
18 + 0.085(18) = 18 + 1.53 = 19.53
The price of the shirt after taxes are applied is $19.53

Essential Question Check-In

Question 13.
How can you determine the total cost of an item including tax if you know the price of the item and the tax rate?
Type below:
_____________

Answer: You can find the total cost of an item including tax by first multiplying the price of the item by the tax rate in decimal form to get the amount of sales tax. Then add the amount of sales tax to the price to get the total cost.

Applications of Percent – Independent Practice – Page No. 157

Question 14.
Emily’s meal costs $32.75 and Darren’s meal costs $39.88. Emily treats Darren by paying for both meals and leaves a 14% tip. Find the total cost.
$ _______

Answer: 82.80

Explanation:
Emily’s meal costs $32.75 and Darren’s meal costs $39.88.
So, the total cost of the meals before tip is $32.75 + $39.88 = $72.63
Emily treats Darren by paying for both meals and leaves a 14% tip.
$72.63 = 0.14(72.63) ≈ $10.17
Round to two decimal places since dollar amounts must be rounded to the nearest cent.
The total cost that Dareen pays is then cost before tip + amount of tip = $72.63 + $10.17 = $82.80

Question 15.
The Jayden family eats at a restaurant that is having a 15% discount promotion. Their meal costs $78.65, and they leave a 20% tip. If the tip applies to the cost of the meal before the discount, what is the total cost of the meal?
$ _______

Answer: 82.58

Explanation:
The Jayden family eats at a restaurant that is having a 15% discount promotion.
The total cost of the meal = cost of meal + tip amount – discount amount
Their meal costs $78.65, and they leave a 20% tip.
We need to find the tip amount and the discount amount using the given cost of the meal, tip percent, and discount percent.
20% of 78.65 = 0.20 × 78.65 = $15.73
Since the cost of the meal before the discount is $78.65 and the discount percentage is 15%, then the amount of the discount is
15% of 78365 = 0.15 × $78.65 ≈ $11.80
The total cost is then
78.65 + 15.73 – 11.80 = $82.58

Question 18.
Kedar earns a monthly salary of $2,200 plus a 3.75% commission on the amount of his sales at a men’s clothing store. What would he earn this month if he sold $4,500 in clothing? Round to the nearest cent.
$ _______

Answer: 2368.75

Explanation:
Given,
Kedar earns a monthly salary of $2,200 plus a 3.75% commission on the amount of his sales at a men’s clothing store.
4500 × 0.0375 = 168.75
The total earnings can be known by adding his monthly salary and his commission.
2200 + 168.75 = 2368.75

Question 19.
Danielle earns a 7.25% commission on everything she sells at the electronics store where she works. She also earns a base salary of $750 per week. How much did she earn last week if she sold $4,500 in electronics merchandise? Round to the nearest cent.
$ _______

Answer: 1076.25

Explanation:
Danielle earns a 7.25% commission on everything she sells at the electronics store where she works.
She also earns a base salary of $750 per week.
The amount she made in the commission is 4500 × 0.0725 = 326.25
We can find the total earnings by adding her weekly pay and commission.
750 + 326.25 = 1076.25
Thus she earns $1076.25 last week if she sold $4,500 in electronics merchandise.

Question 20.
Francois earns a weekly salary of $475 plus a 5.5% commission on sales at a gift shop. How much would he earn in a week if he sold $700 in goods? Round to the nearest cent.
$ _______

Answer: 513.50

Explanation:
Given that, Francois earns a weekly salary of $475 plus a 5.5% commission on sales at a gift shop.
The amount he made in commission
700 × 0.055 = 38.50
We can find the total amount he earned by adding his weekly pay and commission
475 + 38.50 = $513.50

Question 21.
Sandra is 4 feet tall. Pablo is 10% taller than Sandra, and Michaela is 8% taller than Pablo
a. Explain how to find Michaela’s height with the given information.
Type below:
_____________

Answer:
First, we have to find 10% of Sandra’s height: 0.10 × 4 = 0.4
This means that Pablo is then 4 + 0.4 = 4.4 feet tall.
Next find 8% of Pablo’s height: 4.4 × 0.08 = 0.352
This means that Michaela is 4.4 + 0.353 = 4.752 feet tall.

Question 21.
b. What is Michaela’s approximate height in feet and inches?
_______ feet _______ inches

Answer:
Convert from feet to inches.
1 feet = 12 inches
4.752 = 4 + 0.752
0.752 = 12 × 0.752 = 9 inches
4 feet = 12 × 4 = 48 inches
Thus the approximate height of Michaela is 4 feet 9 inches.

Question 22.
Eugene wants to buy jeans at a store that is giving $10 off everything. The tag on the jeans is marked 50% off. The original price is $49.98.
a. Find the total cost if the 50% discount is applied before the $10 discount.
$ _______

Answer: $14.99

Explanation:
Given that,
Eugene wants to buy jeans at a store that is giving $10 off everything.
The tag on the jeans is marked 50% off. The original price is $49.98.
0.5 × 49.98 = 24.99
Now subtract $10 discount.
24.99 – 10 = 14.99
The total cost if the 50% discount is applied before the $10 discount is $14.99

Question 22.
b. Find the total cost if the $10 discount is applied before the 50% discount.
$ _______

Answer: $19.99

Explanation:
We have to find the price after the $10 discount and then find 50% of that price to find the discounted price.
49.98 – 10 = 39.98
0.5 × 39.98 = 19.99
Thus the total cost if the $10 discount is applied before the 50% discount is $19.99

Applications of Percent – Page No. 158

Question 23.
Multistep
Eric downloads the coupon shown and goes shopping at Gadgets Galore, where he buys a digital camera for $95 and an extra battery for $15.99.
a. What is the total cost if the coupon is applied to the digital camera?

$ _______

Answer: 101.49

Explanation:
Use the formula for the discount price:
DP = P – x.P
Price for the digital camera:
DP = 95 – 0.1(95)
DP = 95 – 9.5
DP = $85.5
Total cost = 85.5 + 15.99 = $101.49

Question 23.
b. What is the total cost if the coupon is applied to the extra battery?
$ _______

Answer: 109.391

Explanation:
Use the formula for the discount price:
DP = P – x.P
Price for the digital camera:
DP = 15.99 – 0.1(15.99)
DP = 15.99 – 1.599
DP = $14.399
Total cost = 95 + 14.399 = $109.391

Question 23.
c. To which item should Eric apply the discount? Explain.
____________

Answer: He should apply the discount to the digital camera because then the total cost is lower.

Question 23.
d. Eric has to pay 8% sales tax after the coupon is applied. How much is his total bill?
$ _______

Answer:
Use the formula for Discount price
If he uses a coupon for the digital camera then his total cost will be
T = DP + 0.08 × DP
T = 101.49 + 8.1192
T = $109.6029
If he uses a coupon for the extra battery his total cost will be
T = DP + 0.08 × DP
T = 109.391 + 0.08(109.391)
T = $118.14228

Question 24.
Two stores are having sales on the same shirts. The sale at Store 1 is “2 shirts for $22” and the sale at Store 2 is “Each $12.99 shirt is 10% off”.
a. Explain how much will you save by buying at Store 1.
$ _______

Answer:
For store 1, the shirts are 2 for $22. Each shirt then costs $22 ÷ 2 = $11
At store 2, each shirt is 10% off of $12.99 so each shirt costs:
$12.99 – 0.1(12.99) = $12.99 – $1.30 = $11.69
You will then save $11.69 – $11.00 = 0.69 per shirt if you buy them from Store 1.

Question 24.
b. If Store 3 has shirts originally priced at $20.98 on sale for 55% off, does it have a better deal than the other stores? Justify your answer.
_______

Answer:
If Store 3 sells shirts at 55% off of $20.98, then each shirt costs:
$20.98 – 0.55($20.98) = $20.98 – $11.54 = $9.44
This is lower than the costs per shirt of Store 1 and Store 2 so it has a better deal.

H.O.T.

Focus on Higher Order Thinking

Question 26.
Multistep
In chemistry class, Bob recorded the volume of a liquid as 13.2 mL. The actual volume was 13.7 mL. Use the formula to find the percent error of Bob’s measurement to the nearest tenth of a percent.
_______ %

Answer: 3.6%

Explanation:

|13.2 – 13.7|/13.7 = |-0.5|/13.7
0.5/13.7 ≈ 0.036 = 3.6%

MODULE QUIZ – 5.1 Percent Increase and Decrease – Page No. 159

Find the percent change from the first value to the second.

Question 1.
36; 63
_______ %

Answer: 75%

Explanation:
Use the formula percent change = amount of change/first value
amount of change = 27
First value = 36
(63 – 36)/36 = 27/36 = 0.75 = 75%

5.2 Markup and Markdown

Use the original price and the markdown or markup to find the retail price.

Question 5.
Original price: $60; Markup: 15%
$ _______

Answer: $69

Explanation:
Use the formula retail price = original price + markup
60 + 60 × 0.15 = 60 + 9= 69

Question 8.
Original price: $125; Markdown: 30%
$ _______

Answer: 87.50

Explanation:

Use the formula retail price = original price + markup

125 – 125 × 0.3 = 125 – 37.50 = 87.50

5.3 Applications of Percent

Question 9.
Mae Ling earns a weekly salary of $325 plus a 6.5% commission on sales at a gift shop. How much would she make in a work week if she sold $4,800 worth of merchandise?
$ _______

Answer: 637

Explanation:

Mae Ling weekly earnings is equal to her weekly salary plus her commission.

Since she earns 6.5 % commission on sales, if she sold $4800 worth of merchandise, her commission earnings  would be 6.5 % of 4800 = 0.065 × 4800 = $312.

Since her weekly salary is 325, then her total weekly earnings is $325 + $312 = $637

Question 10.
Ramon earns $1,735 each month and pays $53.10 for electricity. To the nearest tenth of a percent, what percent of Ramon’s earnings are spent on electricity each month?
_______ %

Answer: 3.1%

Explanation:

Divide the electric payment by his monthly pay

53.10/1735 = 0.031 = 3.1%

Question 11.
James, Priya, and Siobhan work in a grocery store. James makes $7.00 per hour. Priya makes 20% more than James, and Siobhan makes 5% less than Priya. How much does Siobhan make per hour?
$ _______

Answer: 7.98 per hour

Explanation:

Since James makes $7 per hour and priya makes 20% more than this, find 20% of 7 and then add that to 7 to find the pay per hour for Priya.

7 + 0.2(7) = 7 + 1.40 = 8.40

Since Priya makes $8.40 per hour and Siobhan makes 5% less than this, find 5% of 8.40 and subtract that from 8.40 to find the pay per hour of Siobhan.

8.40 – 0.05(8.40) = 8.40 – 0.42 = 7.98

Question 12.

The Hu family goes out for lunch, and the price of the meal is $45. The sales tax on the meal is 6%, and the family also leaves a 20% tip on the pre-tax amount. What is the total cost of the meal?
$ _______

Answer: 56.70

Explanation:

Find the amount of tax

45 × 0.06 = 2.70

Find the amount of tip

45 × 0.20 = 9

Find the total cost by adding the cost of the meal, the tax, and the tip.

45 + 2.70 + 9 = $56.70

Essential Question

Question 13.
Give three examples of how percents are used in the real-world. Tell whether each situation represents a percent increase or a percent decrease.
Type below:
____________

Answer:

One example could be giving a tip when you eat at a restaurant. Since the cost increases, it represents a percent increase.

Second example is tax on purchase. Since the price increases it is a percent increase.

Third example is using a coupon when buying an item. Since the price decreases, it is a percent decrease.

Selected Response – Page No. 160

Question 1.
Zalmon walks \(\frac{3}{4}\) of a mile in \(\frac{3}{10}\) of an hour. What is his speed in miles per hour?
Options:
a. 0.225 miles per hour
b. 2.3 miles per hour
c. 2.5 miles per hour
d. 2.6 miles per hour

Answer: 2.5 miles per hour

Explanation:
Given that,
Zalmon walks \(\frac{3}{4}\) of a mile in \(\frac{3}{10}\) of an hour.
Divide the number of miles by the number of hours to get his speed in miles per hour.
\(\frac{3}{4}\) ÷ \(\frac{3}{10}\)
\(\frac{3}{4}\) ÷ \(\frac{10}{3}\) = \(\frac{5}{2}\)
Convert the fraction into decimal form.
\(\frac{5}{2}\) = 2.5 miles per hour
Thus the correct answer is option C.

Question 2.
Find the percent change from 70 to 56.
Options:
a. 20% decrease
b. 20% increase
c. 25% decrease
d. 25% increase

Answer: 20% increase

Explanation:
Use the percent change = amount of change/original amount.
Since the number decreased from 70 to 56, it is a percent decrease.
= (70 – 56)/70 = \(\frac{14}{70}\) = 0.2 = 20%
Thus the correct answer is option A.

Question 3.
The rainfall total two years ago was 10.2 inches. Last year’s total was 20% greater. What was last year’s rainfall total?
Options:
a. 8.16 inches
b. 11.22 inches
c. 12.24 inches
d. 20.4 inches

Answer: 12.24 inches

Explanation:
Given,
The rainfall total two years ago was 10.2 inches. Last year’s total was 20% greater.
Find 20% of 10.2
10.2 × 0.20 = 2.04
Add the value to the original amount of 10.2
10.2 + 2.04 = 12.24
Therefore the correct answer is option C.

Question 4.
A pair of basketball shoes was originally priced at $80 but was marked up by 37.5%. What was the retail price of the shoes?
Options:
a. $50
b. $83
c. $110
d. $130

Answer: $110

Explanation:
A pair of basketball shoes was originally priced at $80 but was marked up by 37.5%.
Use the formula retail price = original price + markup
80 + 80 × 0.375 = 80 + 30 = 110
Thus the correct answer is option C.

Question 5.
The sales tax rate in Jan’s town is 7.5%. If she buys 3 lamps for $23.59 each and a sofa for $769.99, how much sales tax does she owe?
Options:
a. $58.85
b. $63.06
c. $67.26
d. $71.46

Answer: $63.06

Explanation:
The sales tax rate in Jan’s town is 7.5%.
If she buys 3 lamps for $23.59 each and a sofa for $769.99
Total cost before tax is 3 × 23.59 + 769.99
= 70.77 + 769.99 = 840.76
Find the amount of tax by multiplying the tax rate and total cost from the above solution and then round to 2 decimal place.
840.76 × 0.075 = 63.06
Thus the correct answer is option B.

Question 6.
The day after a national holiday, decorations were marked down 40%. Before the holiday, a patriotic banner cost $5.75. How much did the banner cost after the holiday?
Options:
a. $1.15
b. $2.30
c. $3.45
d. $8.05

Answer: $3.45

Explanation:
The day after a national holiday, decorations were marked down 40%. Before the holiday, a patriotic banner cost $5.75.
use the formula retail price = original price – markdown
5.75 – 5.75 × 0.4 = 5.75 – 2.30 = 3.45
Thus the correct answer is option C.

Question 7.
Dustin makes $2,330 each month and pays $840 for rent. To the nearest tenth of a percent, what percent of Dustin’s earnings are spent on rent?
Options:
a. 84%
b. 63.9%
c. 56.4%
d. 36.1%

Answer: 36.1%

Explanation:
Dustin makes $2,330 each month and pays $840 for rent.
Divide his rent by his monthly income. round to three decimal places and then convert to percent form.
840/2330 = 0.361 = 36.1%
Thus the correct answer is option D.

Question 8.
A scuba diver is positioned at -30 feet. How many feet will she have to rise to change her position to -12 feet?
Options:
a. -42 ft
b. -18 ft
c. 18 ft
d. 42 ft

Answer: 18 ft

Explanation:
Given,
A scuba diver is positioned at -30 feet.
-12 – (-30) = 12 + 30 = 18 feet
Thus the correct answer is option C.

Question 9.
A bank offers an annual simple interest rate of 8% on home improvement loans. Tobias borrowed $17,000 over a period of 2 years. How much did he repay altogether?
Options:
a. $1360
b. $2720
c. $18360
d. $19720

Answer: $19720

Explanation:
Given that,
A bank offers an annual simple interest rate of 8% on home improvement loans.
Tobias borrowed $17,000 over a period of 2 years
Find the amount of interest he paid using the formula
I = prt
where p is the amount borrowed
r is the interest rate
t is the number of years
17000 × 0.08 × 2 = 2720
Add the amount borrowed and the amount of interest
17000 + 2720 = 19720.
Thus the correct answer is option D.

Mini-Task

Question 10.
The granola Summer buys used to cost $6.00 per pound, but it has been marked up 15%.
a. How much did it cost Summer to buy 2.6 pounds of granola at the old price?
$ ___________

Answer: $15.60

Explanation:
Multiply 2.6 by the old price of $6
2.6 × 6 = 15.60
It costs $15.60 to buy 2.6 pounds of granola at the old price.

Question 10.
b. How much does it cost her to buy 2.6 pounds of granola at the new price?
$ _______

Answer: $17.94

Explanation:
Find the new price using the formula retail price = original price + markup
Then find the total cost by buying 2.6 pounds at the new price.
6 + 6 × 0.15 = 6 + 0.9 = 6.90
2.6 × 6.90 = 17.94
The new price is $17.94

Question 10.
c. Suppose Summer buys 3.5 pounds of granola. How much more does it cost at the new price than at the old price?
$ _______

Answer: $3.15

Explanation:
3.5 × 6 = 21
3.5 × 6.90 = 24.15
24.15 – 21 = 3.15

Module 5 – Page No. 162

EXERCISES

Question 1.
Michelle purchased 25 audio files in January. In February she purchased 40 audio files. Find the percent increase.
_______ %

Answer: 60%

Explanation:
Given,
Michelle purchased 25 audio files in January. In February she purchased 40 audio files.
Use the percent change = amount of change/original amount.
(40 -25)/25 = 15/25 = 0.6 = 60%
Thus the percent increase is 60%

Question 4.
A sporting goods store marks up the cost s of soccer balls by 250%. Write an expression that represents the retail cost of the soccer balls. The store buys soccer balls for $5.00 each. What is the retail price of the soccer balls?
$ _______

Answer: $17.5

Explanation:
Use the formula retail price = original price + markup to find the expression for an original price of s and a markup percentage of 250%
s + 2.5s = 3.5s
substitute s = 5 into the expression to find the retail price
3.5 × 5 = 17.50
Thus the retail price of the soccer balls is $17.50

Unit 2 Performance Tasks – Page No. 163

Question 1.
Viktor is a bike tour operator and needs to replace two of his touring bikes. He orders two bikes from the sporting goods store for a total of $2,000 and pays using his credit card. When the bill arrives, he reads the following information:
Balance: $2000
Annual interest rate: 14.9%
Minimum payment due: $40
Late fee: $10 if payment is not received by 3/1/2013
a. To keep his good credit, Viktor promptly sends in a minimum payment of $40. When the next bill arrives, it looks a lot like the previous bill.
Balance: $1,984.34
Annual interest rate: 14.9%
Minimum payment due: $40
Late fee: $10 if payment is not received by 4/1/2013
Explain how the credit card company calculated the new balance. Notice that the given interest rate is annual, but the payment is monthly.
Type below:
_____________

Answer:
We have to find the balance after the first bill by subtracting the $40 payment from the original balance of $2000.
Balance after first bill: 2000 – 40 = 1960
Then find the amount of interest charged on the second bill by multiplying the balance of $1960 by the interest rate.
Remember since the interest rate is annually you have to divide it by 12 to get the monthly interest rate.
Interest on the second bill: 1960 × 0.149/12 = 24.34
And then add this interest amount to the balance of $1960 to get the balance on the second bill.
New Balance: 1960 + 24.34 = 1984.34

Question 1.
b. Viktor was upset about the new bill, so he decided to send in $150 for his April payment. The minimum payment on his bill is calculated as 2% of the balance (rounded to the nearest dollar) or $20, whichever is greater. Fill out the details for Viktor’s new bill.
Type below:
_____________

Answer:
Find the balance after the $150 payment. The interest rate hasn’t changed so the annual interest rate on this new bill is the same as the previous bills.
balance after payment: 1984.34 – 150 = 1834.34
annual interest rate: 14/9%
Find the interest charged on the third bill. find the balance on the third bill by adding the interest charged to the balance of $1834.34.
interest on the third bill: 1834.34 × 0.149/12 = 22.78
balance: 1834.34 + 22.78 = 1857.12
To find what the minimum payment will be, first find 2% of the balance.
2% of balance: 0.02 × 1857.12 = 37.14
Minimum payment due: $37.00
Since this is greater than $20, the minimum payment is 2% of the balance rounded to the nearest dollar giving $37 as the payment.
The later fee date is one month after the late fee date of 04/01/2013 on the previous bill which gives 05/01/2013.

Question 1.
c. Viktor’s bank offers a credit card with an introductory annual interest rate of 9.9%. He can transfer his current balance for a fee of $40. After one year, the rate will return to the bank’s normal rate, which is 13.9%. The bank charges a late fee of $15. Give two reasons why Viktor should transfer the balance and two reasons why he should not
Type below:
_____________

Answer: Two reasons he should transfer is that the lower introductory rate would mean less interest charged in the first year and a lower normal rate would mean less interest charged after that first year as well. Two reasons he shouldn’t transfer the balance is that he would have to pay a transfer fee of $40 and that the late fee is $15 instead of $10 if he transfers the balance.

Unit 2 Performance Tasks (con’td) – Page No. 164

Question 2.
The table below shows how far several animals can travel at their maximum speeds in a given time.
a. Write each animal’s speed as a unit rate in feet per second.

Elk: _________ feet per second
Giraffe: _________ feet per second
Zebra: _________ feet per second

Answer:
By seeing the above table we can find the unit rates by dividing the distance traveled by the time in seconds.
elk: 33 ÷ 1/2 = 33 ×  2 = 66 feet per second
Giraffe: 115 ÷ 2 1/2 = 115 ÷ 5/2 = 115 2/5 = 46 feet per second
Zebra: 117 ÷ 2 = 58.5 feet per second

Question 2.
b. Which animal has the fastest speed?
_____________

Answer: The elk had the greatest unit rate so it has the fastest speed.

Question 2.
c. How many miles could the fastest animal travel in 2 hours if it maintained the speed you calculated in part a? Use the formula d = rt and round your answer to the nearest tenth of a mile. Show your work.
Elk: _________ miles
Giraffe: _________ miles
Zebra: _________ miles

Answer:
Elk: 90 miles
Giraffe: 62 miles
Zebra: 72 miles

Explanation:

There are 60 seconds in a minute and 60 minutes in an hour so there are 2 × 60 × 60 = 7200 seconds in 2 hours.
Multiply the unit rate of the elk by 7200 seconds to get the distance traveled in feet.
There are 5280 feet in 1 mile so divide the distance in feet by 5280 to get the distances in miles.
Elk:
66 × 7200 =  475200 feet
Now convert from feet to miles
475200 feet = 90 miles
Giraffe: 46 feet per second
62 × 7200 = 331200 feet
Now convert from feet to miles.
331200 = 62 miles
Zebra: 58.5 feet per second
58.5 × 7200 = 421200 feet
Now convert from feet to miles.
421200 feet = 72 miles

Question 3.
d. The data in the table represents how fast each animal can travel at its maximum speed. Is it reasonable to expect the animal from part b to travel that distance in 2 hours? Explain why or why not.
______

Answer: It is not reasonable. An animal can only travel at its maximum speed for a short amount of time which is usually only for a couple of minutes.

Selected Response – Page No. 165

Question 1.
If the relationship between distance y in feet and time x in seconds is proportional, which rate is represented by \(\frac{y}{x}\) = 0.6?
Options:
a. 3 feet in 5 s
b. 3 feet in 9 s
c. 10 feet in 6 s
d. 18 feet in 3 s

Answer: 3 feet in 5 s

Explanation:
\(\frac{y}{x}\) = 0.6
0.6 = \(\frac{6}{10}\)
Since \(\frac{6}{10}\) = \(\frac{3}{5}\), it represents a rate of 3 feet in 5 seconds,
Therefore the correct answer is option A.

Question 2.
The Baghrams make regular monthly deposits in a savings account. The graph shows the relationship between the number x of months and the amount y in dollars in the account.
What is the equation for the deposit?

Options:
a. \(\frac{y}{x}\) = $25/month
b. \(\frac{y}{x}\) = $40/month
c. \(\frac{y}{x}\) = $50/month
d. \(\frac{y}{x}\) = $75/month

Answer: \(\frac{y}{x}\) = $50/month

Explanation:
By seeing the above graph we can say that the point is (2, 100). This means that \(\frac{y}{x}\) = \(\frac{100}{2}\) = 50.
Thus the correct answer is option C.

Question 3.
What is the decimal form of −4 \(\frac{7}{8}\)?
Options:
a. -4.9375
b. -4.875
c. -4.75
d. -4.625

Answer: -4.875

Explanation:
Given the fraction
−4 \(\frac{7}{8}\)
First divide \(\frac{7}{8}\) = 0.875
4 + 0.875 = 4.875
So, −4 \(\frac{7}{8}\) = -4.875
Therefore the answer is option B.

Question 4.
Find the percent change from 72 to 90.
Options:
a. 20% decrease
b. 20% increase
c. 25% decrease
d. 25% increase

Answer: 25% increase

Explanation:
Use the formula percent change = amount of change/original amount.
the value increased from 72 to 90 so it is a percent increase.
(90-72)/72 = 18/72 = 0.25 = 25%
Thus the correct answer is option D.

Question 5.
A store had a sale on art supplies. The price p of each item was marked down by 60%. Which expression represents the new price?
Options:
a. 0.4p
b. 0.6p
c. 1.4p
d. 1.6p

Answer: 0.4p

Explanation:
Given that,
A store had a sale on art supplies.
The price p of each item was marked down by 60%
Use the formula sale price = original price – markdown
p is the original price and the markdown percent is 40% then combine the like terms.
p – 0.6p = 0.4p
Therefore the correct answer is option A.

Question 6.
Clarke borrows $16,000 to buy a car. He pays simple interest at an annual rate of 6% over a period of 3.5 years. How much does he pay altogether?
Options:
a. $18800
b. $19360
c. $19920
d. $20480

Answer: $19360

Explanation:
Given,
Clarke borrows $16,000 to buy a car.
He pays simple interest at an annual rate of 6% over a period of 3.5 years.
Find the total amount of interest using the formula
I = prt
where p is the amount borrowed
r is the rate of interest
t is the number of years
16000 × 0.06 × 3.5 = 3360
Now add the amount of interest to the amount borrowed to find the total amount
16000 + 3360 = 19,360
Thus the correct answer is option B.

Question 7.
To which set or sets does the number 37 belong?
Options:
a. integers only
b. rational numbers only
c. integers and rational numbers only
d. whole numbers, integers, and rational numbers

Answer: whole numbers, integers, and rational numbers

Explanation:
37 can be written as 37/1 so it is a rational number. 37 doesn’t have a decimal or fraction so it is an integer. Since it is a positive integer, it is also a whole number.
Thus a suitable answer is option D.

Page No. 166

Question 8.
In which equation is the constant of proportionality 5?
Options:
a. x = 5y
b. y = 5x
c. y = x + 5
d. y = 5 – x

Answer: y = 5x

Explanation:
Directly proportional equations are of the form y = kx
where k is the constant of proportionality.
If k = 5, then the equation is y = 5x.
Thus the correct answer is option B.

Question 9.
Suri earns extra money by dog walking. She charges $6.25 to walk a dog once a day 5 days a week and $8.75 to walk a dog once a day 7 days a week. Which equation represents this relationship?
Options:
a. y = 7x
b. y = 5x
c. y = 2.50x
d. y = 1.25x

Answer: y = 1.25x

Explanation:
Given that,
Suri earns extra money by dog walking. She charges $6.25 to walk a dog once a day 5 days a week and $8.75 to walk a dog once a day 7 days a week.
Since 6.25/5 = 1.25
So, the equation is y = 1.25x
where x is the number of days and y is the total charge.
So, the correct answer is option D.

Question 10.
Randy walks \(\frac{1}{2}\) mile in each \(\frac{1}{5}\) hour. How far will Randy walk in one hour?
Options:
a. \(\frac{1}{2}\) miles
b. 2 miles
c. 2 \(\frac{1}{2}\) miles
d. 5 miles

Answer: 2 \(\frac{1}{2}\) miles

Explanation:
Given,
Randy walks \(\frac{1}{2}\) mile in each \(\frac{1}{5}\) hour.
\(\frac{1}{2}\) ÷ \(\frac{1}{5}\)
\(\frac{1}{2}\) × \(\frac{5}{1}\) = \(\frac{5}{2}\)
Convert the fraction to the improper fractions.
\(\frac{5}{2}\) = 2 \(\frac{1}{2}\) miles
Therefore the correct answer is option C.

Question 11.
On a trip to Spain, Sheila bought a piece of jewelry that cost $56.75. She paid for it with her credit card, which charges a foreign transaction fee of 3%. How much was the foreign transaction fee?
Options:
a. $0.17
b. $1.07
c. $1.70
d. $17.00

Answer: $1.70

Explanation:
On a trip to Spain, Sheila bought a piece of jewelry that cost $56.75.
She paid for it with her credit card, which charges a foreign transaction fee of 3%
Find the foreign transaction fee amount by multiplying the cost by the foreign transaction fee percentage.
56.75 × 0.03 = 1.70
Thus the correct answer is option C.

Question 12.
A baker is looking for a recipe that has the lowest unit rate for flour per batch of muffins. Which recipe should she use?
Options:
a. \(\frac{1}{2}\) cup flour for \(\frac{2}{3}\) batch
b. \(\frac{2}{3}\) cup flour for \(\frac{1}{2}\) batch
c. \(\frac{3}{4}\) cup flour for \(\frac{2}{3}\) batch
d. \(\frac{1}{3}\) cup flour for \(\frac{1}{4}\) batch

Answer: \(\frac{1}{2}\) cup flour for \(\frac{2}{3}\) batch

Explanation:
a. \(\frac{1}{2}\) ÷ \(\frac{2}{3}\) = \(\frac{1}{2}\) × \(\frac{3}{2}\) = \(\frac{3}{4}\)
b. \(\frac{2}{3}\) ÷ \(\frac{1}{2}\) = \(\frac{2}{3}\) × \(\frac{2}{1}\) = \(\frac{4}{3}\) = 1 \(\frac{1}{3}\)
c. \(\frac{3}{4}\) ÷ \(\frac{2}{3}\) = \(\frac{3}{4}\) × \(\frac{3}{2}\) = \(\frac{9}{8}\) = 1 \(\frac{1}{8}\)
d. \(\frac{1}{3}\) ÷ \(\frac{1}{4}\) = \(\frac{1}{3}\) ÷ \(\frac{4}{1}\) = 1 \(\frac{1}{3}\)
Thus the correct answer is option A.

Mini-Task

Question 13.
Kevin was able to type 2 pages in 5 minutes, 3 pages in 7.5 minutes, and 5 pages in 12.5 minutes.
a. Make a table of the data.
Type below:
___________

Answer:

Number of Pages	2	3	5
Minutes	5	7.5	12.5

Question 13.
b. Graph the relationship between the number of pages typed and the number of minutes.

Type below:
___________

Answer:

Question 13.
c. Explain how to use the graph to find the unit rate.
Type below:
___________

Answer: The unit rate is 2.5 pages per minute

Explanation:
By using the graph we need to find the slope of the line.
We can do this by using the formula of a slope:
m = (y2-y1)/(x2-x1) = (7.5-5)/(3-2) = 2.5
Thus the unit rate is 2.5 pages per minute.

Conclusion:

Hope the answers provided in Go Math Answer Key Grade 7 Chapter 5 Percent Increase and Decrease are quite satisfactory for all the students. Refer to our Go Math 7th Grade Chapter 5 Percent Increase and Decrease to get the solutions with the best explanations. After your preparation test your math skills by solving the questions in the performance tasks.

Students who are in search of the HMH Go Math Chapter 4 Rates and Proportionality Answer Key can get them on this page. With the help of the Go Math Grade 7 Answer Key 4th Chapter Rates and Proportionality parents can teach simple methods to solve the problems to their children. So, Download Go Math Grade 7 Rates and Proportionality Chapter 4 pdf for free of cost.

Go Math Grade 7 Answer Key Chapter 4 Rates and Proportionality

It is important for the students to learn the concepts given in Go Math Grade 7 Chapter 4 Answer Key to score the highest marks in the exams. The quick way of solving the problems will help you to save time in the exam. The topics covered in this chapter are Unit rates, Constant Rates of Change, Proportional Relationships, and Graphs. Click on the below-provided links and go through all the questions and answers.

Chapter 4 – Unit Rates

• Unit Rates – Guided Practice – Page No. 120
• Unit Rates – Independent Practice – Page No. 121
• Unit Rates – Page No. 122

Chapter 4 – Constant Rates of Change

• Constant Rates of Change – Guided Practice – Page No. 126
• Constant Rates of Change – Page No. 127
• Constant Rates of Change – Page No. 128

Chapter 4 – Proportional Relationships and Graphs

• Proportional Relationships and Graphs – Guided Practice – Page No. 132
• Proportional Relationships and Graphs – Guided Practice – Page No. 133
• Proportional Relationships and Graphs – Page No. 134

Chapter 4 – Module 4

• MODULE QUIZ – 4.1 Unit Rates – Page No. 135
• MIXED REVIEW – Selected Response – Page No. 136
• Module 4 – Page No. 161

Unit Rates – Guided Practice – Page No. 120

Question 1.
Brandon enters bike races. He bikes 8 \(\frac{1}{2}\) miles every \(\frac{1}{2}\) hour. Complete the table to find how far Brandon bikes for each time interval

Type below:
____________

Answer:
1 hour: 8 \(\frac{1}{2}\) + 8 \(\frac{1}{2}\) = 17
1 \(\frac{1}{2}\) hour: 17 + 8 \(\frac{1}{2}\) = 25 \(\frac{1}{2}\)
2 hour: 25 \(\frac{1}{2}\) + 8 \(\frac{1}{2}\) = 34
2 \(\frac{1}{2}\) hour: 34 + 8 \(\frac{1}{2}\) = 42 \(\frac{1}{2}\)

Find each unit rate.

Question 2.
Julio walks 3 \(\frac{1}{2}\) miles in 1 \(\frac{1}{4}\) hours.
________ \(\frac{□}{□}\)

Answer: 2 \(\frac{4}{5}\)

Explanation:
Divide the number of miles by the number of hours to find the unit rate in miles per hour.
3 \(\frac{1}{2}\) ÷ 1 \(\frac{1}{4}\) = \(\frac{7}{2}\)/\(\frac{5}{4}\)
\(\frac{7}{2}\) × \(\frac{4}{5}\) = \(\frac{14}{5}\)
Convert from improper fraction to mixed fraction.
\(\frac{14}{5}\) = 2 \(\frac{4}{5}\) miles per hour

Question 5.
A fertilizer covers \(\frac{5}{8}\) square foot in \(\frac{1}{4}\) hour.
________ \(\frac{□}{□}\)

Answer: 2 \(\frac{1}{2}\) square feet per hour

Explanation:
Given,
A fertilizer covers \(\frac{5}{8}\) square foot in \(\frac{1}{4}\) hour.
Divide the number of square feet, which is \(\frac{5}{8}\) by the number of hours, which is \(\frac{1}{4}\), to find the unit rate in square feet per hour.
\(\frac{5}{8}\) ÷ \(\frac{1}{4}\)
\(\frac{5}{8}\) × \(\frac{4}{1}\) =\(\frac{5}{2}\)
Convert from improper fraction to mixed fraction.
\(\frac{5}{2}\) = 2 \(\frac{1}{2}\) square feet per hour
Thus A fertilizer covers 2 \(\frac{1}{2}\) square feet per hour.

Find each unit rate. Determine which is lower.

Question 6.
Brand A: 240 mg sodium for \(\frac{1}{3}\) pickle or Brand B: 325 mg sodium for \(\frac{1}{2}\) pickle.
____________

Answer:
Find the unit rates in mg per pickle for each brand by dividing the number of mg by the number of pickles.
Brand A: 240 mg ÷ \(\frac{1}{3}\) = 240 × 3 = 720
Brand B: 325 mg ÷ \(\frac{1}{2}\) = 325 × 2 = 650
650 is less than 720 so Brand B has a lower unit rate.

Essential Question Check-In

Question 8.
How can you find a unit rate when given a rate?
Type below:
____________

Answer: To find a unit rate when given a rate such as 25 miles per 5 minutes, divide the first quantity by the second quantity.
In the example I gave, this would mean the unit rate is 25 ÷ 5 = 5 miles per minute.

Unit Rates – Independent Practice – Page No. 121

Question 9.
The information for two pay-as-you-go cell phone companies is given.
a. What is the unit rate in dollars per hour for each company?
On Call: ____________ dollars per hour
Talk Time: ____________ dollars per hour

Answer:
Divide the cost by the number of hours for each company to find the unit rates.
On Call: 10 ÷ 3.5 = 2 \(\frac{6}{7}\) ≈ 2.86
Talk Time: 1.25 ÷ \(\frac{1}{2}\) = 2.50

Question 9.
b. Analyze Relationships
Which company offers the best deal? Explain your answer.
____________

Answer: Talk time has the lowest unit rate so it offers the best deal.

Question 9.
c. What If?
Another company offers a rate of $0.05 per minute. How would you find the unit rate per hour?
____________ dollars per hour

Answer:
Since there are 60 minutes in 1 hour, $.0.05 per minute is
60 × 0.05 = $3 per hour.
Thus the unit rate per hour is $3.

Question 9.
d. Draw Conclusions
Is the rate in part c a better deal than On Call or Talk Time? Explain.
____________

Answer:
The unit rate in part c is greater than the unit rates from part a so it is not a better deal than the other two companies.

Question 10.
Represent Real-World Problems
Your teacher asks you to find a recipe that includes two ingredients with a rate of \(\frac{2 \text { units }}{3 \text { units }}\).
a. Give an example of two ingredients in a recipe that would meet this requirement.
Type below:
____________

Answer: A rate of 2/3 units means that there need to be 2 units of 1 ingredient for every 3 units of a second ingredient.
One example could then be 2 eggs per 3 cups of flour.
Another example could 2 teaspoons of vanilla per 3 teaspoons of sugar.

Question 10.
b. If you needed to triple the recipe, would the rate change? Explain.
____________

Answer: No, the rate would not change. Using the example I gave in
part a) of 2 eggs per 3 cups of flour, tripling the recipe would require using 3(2 eggs) = 6 eggs
3(3 cups of sugar) = 9 cups of flour.
Since 6 eggs/9 cups of flour = 2 eggs/3 cups of flour, the rate is still the same.

Question 11.
A radio station requires DJs to play 2 commercials for every 10 songs they play. What is the unit rate of songs to commercials?
____________ songs per commercial

Answer: 10 ÷ 2 = 5
Divide the number of songs by the number of commercials.
Thus the radio requires 5 songs per commercial.

Question 12.
Multistep
Terrance and Jesse are training for a long-distance race. Terrance trains at a rate of 6 miles every half hour and Jesse trains at a rate of 2 miles every 15 minutes.
a. What is the unit rate in miles per hour for each runner?
Terrance: ____________ mi per hour
Jesse: ____________ mi per hour

Answer:
Find the unit rates for each runner by dividing the number of miles by the number of hours. Remember that 15 minutes is 1/4 of an hour since there are 60 minutes in an hour
15 ÷ 60 = 1/4
a) Terrance: 6 ÷ 1/2 = 6 × 2 = 12 miles per hour.
Jesse: 2 ÷ 1/4 = 2 × 4 = 8 miles per hour.

Question 12.
b. How long will each person take to run a total of 50 miles at the given rates?
Terrance: ______ \(\frac{□}{□}\)
Jesse: ______ \(\frac{□}{□}\)

Answer:
Divide the number of miles by the unit rates found in part a to find the time.
Terrance: 50 ÷ 12 = 50/12 = 4 \(\frac{1}{6}\) hours
Jesse: 50 ÷ 8 = 50/8 = 6 \(\frac{1}{4}\) hours

Question 12.
c. Sandra runs at a rate of 8 miles in 45 minutes. How does her unit rate compare to Terrance’s and to Jesse’s?
______ \(\frac{□}{□}\) mi per hour

Answer:
We need to find the unit rate for Sandra by dividing the number of miles by the number of hours.
Remember that 45 minutes is 3/4 of an hour.
Since 45/60 = 3/4.
Sandra’s unit rate is smaller than Terrance’s but larger than Jesse’s.

Unit Rates – Page No. 122

Question 14.
Justify Reasoning
An online retailer sells two packages of protein bars.

a. Which package has the better price per bar?
____________

Answer:
Find the unit rates per bar by dividing the costs by the number of bars. the 12-pack has a better price per bar.
10-pack: 15.37 ÷ 10 = 1.537 ≈ 1.54
12-pack: 15.35 ÷ 12 ≈ 1.30

Question 14.
b. Which package has the better price per ounce?
____________

Answer:
First, find the total number of ounces by multiplying the number of bars times the number of ounces per bar. then find the unit rates per ounce by dividing the costs by the total number of ounces the 10-pack has the better price per ounce.
10-pack: 10 × 2.1 = 21 ounces
12-pack: 12 × 1.4 = 16.8 ounces
10-pack: 15.37 ÷ 21 ≈ 0.73
12-pack: 15.35 ÷ 16.8 ≈ 0.91

Question 14.
c. Which package do you think is a better buy? Justify your reasoning.
____________

Answer:
The 10-pack is a better deal since the price per ounce is a better measure to use than price per bar. The number of bars doesn’t tell you how you are actually buying since the bars can be very small meaning the number of ounces you are actually buying is small.

Question 15.
Check for Reasonableness
A painter painted about half a room in half a day. Coley estimated the painter would paint 7 rooms in 7 days. Is Coley’s estimate reasonable? Explain.
____________

Answer:
If a painter can paint half a room in a half day, then he can paint 1 room in 1 day.
This would be equivalent to painting 7 rooms in 7 days so his estimate is reasonable.
7 rooms ÷ 7 days = 1/2 room ÷ 1/2 days

Question 16.
Communicate Mathematical
Ideas If you know the rate of a water leak in gallons per hour, how can you find the number of hours it takes for 1 gallon to leak out? Justify your answer.
Type below:
____________

Answer: If you know the rate in gallons per hour, then the rate in hours per gallon is the reciprocal of the rate in gallons per hour.
Example:
If water is leaking at a rate of 5 gallons per hour, then it is leaking at 1/5 hour per gallon.

Constant Rates of Change – Guided Practice – Page No. 126

Question 1.
Based on the information in the table, is the relationship between time and the number of words typed a proportional relationship?

The relationship ____________ proportional

Answer: is proportional

Explanation:
Since 45 ÷ 1 = 45, 90 ÷ 2 = 45, 135 ÷ 3 = 45 and 180 ÷ 4 = 45, the relationship is proportional.
Thus the relationship for the above table is proportional.

Find the constant of proportionality k. Then write an equation for the relationship between x and y.

Question 2.

k = _______

Answer: 5

Explanation:
The equation is of the form y = kx so k = y/x.
Substituting values of x and y from the table gives k = 10/2 = 5.
Plugging this value into y = kx gives the equation y = 5x.
The relationship between x and y is y = 5x.

Question 3.

k = \(\frac{□}{□}\)

Answer: k = \(\frac{1}{4}\)

Explanation:
The equation is of the form y = kx so k = y/x.
Substituting values of x and y from the table gives k = 2/8 = \(\frac{1}{4}\).
Plugging this value into y = kx gives the equation y = \(\frac{1}{4}\)x.

Essential Question Check-In

Question 4.
How can you represent a proportional relationship using an equation?
Type below:
____________

Answer: y = kx

Explanation:
A proportional relationship can always be represented by an equation of the form y = kx
where ks is the constant of proportionality and represents the rate of the change in the y quantity in relation to the x quantity.

Constant Rates of Change – Page No. 127

Information on three car-rental companies is given.

Question 5.
Write an equation that gives the cost y of renting a car for x days from Rent-All.
y = _______ x

Answer: 18.50

Explanation:
Find the constant of proportionality by dividing the total costs by the number of days.
k = 55.50/3 = 18.50
The equation is y = 18.50x

Question 6.
What is the cost per day of renting a car from A-1?
$ _______ per day

Answer: $21.98

Explanation:
Since the cost of each half day is $10.99, the cost for each day is 2 × 10.99 = 21.98
The cost per day of renting a car from A-1 is $21.98

Question 7.
Analyze Relationships
Which company offers the best deal? Why?
The company that offers the best deal is ____________

Answer: Rent all

Explanation:
The costs per day were $18.50 for Rent-All, $21.98 for A-1 Rentals, and $19.25 for Car Town so Reant All offers the best deal since it offers the lowest cost per day.
Thus the company that offers the best deal is Rent-All.

Question 8.
Critique Reasoning
A skydiver jumps out of an airplane. After 0.8 seconds, she has fallen 100 feet. After 3.1 seconds, she has fallen 500 feet. Imtiaz says that the skydiver should fall about 187.5 feet in 1.5 seconds. Is his answer reasonable? Explain.
_______

Answer: No. He assumed the rate of descent was proportional but the rate is increasing as time increased.

Explanation:
Since 100 ÷ 0.8 = 125, the skydiver fell at a speed of 125 ft per second for the first 0.8 seconds.
Since 500 ÷ 3.1 = 161, the skydiver fell at a speed of about 161 ft per second for the first 3.1 seconds.
The rate of descent is then increased as time increases and is not proportional since 125 ≠ 161.
Since 187.5 ÷ 1.5 = 125, he assumed the rate of descent was proportional. His estimate is then not reasonable.
The actual rate of descent should be between 125 and 161.3 since 1.5 seconds is between 0.8 and 3.1 seconds.

Steven earns extra money babysitting. He charges $31.25 for 5 hours and $50 for 8 hours.

Question 9.
Explain why the relationship between how much Steven charges and time is a proportional relationship.
Type below:
____________

Answer: The relationship is proportional since the ratios are equal.

Explanation:
Since 31.25 ÷ 5 = 6.25 and 50 ≈ 8 = 6.25,
the relationship is proportional since the ratios are equal.

Question 10.
Interpret the Answer
Explain what the constant rate of change means in the context.
Type below:
____________

Answer: The rate of change means he charges $6.25.

Explanation:
The constant rate of change of 6.25 means he charges $6.25 per hour
since the rate was found by dividing the charge by the number of hours.
The rate of change means he charges $6.25.

Question 11.
Write an equation to represent the relationship. Tell what the variables represent.
Type below:
____________

Answer: The equation is y = 6.25x
where x is the number of hours and ys is the total charge.

Explanation:
The rate of change is 6.25 so k = 6.25.
This gives an equation of y = 6.25x where x is the number of hours and y is the total charge.

Question 12.
How much would Steven charge for 3 hours?
$ _______

Answer: $18.75

Explanation:
y = 6.25 × 3 = 18.75
Thus Steve charges $18.75 for 3 hours.

Constant Rates of Change – Page No. 128

A submarine dives 300 feet every 2 minutes, and 6,750 feet every 45 minutes.

Question 13.
Find the constant rate at which the submarine dives. Give your answer in feet per minute and in feet per hour.
____________ feet per minute
____________ feet per hour

Answer: 150 feet per minute, 9000 feet per hour

Explanation:
Since 300 ÷ 2 = 150, the submarine is diving at 150 feet per minute.
Since 45 minutes = 3/4 of an hour and 6750 ÷ 3/4 = 9000,
the submarine is diving at a rate of 9000 feet per hour.

Question 15.
Draw Conclusions
If you wanted to find the depth of a submarine during a dive, would it be more reasonable to use an equation with the rate in feet per minute or feet per hour? Explain your reasoning.
____________

Answer: Feet per minute

Explanation:
Since a submarine would only dive for a few minutes at a time and not dive for hours at a time, it is more reasonable to use the rate in feet per minute.

H.O.T.

Focus on Higher Order Thinking

Question 16.
Make a Conjecture
There is a proportional relationship between your distance from a thunderstorm and the time from when you see lightning and hear thunder. If there are 9 seconds between lightning and thunder, the storm is about 3 kilometers away. If you double the amount of time between lightning and thunder, do you think the distance in kilometers also double? justify your reasoning.
_______

Answer: Yes the distance will also double. If the relationship is proportional then distance/time = k
where k is the constant of proportionality.
Since the time was 9 seconds for 3km, then for 18 seconds the distance would be 6 km since 3/9 = 6/18.
6 is double 3 so the distance doubles when the time doubles.

Question 17.
Communicate Mathematical Ideas
A store sells 3 ears of corn for $1. They round prices to the nearest cent as shown in the table. Tell whether you would describe the relationship between cost and number of ears of corn as a proportional relationship. Justify your answer.

_______

Answer:
Since 0.33 ÷ 1 = 0.33, 0.67 ÷ 2 = 0.335, 1.00 ÷ 3 = 0.33.., 1.34 ÷ 4 = 0.335, the relationship is approximately proportional since all the ratios are approximately equal. The difference in the ratios com from rounding the amount charged to the nearest cent.

Question 18.
Jack is 12 and his sister Sophia is 16. Jack says that the relationship between his age and Sophia’s age is proportional and the constant of proportionality is \(\frac{12}{16}\) Do you agree? Explain.
____________

Answer:
Given that current age of Jack = 12 years
Given that current age of Sophia = 16 years
Jack says that the relationship between his age and Sophia‘s is proportional
If Jack’s age is represented by y and Sophia’s age by x then we can write y=kx as they are in proportion
where k is called constant of proportion
Now let’s plug given ages of each that is y=12 and x=16 into y=kx to find the constant of proportionality
12=k×16
12/16=k
Which is the same as the given value of the constant of proportionality?
Hence Jack is right about his statement.
But if you think about practical life situation then the age of both will not be in proportion
For example, after 1 year Jack’s age will be 13 and Sophie’s age will be 17
then constant of proportionality using new values will be 13/17
Clearly 12/16 and 13/17 are not same.
So in practical life, the age of both will not in proportion.

Question 19.
Luke’s turkey chili recipe calls for 1.5 pounds of ground turkey for every 6 servings. How many servings can he make if he has 5 pounds of ground turkey? Show your work.
____________ servings

Answer: 20 servings

Explanation:
Given,
Luke’s turkey chili recipe calls for 1.5 pounds of ground turkey for every 6 servings.
So if 1.5 pounds of turkey can get you 6 servings
1.5 = 6
3 = 12
4.5 = 18
5 =?
to find what 5 pounds are equal to we must do 1.5/6 to find the unit rate of 0.25. We then add 0.25 to 18.25 servings or if you round you can get about 18 servings.
18 + 0.25 × 8 = 18 + 2 = 20

Proportional Relationships and Graphs – Guided Practice – Page No. 132

Complete each table. Tell whether the relationship is a proportional relationship. Explain why or why not.

Question 1.
A student reads 65 pages per hour.

____________

Answer:
Given that,
A student reads 65 pages per hour.
3 hours: 3 × 65 = 195 pages
5 hours: 5 × 65 = 325 pages
10 hours: 10 × 65 = 650 pages
We need to find the number of hours for 585 pages by dividing the number of pages by 65 since the students read 65 pages per hour:
585 pages: 585 ÷ 65 = 9 hours

A relationship is proportional if the quotient of each ordered pair is constant. Since the student is reading at a constant rate of 65 pages per hour, and the quotient of each ordered pair in the table is 65, the relationship is proportional.

Question 2.
A babysitter makes $7.50 per hour.

____________

Answer:
2 hours = 2 × 7.50 = 15
22.50 = 22.50 ÷ 7.50 = 3 hours
5 hours = 5 × 7.50 = 37.50
6. = 60 ÷ 7.50 = 8 hours

Tell whether the relationship is a proportional relationship. Explain why or why not.

Question 3.

____________

Answer:
The relationship has points (2, 4) and (8, 10).
Since 4 ÷ 2 = 2 and 10 ÷ 8 = 1.25, the relationship is not proportional since the ratios are not equal.

Question 4.

____________

Answer:
The relationship appears to be proportional since the points appear to form a line that goes through the origin. That line would go through points (1,2), (2, 4), (5, 10), and (8, 16).
Since 2 ÷ 1 = 2, 4 ÷ 2 = 2, 10 ÷ 5 = 2, and 16 ÷ 8 = 2, the relationship is proportional since all the ratios are equal.

Write an equation of the form y = kx for the relationship shown in each graph.

Question 5.

y = ____________ x

Answer: y = 3.5x

Explanation:
One of the points is (8, 28) so k = 28/8 = 7/2. The equation is the y = 7/2 = 3.5x

Question 6.

y = ____________ x

Answer: y = 0.25x

Explanation:
One of the points is (8,2) so k = 2/8 = 1/4.
The equation is the y = 1/4x = 0.25 x.

Essential Question Check-In

Question 7.
How does a graph show a proportional relationship?
Type below:
____________

Answer:
A proportional relationship between two variables, x and y, exists if y = kx.
This equation is a line that passes through the origin and has a slope of k. The slope can be positive or negative. Therefore if the points lie on a line that goes through the origin, the graph shows a proportional relationship. If the points lie on a line that does not go through the origin, that is, has a nonzero y-intercept, then the relationship is not proportional.

Proportional Relationships and Graphs – Guided Practice – Page No. 133

For Exercises 8–12, the graph shows the relationship between time and distance run by two horses.

Question 8.
Explain the meaning of the point (0,0).
Type below:
____________

Answer:
The point (0, 0) represents a distance of 0 miles in 0 min.

Question 10.
Multiple Representations
Write an equation for the relationship between time and distance for each horse.
For Horse A : y = ____________ x
For Horse B : y = ____________ x

Answer: A: y = 1/4x, B: y = 2/5x

Explanation:
The graph has x representing the time in minutes and y representing the distance in miles so the slope of the line has units of miles per minute.
Since horse A runs 4 min per mile, it runs at a rate of 1/4 mi per min.
This gives the equation y = 1/4x.
Since horse B runs at a rate of 2.5 min per mi, it runs at a rate of 1/2.5 = 2/5 miles per min.
The equation is then y = 2/5x.

Question 12.
Analyze Relationships
Draw a line on the graph representing a horse than runs faster than horses A and B.
Type below:
____________

Answer:
To have a line representing the rate of the horse faster than horses A and B, the line should be a little bit steeper than the other two lines. This can be represented in the graph:

Question 13.
A bullet train can travel at 170 miles per hour. Will a graph representing the distance in miles compared to the time in hours show a proportional relationship? Explain.
____________

Answer: Yes
Since the train is traveling at a constant rate, a graph representing the distance in miles compared to the time in hours will show a proportional relationship.

Question 14.
Critical Thinking
When would it be more useful to represent a proportional relationship with a graph rather than an equation?
Type below:
____________

Answer: It is more useful to represent a proportional relationship with a graph when comparing different and various situations.

Question 15.
Multiple Representations
Bargain DVDs cost $5 each at Mega Movie.
a. Graph the proportional relationship that gives the cost y in dollars of buying x bargain DVDs.

Type below:
____________

Answer:
Since each DVD is $5, make sure to graph a line that corresponds to this rate.

Question 15.
b. Give an ordered pair on the graph and explain its meaning in the real-world context.
Type below:
____________

Answer: An ordered pair in the graph is (3, 15) and this means that three DVDs cost $15.

Proportional Relationships and Graphs – Page No. 134

The graph shows the relationship between distance and time as Glenda swims.

Question 16.
How far did Glenda swim in 4 seconds?
______ feet

Answer: 8 ft

Explanation:
The graph goes through the point (4, 8) so she swam 8 ft in 4 sec.

H.O.T.

Focus on Higher Order Thinking

Question 19.
Make a Conjecture
If you know that a relationship is proportional and are given one ordered pair that is not (0,0), how can you find another pair?
Type below:
____________

Answer:
If you are given a point (a, b) that is not (0, 0) and that the relationship is proportional, then you can find k since k = y/x = b/a.
Then you can write the equation as y = b/ax. From there, you can plug in any value for x to find the corresponding y-coordinate.

The tables show the distance traveled by three cars.

Question 20.
Communicate Mathematical Ideas
Which car is not traveling at a constant speed? Explain your reasoning.
____________

Answer:
Since 120 ÷ 2 = 180 ÷ 3 = 300 ÷ 5 = 360 ÷ 6 = 60, Car 1 is traveling at a constant speed.
Since 200 ÷ 5 = 400 ÷ 10 = 600 ÷ 15 = 800 ÷ 20 = 40, Car 2 is traveling at a constant speed.
Since 65 ÷ 1 ≠ 85 ÷ 2, Car 3 is not traveling at a constant speed.

Question 21.
Make a Conjecture
Car 4 is traveling at twice the rate of speed of car 2. How will the table values for car 4 compare to the table values for car 2?
Type below:
____________

Answer:
From problem 20, car 2 is traveling at 40 miles per hour. If car 4 is traveling twice that rate, then it is traveling at 80 miles per hour.
This means all the values for the distances for car 4 will double the values for the distances for car 2.

MODULE QUIZ – 4.1 Unit Rates – Page No. 135

Find each unit rate. Round to the nearest hundredth, if necessary.

Question 1.
$140 for 18 ft²
$ ______

Answer: $7.78 per ft²

Explanation:
Divide the cost of $140 by the number of square feet, 18 sq. ft, using a calculator:
140 ÷ 18 ≈ $7.78 per sq. ft.
If you are required by your teacher to do the division by hand, divide to three decimal points as shown below:
Since the dollar amounts must be rounded to two decimal places, then $7.77.. ≈ $7.78 so the cost per square foot is $7.78 sq. ft.

Question 1.
14 lb for $2.99
$ ______

Answer: $0.21 per lb

Explanation:
Divide the cost by the number of pounds.
2.99 ÷ 14 = $0.21

Circle the better deal in each pair. Then give the unit rate for the better deal.

4.2 Constant Rates of Change

Question 5.
The table shows the amount of money Tyler earns for mowing lawns. Is the relationship a proportional relationship? Why or why not?

____________

Answer: not proportional

Explanation:
Since 15 ÷ 1 = 15 but 48 ÷ 3 = 16, the relationship is not proportional.

Question 6.
On a recent day, 8 euros were worth $9 and 24 euros were worth $27. Write an equation of the form y = kx to show the relationship between the number of euros and the value in dollars.
Type below:
____________

Answer: y = \(\frac{9}{8}\)x

Explanation:
k = \(\frac{value in dollars}{number of euros}\) = \(\frac{9}{8}\)
so, the equation is y = \(\frac{9}{8}\)x

4.3 Proportional Relationships and Graphs

Question 7.
The graph shows the number of servings in different amounts of frozen yogurt listed on a carton. Write an equation that gives the number of servings y in x pints.

Type below:
____________

Answer: y = \(\frac{5}{2}\)x

Explanation:
The graph goes through the point (2, 5) so k = \(\frac{5}{2}\).
This gives an equation of y = \(\frac{5}{2}\)x

Question 8.
A refreshment stand makes 2 large servings of frozen yogurt from 3 pints. Add the line to the graph and write its equation.
Type below:
____________

Answer: y = \(\frac{2}{3}\)x

Explanation:
If the situation states that 2 servings of frozen yogurt can be made from 3 pints, then we can say that k = \(\frac{2}{3}\), and therefore the equation of the line is y = \(\frac{2}{3}\)x. The graph of the line is shown below.

Essential Question

Question 9.
How can you use rates to determine whether a situation is a proportional relationship?
Type below:
____________

Answer: If the rate is constant, then the situation is a proportional relationship. If the rate is not constant, the situation cannot be a proportional relationship.

MIXED REVIEW – Selected Response – Page No. 136

Question 1.
Kori spent $46.20 on 12 gallons of gasoline. What was the price per gallon?
Options:
a. $8.35
b. $3.85
c. $2.59
d. $0.26

Answer: $3.85

Explanation:
Given that,
Kori spent $46.20 on 12 gallons of gasoline.
Divide the cost by the number of gallons to find the price per gallon.
46.20/12 = 3.85
Thus the correct answer is option B.

Question 2.
A rabbit can run short distances at a rate of 35 miles per hour. A fox can run short distances at a rate of 21 miles per half hour. Which animal is faster, and by how much?
Options:
a. The rabbit; 7 miles per hour
b. The fox; 7 miles per hour
c. The rabbit; 14 miles per hour
d. The fox; 14 miles per hour

Answer: The fox; 7 miles per hour

Explanation:
Given that,
A rabbit can run short distances at a rate of 35 miles per hour. A fox can run short distances at a rate of 21 miles per half hour.
If a fox runs 21 miles for half an hour then it can 42 miles per hour.
42 – 35 = 7 miles per hour
The fox is faster by 7 miles per hour.
Therefore the correct answer is option B.

Question 3.
A pet survey found that the ratio of dogs to cats is 25. Which proportion shows the number of dogs if the number of cats is 140?
Options:
a. \(\frac{2 \mathrm{dogs}}{5 \mathrm{cats}}=\frac{140 \mathrm{dogs}}{350 \mathrm{cats}}\)
b. \(\frac{2 \mathrm{dogs}}{5 \mathrm{cats}}=\frac{140 \mathrm{cats}}{350 \mathrm{dogs}}\)
c. \(\frac{2 \mathrm{dogs}}{5 \mathrm{cats}}=\frac{28 \mathrm{dogs}}{140 \mathrm{cats}}\)
d. \(\frac{2 \mathrm{dogs}}{5 \mathrm{cats}}=\frac{56 \mathrm{dogs}}{140 \mathrm{cats}}\)

Answer: \(\frac{2 \mathrm{dogs}}{5 \mathrm{cats}}=\frac{56 \mathrm{dogs}}{140 \mathrm{cats}}\)

Explanation:
Given,
A pet survey found that the ratio of dogs to cats is 25.
Since 5 × 25 = 140 and
2 × 28 = 56
= 56/140
Thus the correct answer is option D.

Question 4.
What is the cost of 2 kilograms of flour if 3 kilograms cost $4.86 and the unit price for each package of flour is the same?
Options:
a. $0.81
b. $2.86
c. $3.24
d. $9.72

Answer: $3.24

Explanation:
We need to find the unit price.
4.86/3 = 1.62
multiply the unit price by 2 to find the cost of 2 kg
1.62 × 2 = 3.24
Therefore the correct answer is option C.

Question 5.
One gallon of paint covers about 450 square feet. How many square feet will 1.5 gallons of paint cover?
Options:
a. 300ft²
b. 451.5ft²
c. 675ft²
d. 900ft²

Answer: 675ft²

Explanation:
Given,
One gallon of paint covers about 450 square feet.
We need to find how many square feet will 1.5 gallons of paint cover.
For that, we have to multiply the number of gallons by the number of square feet covered by each gallon.
1.5 × 450 = 675 sq. ft.
Thus the correct answer is option C.

Question 6.
The graph shows the relationship between the late fines the library charges and the number of days late.

Options:
a. y = 0.25x
b. y = 0.40x
c. y = 0.50x
d. y = 0.75x

Answer: y = 0.25x

Explanation:
The graph shows the relationship between the late fines the library charges and the number of days late
One of the points is (2, 0.5) so k = 0.5/2 = 0.25.
This gives an equation of y = 0.25x
Thus the correct answer is option A.

Mini-Task

Question 7.
School is 2 miles from home along a straight road. The table shows your distance from home as you walk home at a constant rate.
a. Is the relationship in the table proportional?

___________

Answer: no
Since 1.5/10 = 0.15 and 1/20 = 0.05, the relationship is not proportional since the ratios are not equal.

Question 7.
b. Find your distance from school for each time in the table.
Type below:
___________

Answer:
Since the distance between school and home is 2 mi,
the distance from school when the distance from home is 1.5 mi is 2 – 1.5 = 0.5 mi,
for 1 mi its 2 – 1 = 1 mi, and for 0.5 mi it is 2 – 0.5 = 1.5 mi.

Question 7.
c. Write an equation representing the relationship between the distance from school and time walking.
Type below:
___________

Answer: y = -0.05 x + 2

Explanation:
At time t = 0, you are 2 mi from home since the distance from home to school is 2 mi. This means the y-intercept, b is 2.
To find the slope of the line, find the rate of change:
m = (y2 – y1)/(x2 – x1) = (1 – 1.5)/(20 – 10) = -0.5/10 = -0.05
The line is then y = mx + b
y = 0.5x  + 2.

Module 4 – Page No. 161

EXERCISES

Question 3.
The table below shows the proportional relationship between Juan’s pay and the hours he works. Complete the table. Plot the data and connect the points with a line.


Type below:
____________

Answer:
First, find the constant of proportionality.
Let y represent pay and x represents the number of hours worked.
The constant of proportionality = y/x = 40/2 = 20
Hence, for 1 hour of work, he earns $20.
To find how many hours he needs to work $80, we divide 80 by the constant of proportionality.
80/20 = 4
For 1 hour he earns $20, so for 5 hours he earns 5 × 20 = $100.
For 1 hour he earns $20, so for 6 hours he earns 6 × 20 = $120.

Conclusion:

After the preparation of  Go Math Grade 7 Answer Key Chapter 4 Rates and Proportionality, we suggest the students solve the questions given in the Module Quiz. Test yourself by solving the questions given at the end of this chapter. By this, you can enhance your math skills and secure good marks in the exams. If you have any doubts regarding the solutions you can post your comment in the below comment section.

Get expert verified solutions in Go Math Grade 7 Answer Key Chapter 3 Rational Numbers here. So, the students of the 7th class can Download Go Math Grade 7 Answer Key Chapter 3 Rational Numbers pdf for free. Our Go Math 7th Grade Chapter 3 Rational Numbers helps the students to complete their homework in time and also score the highest marks in the exams.

Go Math Grade 7 Answer Key Chapter 3 Rational Numbers

All concepts are covered in one place along with answers for Go Math Grade 7 Chapter 3 Rational Numbers. Follow the steps to Download HMH Go Math Chapter 3 Grade 7 Answer Key pdf to learn simple methods to solve the problems. The quick way of solving problems will help the students to save time. Hence, check the question and find out the complete answers and explanations for every problem.

Chapter 3 – Rational Numbers and Decimals

• Rational Numbers and Decimals – Guided Practice – Page No. 64
• Rational Numbers and Decimals – Independent Practice – Page No. 65
• Rational Numbers and Decimals – Page No. 66

Chapter 3 – Adding Rational Numbers

• Adding Rational Numbers – Guided Practice – Page No. 72
• Adding Rational Numbers – Independent Practice – Page No. 73
• Adding Rational Numbers – Independent Practice – Page No. 74

Chapter 3 – Subtracting Rational Numbers

• Subtracting Rational Numbers – Guided Practice – Page No. 79
• Subtracting Rational Numbers – Independent Practice – Page No. 80
• Subtracting Rational Numbers – Page No. 81
• Subtracting Rational Numbers – H.O.T – Page No. 82

Chapter 3 – Multiply Rational Numbers

• Multiply Rational Numbers – Guided Practice – Page No. 86
• Multiply Rational Numbers – Independent Practice – Page No. 87
• Multiply Rational Numbers – Page No. 88

Chapter 3 – Divide Rational Numbers

• Divide Rational Numbers – Guided Practice – Page No. 92
• Divide Rational Numbers – Independent Practice – Page No. 93
• Divide Rational Numbers – Page No. 94

Chapter 3 – Applying Rational Number Operations

• Applying Rational Number Operations – Guided Practice – Page No. 98
• Applying Rational Number Operations – Independent Practice – Page No. 99
• Applying Rational Number Operations – Page No. 100

Chapter 3 – Module Review

• Module Quiz – 3.1 Rational Numbers and Decimals – Page No. 101
• MODULE 3 MIXED REVIEW – Selected Response – Page No. 102
• Module 3 Review – Rational Numbers – Page No. 106

Chapter 3 – Performance Tasks

• Unit 1 Performance Tasks – Page No. 107
• Unit 1 Performance Tasks – Page No. 108

Chapter 3 – MIXED REVIEW

• Unit 1 MIXED REVIEW – Selected Response – Page No. 109
• Unit 1 MIXED REVIEW – Page No. 110

Rational Numbers and Decimals – Guided Practice – Page No. 64

Write each rational number as a decimal. Then tell whether each decimal is a terminating or a repeating decimal.

Question 1.
\(\frac{3}{5}\) =
___________ decimals

Answer: terminating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.

\(\frac{3}{5}\) = 3 ÷ 5
3/5 = 0.6
The decimal is not repeating so it is a terminating decimal which is 0.6

Question 2.
\(\frac{89}{100}\) =
___________ decimals

Answer: terminating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
\(\frac{89}{100}\) = 0.89
The decimal is not repeating so it is a terminating decimal which is 0.89

Question 5.
\(\frac{7}{9}\) =
___________ decimals

Answer: repeating

Explanation:
To convert fraction decimals, we have to divide the numerator by the denominator.
If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
\(\frac{7}{9}\) = 0.77…
The quotient is a repeating decimal which is 0.77…

Question 6.
\(\frac{9}{25}\) =
___________ decimals

Answer: terminating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
\(\frac{9}{25}\) = 0.36
The decimal is not repeating so it is a terminating decimal which is 0.36

Question 7.
\(\frac{1}{25}\) =
___________ decimals

Answer: terminating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
\(\frac{1}{25}\) = 0.04
The decimal is not repeating so it is a terminating decimal which is 0.04

Question 8.
\(\frac{25}{176}\) =
___________ decimals

Answer: repeating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
\(\frac{25}{176}\) = 0.14204545454
The quotient is a repeating decimal which is 0.14204545454

Question 9.
\(\frac{12}{1000}\) =
___________ decimals

Answer: terminating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
\(\frac{12}{1000}\) =0.012
The decimal is not repeating so it is a terminating decimal which is 0.012

Write each mixed number as a decimal.

Question 10.
11 \(\frac{1}{6}\) =
___________ decimals

Answer: repeating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
11 \(\frac{1}{6}\) = 11.1666666667
The quotient is a repeating decimal which is 11.1666666667

Question 13.
7 \(\frac{3}{15}\) =
___________ decimals

Answer: terminating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
First, convert the mixed fraction to the improper fraction.
7 \(\frac{3}{15}\) = \(\frac{108}{15}\) = 7.2
Thus, the decimal is not repeating so it is a terminating decimal which is 7.2

Question 14.
54 \(\frac{3}{11}\) =
___________ decimals

Answer: repeating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
First, convert the mixed fraction to the improper fraction.
54 \(\frac{3}{11}\) = \(\frac{597}{11}\) = 54.2727…
The quotient is a repeating decimal which is 54.2727…

Question 15.
3 \(\frac{1}{18}\) =
___________ decimals

Answer: repeating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
First, convert the mixed fraction to the improper fraction.
3 \(\frac{1}{18}\) = \(\frac{55}{18}\) = 3.055..
The quotient is a repeating decimal which is 3.055..

Question 16.
Maggie bought 3 \(\frac{2}{3}\) lb of apples to make some apple pies. What is the weight of the apples written as a decimal?
3 \(\frac{2}{3}\) =
___________ decimal

Answer: repeating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
First, convert the mixed fraction to the improper fraction.
3 \(\frac{2}{3}\) = \(\frac{11}{3}\) = 3.66..
The quotient is a repeating decimal which is 3.66..

Question 17.
Harry’s dog weighs 12 \(\frac{7}{8}\) pounds. What is the weight of Harry’s dog written as a decimal?
12 \(\frac{7}{8}\) =
___________ decimals

Answer: terminating

Explanation:
Given that,
Harry’s dog weighs 12 \(\frac{7}{8}\) pounds.
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
First, convert the mixed fraction to the improper fraction.
12 \(\frac{7}{8}\) = \(\frac{103}{8}\) = 12.875

Essential Question Check-In

Question 18.
Tom is trying to write \(\frac{3}{47}\) as a decimal. He used long division and divided until he got the quotient 0.0638297872, at which point he stopped. Since the decimal doesn’t seem to terminate or repeat, he concluded that \(\frac{3}{47}\) is not rational. Do you agree or disagree? Why?
___________

Answer: disagree

Explanation:
We are given the number:
{0, 1, 2, 3, ……45, 46}
When dividing a number by 47 the possible remainders at each step are:
This means that after at most 47 steps we get a remainder which repeats. This means that process and which repeats. This means that the process stops and we get a repeating decimal.

Rational Numbers and Decimals – Independent Practice – Page No. 65

Use the table for 19–23. Write each ratio in the form \(\frac{a}{b}\) and then as a decimal. Tell whether each decimal is a terminating or a repeating decimal.

Question 19.
Basketball players to football players
___________ decimal

Answer: Repeating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
Since the item is asking us to write basketball players to football players, we write the number of basketball players (5) in the numerator and the number of football players (11) in the denominator.
5/11 = 0.4545..
This is a repeating decimal with 45 as the repeating digits.

Question 20.
Hockey players to lacrosse players
___________ decimal

Answer: terminating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
Since the item is asking us to write hockey players to lacrosse players, we write the number of hockey players (6) in the numerator and the number of lacrosse players (10) in the denominator.
Now convert the fraction into the decimal
6/10 = 0.6
This is a terminating decimal which is 0.6.

Question 21.
Polo players to football players
___________ decimal

Answer: Repeating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
Since the item is asking us to write polo players to football players, we write the number of polo players (4) in the numerator and the number of football players (11) in the denominator.
Now we convert this as a decimal.
4/11 = 0.36..
This is a repeating decimal with 36 as the repeating digits.

Question 22.
Lacrosse players to rugby players
___________ decimal

Answer: Repeating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
Since the item is asking us to write lacrosse players to rugby players, we write the number of lacrosse players (10) in the numerator and the number of rugby players (15) in the denominator.
10/15 = 0.66..
This is a repeating decimal with 6 as the repeating digit.

Question 23.
Football players to soccer players
___________ decimal

Answer: terminating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits
Since the item is asking us to write football players to soccer players, we write the number of football players (11) in the numerator and the number of soccer players (11) in the denominator.
11/11 = 1
This is a terminating decimal which is 1.

Question 25.
Yvonne bought 4 \(\frac{7}{8}\) yards of material to make a dress.
a. What is 4 \(\frac{7}{8}\) written as an improper fraction?
\(\frac{□}{□}\)

Answer:
To convert fraction decimals, we have to divide the numerator by the denominator.
If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
Convert from mixed fraction to improper fraction.
4 \(\frac{7}{8}\) = (8 × 4) + 7 = 32 + 7 = 39/8

Question 25.
b. What is 4 \(\frac{7}{8}\) written as a decimal?
______

Answer:
Remember that we need to add the whole number and just convert the fraction part to decimal.
7/8 = 0.875
The fraction is a terminating decimal. Combining the whole number and the decimal part we get,
4 + 0.875 = 4.875

Question 25.
c. Communicate Mathematical Ideas If Yvonne wanted to make 3 dresses that use 4 \(\frac{7}{8}\) yd of fabric each, explain how she could use estimation to make sure she has enough fabric for all of them.
Type below:
_____________

Answer:
Using estimation, we say that 4 \(\frac{7}{8}\) ≈ 5.
We can now multiply 3 by 5, and therefore, she needs 15 yards of fabric.

Rational Numbers and Decimals – Page No. 66

Question 26.
Vocabulary A rational number can be written as the ratio of one _______ to another and can be represented by a repeating or ______ decimal.
Type below:
_____________

Answer:
A rational number can be written as the ratio of one integer to another and can be represented by a repeating or terminating decimal.

Question 29.
Analyze Relationships You are given a fraction in simplest form. The numerator is not zero. When you write the fraction as a decimal, it is a repeating decimal. Which numbers from 1 to 10 could be the denominator?
Type below:
_____________

Answer: {3, 6, 7, 9}

Explanation:
Since the only numbers which can be factors of the denominators lead to a terminating decimal are 1, 2, and 5 and combinations of them, it means that if the denominator has at least one of the other numbers at the denominator, the decimal form will be a repeating decimal.
Among the numbers from 1 to 10, the presence of any of these numbers in the denominator will lead to a repeating decimal:
{3, 6, 7, 9}

Adding Rational Numbers – Guided Practice – Page No. 72

Use a number line to find each sum.

Question 1.
−3 + (−1.5) =
______

Answer: -4.5

Explanation:
Remember if the number being added is positive, move the number of units going to the right and if the number being added is negative, move the number of units to the left.
Since we are adding a negative number, starting from -3, we move 1.5 units to the left. This results in -4.5.

Question 2.
1.5 + 3.5 =
______

Answer: 5

Explanation:
Remember if the number being added is positive, move the number of units going to the right and if the number being added is negative, move the number of units to the left.
Since we are adding a positive number, starting from 1.5 we move 3.5 units to the right. This results in 5.

Question 5.
3 + (−5) =
______

Answer: -2

Explanation:
Remember if the number being added is positive, move the number of units going to the right and if the number being added is negative, move the number of units to the left.
Since we are adding a negative number, starting from 3 we move 5 units to the left. This results in -2.

Question 6.
(−1.5) + 4 =
______

Answer: 2.5

Explanation:
Remember if the number being added is positive, move the number of units going to the right and if the number being added is negative, move the number of units to the left.
Since we are adding a positive number, starting from 1.5 we move 4 units to the left. This results in 2.5

Question 7.
Victor borrowed 21.50 dollars from his mother to go to the theater. A week later, he paid her 21.50 dollars back. How much does he still owe her?
______

Answer: 0

Explanation:
We use positive numbers for the money he receives and negative numbers for the money he returns.
21.50 – 21.50 = 0
The result is zero. This means he doesn’t owe anything to his mother.

Question 8.
Sandra used her debit card to buy lunch for 8.74 on Monday. On Tuesday, she deposited 8.74 back into her account. What is the overall increase or decrease in her bank account?
______

Answer: 0

Explanation:
We use positive numbers for the money she deposits and negative numbers for the money she spends.
-8.74 + 8.74 = 0
The result is zero. This means her bank account didn’t increase or decrease.

Find each sum without using a number line.

Question 9.
2.75 + (−2) + (−5.25) =
______

Answer: -4.50

Explanation:
We are given the expression:
2.75 + (-2) + (-5.25)
We group numbers with the same sign using the associative property.
2.75 – 7.25 = -4.50

Question 13.
4.5 + (−12) + (−4.5) =
______

Answer: -12

Explanation:
We are given the expression
4.5 + (-12) + (-4.5)
We group numbers with the same sign using the associative property.
0 – 12 = -12
The larger number is having a negative sign so the answer is -12.

Question 14.
\(\frac{1}{4}\) + (− \(\frac{3}{4}\)) =
– \(\frac{□}{□}\)

Answer: -0.50

Explanation:
We are given the expression
\(\frac{1}{4}\) + (− \(\frac{3}{4}\))
Convert the fraction to Decimal.
0.25 – 0.75 = -0.50
The larger number is having the negative sign so the sum is -0.50

Question 15.
−4 \(\frac{1}{2}\) + 2 =
– \(\frac{□}{□}\)

Answer: -2.5

Explanation:
We  = are given the expression
−4 \(\frac{1}{2}\) + 2
Convert from fraction to decimal.
-4.5 + 2 = -2.5
The larger number is having the negative sign so the sum is -2.5.

Question 16.
−8 + (−1 \(\frac{1}{8}\)) =
– \(\frac{□}{□}\)

Answer: -9.125

Explanation:
We are given the expression
−8 + (−1 \(\frac{1}{8}\))
Convert from fraction to decimal.
-8 + (-1.125) = – 9.125

Question 17.
How can you use a number line to find the sum of -4 and 6?
Type below:
____________

Answer: 6

Explanation:
Remember if the number being added is positive, move the number of units going to the right and if the number being added is negative, move the number of units to the left.
Since we are adding a positive number, starting from -4 we move 6 units to the right. This results in 2.

Adding Rational Numbers – Independent Practice – Page No. 73

Question 18.
Samuel walks forward 19 steps. He represents this movement with a positive 19. How would he represent the opposite of this number?
_______

Answer: -19
He would represent the opposite of 19 by a negative 19.

Question 20.
A submarine submerged at a depth of -35.25 meters dives an additional 8.5 meters. What is the new depth of the submarine?
_______

Answer: In adding two integers with the same sign add their absolute value and keep the common sign.
When adding two integers with opposite signs subtract the smaller absolute value from the larger one and keep the sign of the number with the larger absolute value.
Since the submarine dove 32.25 meters down this can be interrupted as -32.25. And because it dove an additional 8.5 meters down, we can add -8.5 meters to the previous distance.
Add 32.25 and 8.5 meters
32.25 + 8.5 = 43.75 meters
Thus the submarine’s new depth is 43.75 meters deep or -43.75 meters.

Question 22.
Geography
The average elevation of the city of New Orleans, Louisiana, is 0.5 m below sea level. The highest point in Louisiana is Driskill Mountain about 163.5 m higher than New Orleans. How high is Driskill Mountain?
_______

Answer: 163 meters

Explanation:
We use the positive numbers for the elevation above sea level and the negative numbers for the elevation below sea level.
163.5 – 0.5 = 163 meters
Thus the height of Driskill mountain is 163 meters.

Question 23.
Problem-Solving
A contestant on a game show has 30 points. She answers a question correctly to win 15 points. Then she answers a question incorrectly and loses 25 points. What is the contestant’s final score?
_______

Answer: 20

Explanation:
We use positive numbers for won points and negative numbers for lost points.
30 + 15 + (-25) = 20
Thus the final score is 20.

Financial Literacy

Use the table for 24–26. Kameh owns a bakery. He recorded the bakery income and expenses in a table.

Question 26.
Communicate Mathematical Ideas
If the bakery started with an extra $250 from the profits in December, describe how to use the information in the table to figure out the profit or loss of money at the bakery by the end of August. Then calculate the profit or loss.
Balance: $ _______

Answer: 948.71

Explanation:
If the bakery started with an extra $250 from the profits in December.
We will add this amount to January’s income.
250 + 1205 + 1183 + 1664 + 2413 + 2260 + 2183 = 11,158
We compute the expenses during the 6 months
(-1290) + (-1345.44) + (-1664) + (-2106.24) + (-1958.50) + (-1845.12) = -10209.29
11158 -10209.29 = 948.71
Since the result is a positive number, the bakery has profit.

Adding Rational Numbers – Independent Practice – Page No. 74

Question 27.
Vocabulary
-2 is the ________ of 2.
__________

Answer: additive inverse

Explanation:
When the sum of two numbers with opposite signs is 0, then they are additive inverses of each other.
Therefore, -2 is the additive inverse of 2.

Question 28.
The basketball coach made up a game to play where each player takes 10 shots at the basket. For every basket made, the player gains 10 points. For every basket missed, the player loses 15 points.
a. The player with the highest score sank 7 baskets and missed 3. What was the highest score?
_______ points

Answer: 25

Explanation:
We use the positive numbers for won points and the negative numbers for lost points.
We determine the highest score:
7(10) + 3(-15) = 70 + (-45) = 25

Question 28.
b. The player with the lowest score sank 2 baskets and missed 8. What was the lowest score?
_______ points

Answer: -100

Explanation:
We determine the lowest score:
2(10) + 8(-15) = 20 + (-120) = -100

Question 28.
c. Write an expression using addition to find out what the score would be if a player sank 5 baskets and missed 5 baskets.
Type below:
__________

Answer: -25

Explanation:
We determine the score for 5 baskets and 5 missed baskets:
5(10) + 5(-15) = 50 + (-75)
50 – 75 = -25

H.O.T

FOCUS ON HIGHER ORDER THINKING

Question 29.
Communicate Mathematical Ideas
Explain the different ways it is possible to add two rational numbers and get a negative number.
Type below:
__________

Answer:
The sum of two rational numbers is negative either if both numbers are negative, or they have different signs, but the negative number is the one with the greater absolute value.

Subtracting Rational Numbers – Guided Practice – Page No. 79

Use a number line to find each difference.

Question 1.
5 − (−8) =
_______

Answer: 13

Explanation:
Remember if the number being subtracted is positive, move the number of units going to the left and if the number being subtracted is negative, move the number of units to the right.
Since we are subtracting a negative number, starting from 5, we move 8 units to the right. This results in 13.

Question 4.
−0.5 − 3.5 =
_______

Answer: -4

Explanation:
Remember if the number being subtracted is positive, move the number of units going to the left and if the number being subtracted is negative, move the number of units to the right.
Since we are subtracting a positive number, starting from -0.5, we move 3.5 units to the left. This results in -4

Find each difference.

Question 5.
−14 − 22 =
_______

Answer: -36

Explanation:
We have to determine the difference
-14 – 22 = (-14) + (-22) = -36
−14 − 22 = -36

Question 8.
65 − (−14) =
_______

Answer: 79

Explanation:
We convert subtraction into addition with the opposite number
65 − (−14) = 65 + 14 = 79
The answer is 79.

Question 9.
− \(\frac{2}{9}\) − (−3) =
_______ \(\frac{□}{□}\)

Answer: 2 \(\frac{7}{9}\)

Explanation:
We convert subtraction into addition with the opposite number
− \(\frac{2}{9}\) − (−3) = − \(\frac{2}{9}\) + 3 = 2 \(\frac{7}{9}\)
The answer is 2 \(\frac{7}{9}\)

Question 10.
24 \(\frac{3}{8}\) − (−54 \(\frac{1}{8}\)) =
_______ \(\frac{□}{□}\)

Answer: 78 \(\frac{1}{2}\)

Explanation:
We convert subtraction into addition with the opposite number.
24 \(\frac{3}{8}\) − (−54 \(\frac{1}{8}\)) = 24 \(\frac{3}{8}\) + 54 \(\frac{1}{8}\) = 78 \(\frac{1}{2}\)
Thus the result is 78 \(\frac{1}{2}\).

Question 11.
A girl is snorkeling 1 meter below sea level and then dives down another 0.5 meter. How far below sea level is the girl?
_______

Answer: 1.5 meter

Explanation:
1 m below sea level is represented by the number -14. Since she is diving down 0.5 m, you must subtract -1 – 0.5 = -1.5 m
Thus the girl is 1.5 m long.

Question 15.
Mandy is trying to subtract 4 – 12, and she has asked you for help. How would you explain the process of solving the problem to Mandy, using a number line?
Type below:
____________

Answer: Start at 4 on the number line. Then move 12 places to the left since you are subtracting. This gives -8.

Subtracting Rational Numbers – Independent Practice – Page No. 80

Question 16.
Science
At the beginning of a laboratory experiment, the temperature of a substance is -12.6 °C. During the experiment, the temperature of the substance decreases by 7.5 °C. What is the final temperature of the substance?
_______

Answer: -20.1°C

Explanation:
Remember if the number being subtracted is positive, move the number of units going to the left and if the number being subtracted is negative, move the number of units to the right.
Since the temperature of the substance is -12.6 and it decreases further by 7.5, we can create the expression -12.6 – 7.5.
-12.6 – 7.5 = -20.1
Thus the final temperature is -20.1°C

Question 18.
A city known for its temperature extremes started the day at -5 degrees Fahrenheit. The temperature increased by 78 degrees Fahrenheit by midday and then dropped 32 degrees by nightfall.
a. What expression can you write to find the temperature at nightfall?
Type below:
____________

Answer:
The temperature started at -5 degrees then increased to 78 degrees and then dropped to 32 degrees.
The expression is -5 + 78 – 32

Question 18.
b. What expression can you write to describe the overall change in temperature? Hint: Do not include the temperature at the beginning of the day since you only
Type below:
____________

Answer: The overall change is the increase and decrease combined.
The expression is 78 – 32

Question 18.
c. What is the final temperature at nightfall? What is the overall change in temperature?
Type below:
____________

Answer:
Use the first expression -5 + 78 – 32 = 73 – 32 = 41 degrees
78 – 32 = 46 degrees

Question 19.
Financial Literacy
On Monday, your bank account balance was -$12.58. Because you didn’t realize this, you wrote a check for $30.72 for groceries.
a. What is the new balance in your checking account?
$ _______

Answer:
Subtract the check amount from the initial balance.
-$12.58 – $30.72 = -$43.30

Question 19.
b. The bank charges a $25 fee for paying a check on a negative balance. What is the balance in your checking account after this fee?
$ _______

Answer:
Subtract 25 from the balance from part a.
-$43.30 – $25 = -$68.30

Question 19.
c. How much money do you need to deposit to bring your account balance back up to $0 after the fee?
$ _______

Answer:
Since the account balance is -$68.30, a deposit of $68.30 is required to make the balance $0.

Astronomy

Use the table for problems 20–21.

Question 20.
How much deeper is the deepest canyon on Mars than the deepest canyon on Venus?
_______

Answer: -16,500 feet deeper

Explanation:
Subtract the lowest elevations of Mars and Venus.
-26,000 – (-9500) = -16,500

Question 21.
Persevere in Problem-Solving
What is the difference between Earth’s highest mountain and its deepest ocean canyon? What is the difference between Mars’ highest mountain and its deepest canyon? Which difference is greater? How much greater is it?
Type below:
____________

Answer:
Subtract the highest elevation and the lowest elevation on Earth.
29,035 – (-36,198) = 65,233
Subtract the highest elevation and the lowest elevation on Mars.
96,000 – 65,233 = 30,767
96,000 is greater than 65,233 so the difference for Mars is greater. subtract these two numbers to get how much greater.

Subtracting Rational Numbers – Page No. 81

Question 22.
Pamela wants to make some friendship bracelets for her friends. Each friendship bracelet needs 5.2 inches of string.
a. If Pamela has 20 inches of string, does she have enough to make bracelets for 4 of her friends?
a. _______

Answer: no

Explanation:
Each bracelet needs 5.2 inches so multiply 4 and 5.2 inches to see how many total inches she needs this is greater than 20 so she does not have enough.
4 × 5.2 = 20.8 inches

Question 22.
b. If so, how much string would she have left over? If not, how much more string would she need?
_______ in.

Answer: She needs 0.8 inches more

Question 23.
Jeremy is practicing some tricks on his skateboard. One trick takes him forward 5 feet, then he flips around and moves backward 7.2 feet, then he moves forward again for 2.2 feet.
a. What expression could be used to find how far Jeremy is from his starting position when he finishes the trick?
Type below:
___________

Answer: 5 – 7.2 + 2.2

Explanation:
He moves 5 feet forward, back 7.2 feet, and then forward 2.2 feet.

Question 23.
b. How far from his starting point is he when he finishes the trick? Explain
_______ ft.

Answer: 0 ft

Explanation:
Since the distance just pulls hi back and forth at the same amount of distance.
5 – 7.2 + 2.2 = 0 ft

Question 24.
Esteban has $20 from his allowance. There is a comic book he wishes to buy that costs $4.25, a cereal bar that costs $0.89, and a small remote control car that costs $10.99.
a. Does Esteban have enough to buy everything?
a. _______

Answer:
Find the total amount of money he wants to spend this is less than 20 so he has enough
4.25 + 0.89 + 10.99 = 16.13
Thus Esteban had enough money.

Question 24.
b. If so, how much will he have left over? If not, how much does he still need?
$ _______

Answer:
Subtract the amount he wants to spend from the amount he has to find how much he has left.
20 – 16.13 = 3.87
Thus $3.87 left.

Subtracting Rational Numbers – H.O.T – Page No. 82

Focus on Higher Order Thinking

Question 26.
Problem-Solving
The temperatures for five days in Kaktovik, Alaska, are given below.
-19.6 °F, -22.5 °F, -20.9 °F, -19.5 °F, -22.4 °F
Temperatures for the following week are expected to be twelve degrees lower every day. What are the highest and lowest temperatures expected for the corresponding 5 days next week?
Type below:
____________

Answer:
The highest temperature for the first five days was -19.5 degrees so the highest temperature the following week is 12 degrees less than that the lowest temperature the first week was -22.9 degrees so the lowest temperature the second week is 12 degrees below that
high: -19.5 – 12 = -31.5°F
low: -22.5 – 12 = -34.5°F

Question 27.
Make a Conjecture
Must the difference between two rational numbers be a rational number? Explain.
_______

Answer:
Yes, the difference between two rational numbers must be rational. Subtracting two fractions equals a fraction of an integer. Integers are rational numbers so even if the answer isn’t a fraction, it is still a rational number.

Multiply Rational Numbers – Guided Practice – Page No. 86

Use a number line to find each product.

Question 1.
5(−\(\frac{2}{3}\)) =
_______ \(\frac{□}{□}\)

Answer: -3 \(\frac{1}{3}\)

Explanation:
Remember if the number being multiplied is positive, starting from zero moves the number of units by how many times it is multiplied going to the right, and if the number being multiplied is negative, starting from zero, move the number of units by how many times it is multiplied going to the left.
Since we are multiplying −\(\frac{2}{3}\) by 5, starting from 0, we move \(\frac{2}{3}\) units to the left five times. This results in -3 \(\frac{1}{3}\)

Question 2.
3(−\(\frac{1}{4}\)) =
\(\frac{□}{□}\)

Answer: –\(\frac{3}{4}\)

Explanation:
Remember if the number being multiplied is positive, starting from zero move the number of units by how many times it is multiplied going to the right and if the number being multiplied is negative, starting from zero, move the number of units by how many times it is multiplied going to the left.
Since we are multiplying −\(\frac{1}{4}\) by 3, starting from 0, we move −\(\frac{1}{4}\) units to the left three times. This results in –\(\frac{3}{4}\).

Question 3.
−3(−\(\frac{4}{7}\)) =
_______ \(\frac{□}{□}\)

Answer: 1 \(\frac{5}{7}\)

Explanation:
Remember if the number being multiplied is positive, starting from zero move the number of units by how many times it is multiplied going to the right and if the number being multiplied is negative, starting from zero, move the number of units by how many times it is multiplied going to the left.
Since we are multiplying −\(\frac{4}{7}\) by -3, let us first multiply −\(\frac{4}{7}\) by 3. Starting from 0, we move \(\frac{4}{7}\) units to the left three times.
This results in -1 \(\frac{5}{7}\)
Therefore the opposite of this is 1 \(\frac{5}{7}\).

Question 4.
−\(\frac{3}{4}\)(−4) =
______

Answer: 3

Explanation:
Remember if the number being multiplied is positive, starting from zero move the number of units by how many times it is multiplied going to the right and if the number being multiplied is negative, starting from zero, move the number of units by how many times it is multiplied going to the left.
Since we are multiplying −\(\frac{3}{4}\) by -4, let us first multiply −\(\frac{3}{4}\) by 4. Starting from 0, we move \(\frac{3}{4}\) units to the left three times. This results in -3. Therefore the opposite of this is 3.

Question 5.
4(−3) =
______

Answer: -12

Explanation:
Remember if the number being multiplied is positive, starting from zero move the number of units by how many times it is multiplied going to the right and if the number being multiplied is negative, starting from zero, move the number of units by how many times it is multiplied going to the left.
Since we are multiplying -3 by 4, starting from 0, we move 3 units to the left four times. This results in -12.

Question 6.
(−1.8)5 =
______

Answer: -9

Explanation:
Remember if the number being multiplied is positive, starting from zero move the number of units by how many times it is multiplied going to the right and if the number being multiplied is negative, starting from zero, move the number of units by how many times it is multiplied going to the left.
Since we are multiplying -1.8 by 5, starting from 0, we move 1.8 units to the left five times. This results in -9.

Question 7.
−2(−3.4) =
______

Answer: 6.8

Explanation:
Remember if the number being multiplied is positive, starting from zero move the number of units by how many times it is multiplied going to the right and if the number being multiplied is negative, starting from zero, move the number of units by how many times it is multiplied going to the left.
Since we are multiplying -2 by -3.4, starting from 0, starting from 0, we move 3.4 units to the left two times. This results in -6.8. Therefore, the opposite of this is 6.8.

Question 8.
0.54(8) =
______

Answer: 4.32

Explanation:
Remember if the number being multiplied is positive, starting from zero move the number of units by how many times it is multiplied going to the right and if the number being multiplied is negative, starting from zero, move the number of units by how many times it is multiplied going to the left.
Since we are multiplying 0.54 by 8, starting from 0, we move 0.54 units to the right eight times. This results in 4.32.

Question 9.
−5(−1.2) =
______

Answer: 6

Explanation:
Remember if the number being multiplied is positive, starting from zero move the number of units by how many times it is multiplied going to the right and if the number being multiplied is negative, starting from zero, move the number of units by how many times it is multiplied going to the left.
Since we are multiplying -1.2 by -5, Starting from 0, we move 1.2 units to the left five times. This results in -6. Therefore the opposite of this is 6.

Question 10.
−2.4(3) =
______

Answer: -7.2

Explanation:
Remember if the number being multiplied is positive, starting from zero moves the number of units by how many times it is multiplied going to the right, and if the number being multiplied is negative, starting from zero, move the number of units by how many times it is multiplied going to the left.
Since we are multiplying -2.4 by 3, starting from 0, we move 2.4 units to the left three times. This results in -7.2

Multiply.

Question 13.
\(-\frac{1}{8} \times 5 \times \frac{2}{3}\) =
\(\frac{□}{□}\)

Answer: –\(\frac{5}{12}\)

Explanation:
Use the commutative property to switch the order of the first two fractions.
\(-\frac{1}{8} \times 5 \times \frac{2}{3}\) = –\(\frac{1}{8}\) × \(\frac{2}{3}\) × 5
–\(\frac{1}{12}\) × 5 = –\(\frac{5}{12}\)

Question 14.
\(-\frac{2}{3}\left(\frac{1}{2}\right)\left(-\frac{6}{7}\right)\) =
\(\frac{□}{□}\)

Answer: \(\frac{2}{7}\)

Explanation:
Multiply the first two fractions by canceling the 2s.
\(-\frac{2}{3}\left(\frac{1}{2}\right)\left(-\frac{6}{7}\right)\) = –\(\frac{1}{3}\)(-\(\frac{6}{7}\))
Multiply by canceling the 3 and 6 to get a 2 in the numerator two negatives make a positive.
So the answer is \(\frac{2}{7}\)

Question 15.
The price of one share of Acme Company declined $3.50 per day for 4 days in a row. What is the overall change in the price of one share?
$ _______

Answer: -$14

Explanation:
Given that,
The price of one share of Acme Company declined $3.50 per day for 4 days in a row.
-$3.50 × 4 = -$14.00
Thus the overall change in the price of one share is -$14.

Question 16.
In one day, 18 people each withdrew $100 from an ATM machine. What is the overall change in the amount of money in the ATM machine?
$ _______

Answer: The overall change in the amount of money in the ATM machine is the product of the amount people withdrew times the number of people. This gives -100(18) = -1800.
Therefore the overall change in the amount of money in the ATM machine is -$1800.

Question 17.
Explain how you can find the sign of the product of two or more rational numbers.
Type below:
____________

Answer: If the product has an even number of negative signs, then the product is positive. If the product has an odd number of negative signs, then the product is negative.

Multiply Rational Numbers – Independent Practice – Page No. 87

Question 18.
Financial Literacy
Sandy has $200 in her bank account.
a. If she writes 6 checks for exactly $19.98, what expression describes the change in her bank account?
_______

Answer: The change in her bank account is equal to the product of the check amounts and the number of checks.
This gives the expression 6(-19.98)

Question 18.
b. What is her account balance after the checks are cashed?
$ _______

Answer: She started with $200 and her account balance changes by 6(-19.98) dollars so her account balance is 200 – 6(-19.98) = 200 – 119.88 = 80.12

Question 19.
Communicating Mathematical Ideas
Explain, in words, how to find the product of -4(-1.5) using a number line. Where do you end up?
Type below:
____________

Answer:
First, find the value of -4(-1.5) by starting at 0 on the number line and moving 1.5 units left four times.
This gives a value of 4(-1.5) = -6
Since -4(-1.5) is the opposite of 4(-1.5), the answer is 6.

Question 22.
Multistep
For Home Economics class, Sandra has 5 cups of flour. She made 3 batches of cookies that each used 1.5 cups of flour. Write and solve an expression to find the amount of flour Sandra has left after making the 3 batches of cookies.
_______ cups

Answer: 0.5 cups

Explanation:
Sandra has a total of 5 cups of flour. Since she used 1.5 cups per batch of the cookie, and there are 3 batches, we can subtract the product of the cups and the number of batches
1.5 × 3 = 4.5
Therefore the expression should be 5 – 4.5 = 0.5
Thus Sandra has 0.5 cups of flour left.

Question 23.
Critique Reasoning
In class, Matthew stated,“I think that a negative is like the opposite. That is why multiplying a negative times a negative equals a positive. The opposite of negative is positive, so it is just like multiplying the opposite of a negative twice, which is two positives.”
Do you agree or disagree with his reasoning? What would you say in response to him?
_______

Answer: I agree with him. The product of two negatives is positive because the product of two positives is positive and negatives are opposites of positives.

Question 24.
Kaitlin is on a long car trip. Every time she stops to buy gas, she loses 15 minutes of travel time. If she has to stop 5 times, how late will she be getting to her destination?
_______ minutes

Answer: 75 minutes

Explanation:
Multiply the number of stops by the length of each stop to find the time she will be late.
5 × 15 = 75
Thus Kaitlin will be 75 minutes late to reach her destination.

Multiply Rational Numbers – Page No. 88

Question 25.
The table shows the scoring system for quarterbacks in Jeremy’s fantasy football league. In one game, Jeremy’s quarterback had 2 touchdown passes, 16 complete passes, 7 incomplete passes, and 2 interceptions. How many total points did Jeremy’s quarterback score?

_______ pts

Answer: 13.5 points

Explanation:
Write the expression for the total number of points
2(6) + 16(0.5) + 7(-0.5) + 2(-1.5)
= 12 + 8 – 3.5 – 3
= 20 – 6.5
= 13.5
Thus Jeremy’s quarterback scored 13.5 points.

H.O.T

Focus On Higher Order Thinking

Question 26.
Represent Real-World Problems
The ground temperature at Brigham Airport is 12 °C. The temperature decreases by 6.8 °C for every increase of 1 kilometer above the ground. What is the overall change in temperature outside a plane flying at an altitude of 5 kilometers above Brigham Airport?
_______ °C

Answer: -22°C

Explanation:
Remember if the number being multiplied is positive, starting from zero moves the number of units by how many times it is multiplied going to the right and if the number being multiplied is negative, starting from zero, move the number of units by how many times it is multiplied going to the left.
Note that the ground temperature is 12°C. Since the temperature decreases by 6.8°C for every kilometer above ground, and the given height of the plane is 5 kilometers,
We can subtract the product of the temperature and the distance 5(6.8) from the ground temperature.
Therefore the expression should be 12 – 5(6.8)
= 12 – 34
= -22
Thus the temperature outside a plane flying at an altitude of 5 kilometers above Brigham Airport is -22°C

Question 27.
Identify Patterns
The product of four numbers, a, b, c, and d, is a negative number. The table shows one combination of positive and negative signs of the four numbers that could produce a negative product. Complete the table to show the seven other possible combinations.

Type below:
_____________

Answer:
In multiplying numbers, an odd number of negative signs produces a negative product.

Question 28.
Reason Abstractly
Find two integers whose sum is -7 and whose product is 12. Explain how you found the numbers.
Type below:
_____________

Answer: -3 and -4

Explanation:
Let x and y be the two numbers. Write the equations using the given information
x + y = -7
xy = 12
Since the two numbers multiply to a positive and add to a negative the two numbers must be negative. Find the pairs of negative numbers that multiply by 12.
-1 and -12, -2 and -6, and -3 and -4.
Thus the pairs that have a sum of -7 and a product of 12 are -3 and -4.

Divide Rational Numbers – Guided Practice – Page No. 92

Find each quotient.

Question 1.
\(\frac{0.72}{-0.9}\) =
_______

Answer: -0.8

Explanation:
We have to find the quotient:
\(\frac{0.72}{-0.9}\)
We determine the sign of the quotient.
The quotient will be negative because the numbers have different signs.
\(\frac{0.72}{-0.9}\) = -0.8

Question 2.
\(\left(-\frac{\frac{1}{5}}{\frac{7}{5}}\right)\) =
\(\frac{□}{□}\)

Answer: – \(\frac{1}{7}\)

Explanation:
We have to find the quotient:
\(\left(-\frac{\frac{1}{5}}{\frac{7}{5}}\right)\)
We determine the sign of the quotient.
The quotient will be negative because the numbers have different signs.
\(\left(-\frac{\frac{1}{5}}{\frac{7}{5}}\right)\) = – \(\frac{5}{35}\) = – \(\frac{1}{7}\)

Question 3.
\(\frac{56}{-7}\) =
_______

Answer: -8

Explanation:
We have to find the quotient:
\(\frac{56}{-7}\)
We determine the sign of the quotient.
The quotient will be negative because the numbers have different signs.
7 divides 56 eight times.
Thus the quotient of \(\frac{56}{-7}\) = -8

Question 6.
\(\frac{-91}{-13}\) =
_______

Answer: 7

Explanation:
We have to find the quotient:
\(\frac{-91}{-13}\)
We determine the sign of the quotient.
The quotient will be positive because the numbers have the same signs.
13 divides 91 seven times.
\(\frac{-91}{-13}\) = 7
Thus the quotient is 7.

Question 7.
\(\frac{-\frac{3}{7}}{\frac{9}{4}}\) =
\(\frac{□}{□}\)

Answer: –\(\frac{4}{21}\)

Explanation:
We have to find the quotient:
\(\frac{-\frac{3}{7}}{\frac{9}{4}}\)
We determine the sign of the quotient.
The quotient will be negative because the numbers have different signs.
\(\frac{-\frac{3}{7}}{\frac{9}{4}}\) = -3/7 × 4/9 = -12/63
-12/63 = -4/21
\(\frac{-\frac{3}{7}}{\frac{9}{4}}\) = –\(\frac{4}{21}\)

Question 8.
– \(\frac{12}{0.03}\) =
_______

Answer: -400

Explanation:
We have to find the quotient:
– \(\frac{12}{0.03}\)
We determine the sign of the quotient.
The quotient will be negative because the numbers have different signs.
– \(\frac{12}{0.03}\) = -400
So the quotient is -400.

Question 9.
A water pail in your backyard has a small hole in it. You notice that it has drained a total of 3.5 liters in 4 days. What is the average change in water volume each day?
_______ liters per day

Answer: -0.875 litres/day

Explanation:
Given that,
A water pail in your backyard has a small hole in it. You notice that it has drained a total of 3.5 liters in 4 days.
The average change of water volume each day is the quotient.
So, divide -3.5 by 4.
The quotient will be negative because the numbers have different signs.
-3.5/4 = -0.875
Thus the water volume diminishes by 0.875 liters each day.

Question 10.
The price of one share of ABC Company declined a total of $45.75 in 5 days. What was the average change in the price of one share per day?
$ _______

Answer: -$9.15

Explanation:
The price of one share of ABC Company declined a total of $45.75 in 5 days.
We use negative numbers for the price going down.
The average change in the price of one share per day is the quotient.
-45.75/5
We determine the sign of the quotient.
The quotient will be negative because the numbers have different signs.
-45.75/5 = -9.15
Thus the price of one share diminishes by $9.15 per day.

Question 11.
To avoid a storm, a passenger-jet pilot descended 0.44 miles in 0.8 minutes. What was the plane’s average change of altitude per minute?
_______

Answer: -0.55 miles/min

Explanation:
We use negative numbers for the altitude going down.
The plane’s average change of altitude per minute is the quotient:
-0.44/0.8
We determine the sign of the quotient
The quotient will be negative because the numbers have different signs.
-0.44/0.8 = -0.55
Therefore the plane descends by 0.55 miles per minute.

Essential Question Check-In

Question 12.
Explain how you would find the sign of the quotient \(\frac{32 \div(-2)}{-16 \div 4}\).
Type below:
___________

Answer:
Positive

Explanation:
Given,
\(\frac{32 \div(-2)}{-16 \div 4}\)
Since all the operations are of multiplication and division, the sign is given by the number of negative signs.
If the number of negative signs is even, the quotient is positive while if the number of negative signs is odd, the quotient is negative.
In this case, the number of negative signs is 2, therefore even, so the quotient is positive.
\(\frac{32 \div(-2)}{-16 \div 4}\) = -16/-4 = 4
Thus the solution is positive.

Divide Rational Numbers – Independent Practice – Page No. 93

Question 13.
\(\frac{5}{-\frac{2}{8}}\) =
_______

Answer: -20

Explanation:
We are given the expression:
\(\frac{5}{-\frac{2}{8}}\)
The quotient will be negative because the numbers have different signs.
\(\frac{5}{-\frac{2}{8}}\) = 5 ÷ (-2/8)
We rewrite using the multiplication by multiplying with the reciprocal:
5 × -8/2 = 5 × -4 = -20
Thus the quotient for \(\frac{5}{-\frac{2}{8}}\) is -20.

Question 15.
\(\frac{(-120)}{(-6)}\) =
_______

Answer: 20

Explanation:
We have to find the quotient:
\(\frac{(-120)}{(-6)}\)
We determine the sign of the quotient.
The quotient will be positive because the numbers have the same signs.
6 divides 120 twenty times.
\(\frac{(-120)}{(-6)}\) = 20
Thus the quotient for \(\frac{(-120)}{(-6)}\) is 20.

Question 16.
\(\frac{-\frac{4}{5}}{-\frac{2}{3}}\) =
\(\frac{□}{□}\)

Answer: \(\frac{6}{5}\)

Explanation:
We have to find the quotient:
\(\frac{-\frac{4}{5}}{-\frac{2}{3}}\)
We determine the sign of the quotient.
The quotient will be positive because the numbers have the same signs.
(-4/5) × (-3/2) = 12/10 = 6/5
Thus the quotient for \(\frac{-\frac{4}{5}}{-\frac{2}{3}}\) is \(\frac{6}{5}\)

Question 17.
1.03 ÷ (−10.3) =
_______

Answer: -0.1

Explanation:
We have to find the quotient:
1.03 ÷ (−10.3)
We determine the sign of the quotient.
The quotient will be negative because the numbers have different signs.
1.03 ÷ (-10.3) = -0.1

Question 18.
\(\frac{(-0.4)}{80}\) =
_______

Answer: -0.005

Explanation:
We have to find the quotient:
\(\frac{(-0.4)}{80}\)
We determine the sign of the quotient.
The quotient will be negative because the numbers have different signs.
\(\frac{(-0.4)}{80}\) = -0.005
Thus the quotient for \(\frac{(-0.4)}{80}\) is -0.005.

Question 19.
\(1 \div \frac{9}{5}\) =
\(\frac{□}{□}\)

Answer: \(\frac{5}{9}\)

Explanation:
We have to find the quotient:
\(1 \div \frac{9}{5}\)
We determine the sign of the quotient.
The quotient will be positive because the numbers have the same signs
\(1 \div \frac{9}{5}\) = 1 × 5/9 = 5/9
Thus the quotient for \(1 \div \frac{9}{5}\) is \(\frac{5}{9}\)

Question 21.
\(\frac{-10.35}{-2.3}\) =
_______

Answer: 4.5

Explanation:
We have to find the quotient:
\(\frac{-10.35}{-2.3}\)
We determine the sign of the quotient.
The quotient will be positive because the numbers have the same signs
\(\frac{-10.35}{-2.3}\) = 4.5
So, the quotient for \(\frac{-10.35}{-2.3}\) is 4.5

Question 22.
Alex usually runs for 21 hours a week, training for a marathon. If he is unable to run for 3 days, describe how to find out how many hours of training time he loses, and write the appropriate integer to describe how it affects his time.
_______ hours

Answer: -9

Explanation:
Alex usually runs for 21 hours a week, training for a marathon.
If he runs 21 hours a week, he runs 21/3 = 3 hours.
If he doesn’t run for 3 days, then he is losing 3(3) = 9 hours of training time.
Since he is losing hours, the integer is negative so the answer is -9.

Question 25.
Multistep
A seafood restaurant claims an increase of $1,750.00 over its average profit during a week where it introduced a special of baked clams.
a. If this is true, how much extra profit did it receive per day?
$ _______ per day

Answer:
There are 7 days in a week so divide the total profit of $1750 by 7 to find the extra profit per day.
1750/7 = 250
Thus he receives $250 extra profit per day.

Question 25.
b. If it had, instead, lost $150 per day, how much money would it have lost for the week?
$ _______

Answer:
Multiply the daily loss of $150 by 7 to get the weekly loss.
150 × 7 = $1050

Question 25.
c. If its total loss was $490 for the week, what was its average daily change?
$ _______ per day

Answer:
Since the company lost $490, its income changed by -$490.
Divide the change in income by 7 to find the average daily change.
-$490/7 = -$70
Thus the average daily change is -$70 per day.

Question 26.
A hot air balloon descended 99.6 meters in 12 seconds. What was the balloon’s average rate of descent in meters per second?
_______ m/s

Answer: 8.3 meters per second

Explanation:
Given that,
A hot air balloon descended 99.6 meters in 12 seconds.
99.6/12 = 8.3 meters per second.
Thus the balloon’s average rate of descent is 8.3 meters per second.

Divide Rational Numbers – Page No. 94

Question 27.
Sanderson is having trouble with his assignment. His shown work is as follows:
\(\frac{-\frac{3}{4}}{\frac{4}{3}}=-\frac{3}{4} \times \frac{4}{3}=-\frac{12}{12}=-1\)
However, his answer does not match the answer that his teacher gives him. What is Sanderson’s mistake? Find the correct answer.
\(\frac{□}{□}\)

Answer: – \(\frac{9}{16}\)

Explanation:
Sanderson made the mistake of not flipping the bottom fraction when he rewrote the problem as multiplication. The correct work is
\(\frac{-\frac{3}{4}}{\frac{4}{3}}\) = -3/4 × 4/3 = – \(\frac{9}{16}\)

Question 28.
Science
Beginning in 1996, a glacier lost an average of 3.7 meters of thickness each year. Find the total change in its thickness by the end of 2012.
_______ meters

Answer: 59.2 meters

Explanation:
Beginning in 1996, a glacier lost an average of 3.7 meters of thickness each year.
1996 to 2012 is 16 years so the total change in thickness is 16(3.7) = 59.2 meters.

H.O.T

Focus On Higher Order Thinking

Question 29.
Represent Real-World Problems
Describe a real-world situation that can be represented by the quotient -85 ÷ 15. Then find the quotient and explain what the quotient means in terms of the real-world situation.
Quotient: _______

Answer: -5.67

Explanation:
A possible real-world situation could be:
Sam has withdrawn $85 from his bank account over a period of 15 days. Find the average change in his account balance per day.
Answer: -85/15 = -5.67
So, the average rate of change in his account balance is -$5.67 per day.

Question 30.
Construct an Argument
Divide 5 by 4. Is your answer a rational number? Explain.
_______

Answer: A quotient is a rational number because it is a fraction.

Question 31.
Critical Thinking
Should the quotient of an integer divided by a nonzero integer always be a rational number? Why or why not?
_______

Answer:
Remember that in dividing and simplifying rational numbers, the quotient is positive if the signs of the numbers are the same, and negative if the signs of the numbers are different.
The quotient should be a rational number. This is because since the integers can be expressed as a quotient of two integers, then it is a rational number.

Applying Rational Number Operations – Guided Practice – Page No. 98

Question 1.
Mike hiked to Big Bear Lake in 4.5 hours at an average rate of 3 \(\frac{1}{5}\) miles per hour. Pedro hiked the same distance at a rate of 3 \(\frac{3}{5}\) miles per hour. How long did it take Pedro to reach the lake?
_______ hours

Answer: 4 hours

Explanation:
Given that,
Mike hiked to Big Bear Lake in 4.5 hours at an average rate of 3 \(\frac{1}{5}\) miles per hour. Pedro hiked the same distance at a rate of 3 \(\frac{3}{5}\) miles per hour.
4.5h × 3 \(\frac{3}{5}\) miles per hour = 4.5 × 3.2 miles = 14.4 miles
Plug in the distance you found in step 1 and the given rate in the problem to find the number of hours for Pedro.
14.4 miles ÷ 3 \(\frac{3}{5}\) miles per hour = 14.4 ÷ 3.6 hours = 4 hours

Essential Question Check-In

Question 3.
Why is it important to consider using tools when you are solving a problem?
Type below:
___________

Answer: It is important to consider using tools, such as a calculator, when solving problems because some problems involve multiplying and dividing decimals that are too time-consuming to do by hand.

Applying Rational Number Operations – Independent Practice – Page No. 99

Solve, using appropriate tools.

Question 4.
Three rock climbers started a climb with each person carrying 7.8 kilograms of climbing equipment. A fourth climber with no equipment joined the group. The group divided the total weight of climbing equipment equally among the four climbers. How much did each climber carry?
_______ kilograms

Answer: 5.85 kilograms

Explanation:
Given,
Three rock climbers started a climb with each person carrying 7.8 kilograms of climbing equipment.
A fourth climber with no equipment joined the group.
3 × 7.8 = 23.4
The group divided the total weight of climbing equipment equally among the four climbers.
23.4/4 = 5.85 kilograms
Thus each climber carries 5.85 kilograms

Question 6.
Diane serves breakfast to two groups of children at a daycare center. One box of Oaties contains 12 cups of cereal. She needs \(\frac{1}{3}\) cup for each younger child and \(\frac{3}{4}\) cup for each older child. Today’s group includes 11 younger children and 10 older children. Is one box of Oaties enough for everyone? Explain.
________

Answer: Yes

Explanation:
11 × \(\frac{1}{3}\) + 10 × \(\frac{3}{4}\)
\(\frac{11}{3}\) + \(\frac{15}{2}\)
\(\frac{22}{6}\) + \(\frac{45}{6}\) = \(\frac{67}{6}\)
= 11 \(\frac{1}{6}\)

Question 7.
The figure shows how the yard lines on a football field are numbered. The goal lines are labeled G. A referee was standing on a certain yard line as the first quarter ended. He walked 41 \(\frac{3}{4}\) yards to a yard line with the same number as the one he had just left. How far was the referee from the nearest goal line?
1
________ \(\frac{□}{□}\)

Answer: 29 \(\frac{1}{8}\)

Explanation:
The American football field is 100 yds long, 53 1/3 yards wide, and has 10-yard touchdown zones at each end of the field.
Let x = distance of the referee at the end of the quarter from the nearest goal.
The distance between the same yard lines on either side of the centerline is
100 – 2x
This distance is the 41 3/4 yards that the referee walked. Therefore
100 – 2x = 41.75
-2x = 41.75 – 100 = -58.25
x = 29.125 yd
Convert from decimal to fraction.
x = 29 \(\frac{1}{8}\) yards

In 8–10, a teacher gave a test with 50 questions, each worth the same number of points. Donovan got 39 out of 50 questions right. Marci’s score was 10 percentage points higher than Donovan’s.

Question 8.
What was Marci’s score? Explain.
________ %

Answer: 88 %

Explanation:
39/50 = 78/100
78/100 + 10/100 = 88/100 = 44/50
88/100 = 88%

Question 9.
How many more questions did Marci answer correctly? Explain.
________ questions

Answer: 5 questions

Explanation:
Marci got 44 correct and Donovan got 39 correct so she got 44 – 39 = 5 more questions correct.

Question 10.
Explain how you can check your answers for reasonableness.
Type below:
_____________

Answer:
You can check your answers for reasonableness by using estimates.
Donovan scored 39/50 which is about 40/50 = 80/100 = 80%
Ten percentage points higher is than 80% + 10% = 90% = 90/100 = 45/50.
Since Marci’s score was 44/50, it is a reasonable answer.

Applying Rational Number Operations – Page No. 100

For 11–13, use the expression 1.43 × \(\left(-\frac{19}{37}\right)\)

Question 12.
Find the product. Explain your method.
_______

Answer:
Using a calculator, you get that 1.43 × (-19/37) ≈ -0.734

Question 13.
Does your answer to Exercise 12 justify your answer to Exercise 11?
_______

Answer: Yes

Explanation:
-0.734 is close to the estimate of -0.75 so the answer to Exercise 12 justifies the answer to Exercise 11.

H.O.T

Focus On Higher Order Thinking

Question 14.
Persevere in Problem-Solving
A scuba diver dove from the surface of the ocean to an elevation of −79 \(\frac{9}{10}\) feet at a rate of -18.8 feet per minute. After spending 12.75 minutes at that elevation, the diver ascended to an elevation of −28 \(\frac{9}{10}\) feet. The total time for the dive so far was 19 \(\frac{1}{8}\) minutes. What was the rate of change in the diver’s elevation during the ascent?
_______ ft/min

Answer: 24 ft/min

Explanation:
Given that,
A scuba diver dove from the surface of the ocean to an elevation of −79 \(\frac{9}{10}\) feet at a rate of -18.8 feet per minute.
After spending 12.75 minutes at that elevation, the diver ascended to an elevation of −28 \(\frac{9}{10}\) feet.
The total time for the dive so far was 19 \(\frac{1}{8}\) minutes.
−79 \(\frac{9}{10}\) ÷ -18.8 = 4.25 minutes
Find the time it took to ascend by subtracting the descent time and time spent at the descent elevation from the total dive time.
19 \(\frac{1}{8}\) – 4.25 – 12.75 = 2 1/8 minutes
-28 \(\frac{9}{10}\) – (-−79 \(\frac{9}{10}\)) = 51 feet
Find the rate of change by dividing the distance in feet divided by the time.
51/2 1/8 = 24 feet per minute

Question 16.
Represent Real-World Problems
Describe a real-world problem you could solve with the help of a yardstick and a calculator.
Type below:
___________

Answer:
Finding the perimeter of the table. Using the yardstick you can get the side length of the table and add these measurements to get the perimeter.

Module Quiz – 3.1 Rational Numbers and Decimals – Page No. 101

Write each mixed number as a decimal.

Question 1.
4 \(\frac{1}{5}\) =
_______

Answer: 4.2

Explanation:
To convert fractions to decimals, simply divide the numerator to the denominator. If the quotient goes on and on, it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
\(\frac{1}{5}\) = 0.2
4 + 0.2 = 4.2
4 \(\frac{1}{5}\) = 4.2

Question 2.
12 \(\frac{14}{15}\) =
_______

Answer: 12.933..

Explanation:
To convert fractions to decimals, simply divide the numerator to the denominator. If the quotient goes on and on, it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
\(\frac{14}{15}\) = 0.933..
12 + 0.933 = 12.933..
12 \(\frac{14}{15}\) = 12.933..

Question 3.
5 \(\frac{5}{32}\) =
_______

Answer: 5.15625

Explanation:
To convert fractions to decimals, simply divide the numerator to the denominator. If the quotient goes on and on, it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
\(\frac{5}{32}\) = 0.15625
5 + 0.15625 = 5.15625
5 \(\frac{5}{32}\) = 5.15625

3.2 Adding Rational Numbers

Find each sum.

Question 4.
4.5 + 7.1 =
_______

Answer: 11.6

Explanation:
To add or subtract numbers, make sure to align the digits vertically before doing the operation.
Make sure to align ones, tens, hundreds, and thousands of digits before adding
4.5
+7.1
 11.6

3.3 Subtracting Rational Numbers

Find each difference.

Question 7.
14.2 − (−4.9) =
_______

Answer: 19.1

Explanation:
14.2 − (−4.9)
= 14.2 + 4.9 = 19.1

3.4 Multiplying Rational Numbers

Multiply.

Question 8.
\(-4\left(\frac{7}{10}\right)\) =
\(\frac{□}{□}\)

Answer: –\(\frac{14}{5}\)

Explanation:
Multiply the whole number with the numerator. write this product in the numerator and keep the same denominator.
\(-4\left(\frac{7}{10}\right)\) = –\(\frac{14}{5}\)

Question 9.
−3.2(−5.6)(4) =
_______

Answer: 71.68

Explanation:
Multiply the first two numbers. There are two negative signs so the answer will be positive.
−3.2(−5.6)(4) = 17.92 × 4 = 71.68

3.5 Dividing Rational Numbers

Find each quotient.

Question 10.
\(-\frac{19}{2} \div \frac{38}{7}\) =
\(\frac{□}{□}\)

Answer: –\(\frac{7}{4}\)

Explanation:
\(-\frac{19}{2} \div \frac{38}{7}\)
-19/2 × 7/38 = -7/2 × 1/2
= –\(\frac{7}{4}\)

Question 11.
\(\frac{-32.01}{-3.3}\) =
_______

Answer: 9.7

Explanation:
Given,
\(\frac{-32.01}{-3.3}\)
Remember that dividing two negatives gives a positive answer.
-32.01 ÷ -3.3 = 9.7

3.6 Applying Rational Number Operations

Question 12.
Luis bought stock at $83.60. The next day, the price increased by 15.35 dollars. This new price changed by −4 \(\frac{3}{4}\)% the following day. What was the final stock price? Is your answer reasonable? Explain.
$ _______

Answer: $94.25

Explanation:
83.60 + 15.35 = 98.95
98.95 × −4 \(\frac{3}{4}\)% = 98.95 × -0.0475 = 4.70
98.95 – 4.70 = $94.25

Essential Question

Question 13.
How can you use negative numbers to represent real-world problems?
Type below:
___________

Answer:
Negative numbers can be used in real-world problems to represent decreases or values that are below a level considered to be 0.

MODULE 3 MIXED REVIEW – Selected Response – Page No. 102

Question 1.
What is −7 \(\frac{5}{12}\) written as a decimal?
Options:
a. -7.25
b. -7.333…
c. -7.41666…
d. -7.512

Answer: -7.41666…

Explanation:
Given,
−7 \(\frac{5}{12}\)
Convert from fraction to decimal.
5 ÷ 12 = 0.4166..
−7 \(\frac{5}{12}\) = -7.4166….
Thus the correct answer is option C.

Question 2.
Glenda began the day with a golf score of -6 and ended with a score of -10. Which statement represents her golf score for that day?
Options:
a. -6 – (-10) = 4
b. -10 – (-6) = -4
c. -6 + (-10) = -16
d. -10 + (-6) = -16

Answer: -10 – (-6) = -4

Explanation:
Given,
Her golf score for the day can be found by subtracting her ending score and her beginning score which gives
-10 – (-6) = -10 + 6 = -4
So, the correct answer is option B.

Question 3.
A submersible vessel at an elevation of -95 feet descends to 5 times that elevation. What is the vessel’s new elevation?
Options:
a. -475 ft
b. -19 ft
c. 19 ft
d. 475 ft

Answer: -475 ft

Explanation:
Given,
A submersible vessel at an elevation of -95 feet descends to 5 times that elevation.
-95 feet × 5 = -475 feet
Thus the correct answer is option A.

Question 4.
The temperature at 7 P.M. at a weather station in Minnesota was -5 °F. The temperature began changing at the rate of -2.5 °F per hour. What was the temperature at 10 P.M.?
Options:
a. -15 °F
b. -12.5 °F
c. 2.5 °F
d. 5 °F

Answer: -12.5 °F

Explanation:
Find the total change in temperature by multiplying the rate of change per hour times the number of hours from 7 pm to 10 pm.
-5 + (-7.5) = -12.5°F
Thus the correct answer is option B.

Question 5.
What is the sum of -2.16 and -1.75?
Options:
a. 0.41
b. 3.91
c. -0.41
d. -3.91

Answer: -3.91

Explanation:
Both numbers are negative so add their opposites and make the answer negative.
-2.16 + (-1.75) = -(2.16 + 1.75) = -3.91
So, the correct answer is option D.

Question 6.
On Sunday, the wind chill temperature reached -36 °F. On Monday, the wind chill temperature only reached \(\frac{1}{4}\) of Sunday’s wind chill temperature. What was the lowest wind chill temperature on Monday?
Options:
a. -9 °F
b. -36 \(\frac{1}{4}\) °F
c. -40 °F
d. -144 °F

Answer: -9 °F

Explanation:
Given that,
On Sunday, the wind chill temperature reached -36 °F.
On Monday, the wind chill temperature only reached \(\frac{1}{4}\) of Sunday’s wind chill temperature.
-36 × \(\frac{1}{4}\) = -9°F
Thus the correct answer is option A.

Question 7.
The level of a lake was 8 inches below normal. It decreased 1 \(\frac{1}{4}\) inches in June and 2 \(\frac{3}{8}\) inches more in July. What was the new level with respect to the normal level?
Options:
a. -11 \(\frac{5}{8}\) in.
b. -10 \(\frac{5}{8}\) in.
c. -9 \(\frac{1}{8}\) in.
d. -5 \(\frac{3}{8}\) in.

Answer: -11 \(\frac{5}{8}\) in.

Explanation:
The level of a lake was 8 inches below normal. It decreased 1 \(\frac{1}{4}\) inches in June and 2 \(\frac{3}{8}\) inches more in July.
The initial level is below normal so it is represented by a negative number. The level continued to decrease in June and July so those changes are also represented by negative numbers.
Find the sum of these values to find what the new level was with respect to the normal level.
-8 – 1 \(\frac{1}{4}\) – 2 \(\frac{3}{8}\)
= -8 – \(\frac{5}{4}\) – \(\frac{19}{8}\)
= – \(\frac{93}{8}\)
= – 11 \(\frac{5}{8}\)
Thus the correct answer is option A.

Mini-Task

Question 8.
The average annual rainfall for a town is 43.2 inches.
a. What is the average monthly rainfall?
________

Answer:
If the average rainfall us 43.2 inches then the monthly rainfall is 43.2/12 = 3.6 inches since there are 12 months in a year.

Question 8.

b. The difference of a given month’s rainfall from the average monthly rainfall is called the deviation. What is the deviation for each month shown?
May: ___________ inch
June: ___________ inches
July: ___________ inches

Answer:
The deviation for May is 2 3/5 – 3.6 = 2.6 – 3.6 = -1 inches.
The deviation for June is 7/8 – 3.6 = -2.725 inches.
The deviation for July 4 1/4 – 3.6 = 0.65 inches.

Question 8.
c. The average monthly rainfall for the previous 9 months was 4 inches. Did the town exceed its average annual rainfall? If so, by how much?
________

Answer:
If is rained 4 inches for 9 months, the total amount of rain over the 12 month period is than 9(4) + 2 3/5 + 7/8 + 4 1/4
= 36 + 2.6 + 0.875 + 4.25 = 43.725.
Since this is greater than the average annual rainfall of 43.2, the town did exceed is average annual rainfall.
the difference of 43.725 and 43.2 is
43.725 – 43.2 = 0.525
so, it exceeded it by 0.525 inches.

Module 3 Review – Rational Numbers – Page No. 106

EXERCISES

Write each mixed number as a whole number or decimal. Classify each number according to the group(s) to which it belongs: rational numbers, integers, or whole numbers.

Question 1.
\(\frac{3}{4}\)
________

Answer: 0.75, rational

Explanation:
Write as a decimal by dividing 3 by 4. A shortcut with fourths is to think of the fractions in terms of money. 4 quarters make a dollar and 3 quarters is $0.75 so three-fourths is 0.75 in decimal form.
Since \(\frac{3}{4}\) could not be written as a whole number or integer, it is a rational number.

Question 2.
\(\frac{8}{2}\)
________

Answer: 4

Explanation:
\(\frac{8}{2}\) = 4
Since 4 doesn’t have a decimal and is positive, it is a whole number. All whole numbers are also integers and rational numbers so 4 is a rational number, integer, and a whole number.

Question 3.
\(\frac{11}{3}\)
________

Answer: 3.66

Explanation:
Rewrite as a mixed number and then divide 2 by 3 to get the decimal part of the number.
\(\frac{11}{3}\) = 3 2/3 = 3.666…
Since 3.66.. has a decimal, it is not an integer or whole number. Therefore it is a rational number only.

Question 4.
\(\frac{5}{2}\)
________

Answer: 2.5

Explanation:
Write as a mixed number and then divide 1 by 2 to get the decimal part of the number a shortcut is to think of the fraction in terms of money. Half a dollar is $0.50 so one half equals 0.50 = 0.50
Since 2.5 has a decimal, it is not an integer or whole number. Therefore 2.5 is a rational number only.

Find each sum or difference.

Question 5.
−5 + 9.5
________

Answer: 4.5

Explanation:
Rewrite as subtraction and then subtract.
-5 + 9.5 = 4.5

Question 7.
−0.5 + (−8.5)
________

Answer: -9

Explanation:
Both numbers are negative so add their opposites and write the answers as negative.
−0.5 + (−8.5) = -(0.5 + 8.5) = -9

Question 8.
−3 − (−8)
________

Answer: 5

Explanation:
Rewrite as addition since subtracting a negative is the same as adding a positive.
−3 − (−8) = -3 + 8 = 5

Question 9.
5.6 − (−3.1)
________

Answer: 8.7

Explanation:
Rewrite as addition since subtracting a negative is the same as adding a positive.
5.6 − (−3.1) = 5.6 + 3.1 = 8.7

Find each product or quotient

Question 11.
−9 × (−5)
________

Answer: 45

Explanation:
Multiply two negative numbers to make a positive number.
−9 × (−5) = 45

Question 12.
0 × (−7)
________

Answer: 0

Explanation:
Any number multiplied by 0 will be zero.
So, the product is 0.

Question 13.
−8 × 8
________

Answer: -64

Explanation:
Multiply since there is only one negative the answer is negative.
-8 × 8 = -64

Question 14.
\(\frac{-56}{8}\)
________

Answer: -7

Explanation:
Divide since there is only one negative the answer is negative.
8 divides 56 seven times.
\(\frac{-56}{8}\) = -7

Question 15.
\(\frac{-130}{-5}\)

Answer: 26

Explanation:
Divide since there are two negative signs the answer is positive.
\(\frac{-130}{-5}\) = 26

Question 16.
\(\frac{34.5}{1.5}\)
________

Answer: 23

Explanation:
Divide since both the numbers are positive the answer will be positive.
\(\frac{34.5}{1.5}\) = 23
1.5 divides 34.5 23 times.
So, the quotient is 23.

Question 19.
Lei withdrew $50 from her bank account every day for a week. What was the change in her account in that week?
$ ________

Answer: -$350

Explanation:
Lei withdrew $50 from her bank account every day for a week.
Convert from week to days
1 week = 7 days
7 × -50 = -350
The change in her account is -$350.

Question 20.
Dan is cutting 4.75-foot lengths of twine from a 240-foot spool of twine. He needs to cut 42 lengths and says that 40.5 feet of twine will remain. Show that this is reasonable.
Type below:
__________

Answer:
The estimation of 4.75 is 5 and 42 is 40.
5 × 40 = 200
So he will be using about 200 feet.
He has 240 feet so he will have about 240-200 = 40 feet remaining.
Since 40 ≈ 40.5
The answer is reasonable.

Unit 1 Performance Tasks – Page No. 107

Question 1.
Armand is an urban planner, and he has proposed a site for a new town library. The site is between City Hall and the post office on Main Street.

The distance between City Hall and the post office is 612 miles. City Hall is 114 miles closer to the library site than it is to the post office.
a. Write 6 \(\frac{1}{2}\) miles and 1 \(\frac{1}{4}\) miles as decimals
6 \(\frac{1}{2}\) = __________
1 \(\frac{1}{4}\) = __________

Answer:
Write as decimal by dividing 1 by 2 and dividing 1 by 4 a shortcut is to think about ur in terms of money. Half of a dollar is $0.50 and a quarter is $0.25
So 1/2 = 0.50 = 0.5
1/4 = 0.25
6 1/2 = 6.5 and 1 1/4 = 1.25

Question 1.
b. Let d represent the distance from City Hall to the library site. Write an expression for the distance from the library site to the post office.
__________

Answer:
The library is closer to City Hall than the post office is so d is the difference between the distance from City Hall to the Post Office and the distance between City Hall and the Library Site.
d = 6 1/2 – 1 1/4

Question 1.
c. Write an equation that represents the following statement: The distance from City Hall to the library site plus the distance from the library site to the post office is equal to the distance from City Hall to the post office.
Type below:
__________

Answer:
The distance from the City Hall to the library is d, the distance from the library to the post office is 1 1/4 since the library is 1 1/4 miles closer to City Hall than the post office is, the distance from City Hall to the Post Office is 6 1/4
d + 1 1/4 = 6 1/4

Question 1.
d. Solve your equation from part c to determine the distance from City Hall to the library site, and the distance from the post office to the library site.
City Hall to library site: __________ miles
Library site to post office: __________ miles

Answer:
d = 6 1/2 – 1 1/4
d = 6 2/4 – 1 1/4
d = 5 1/4
Thus the distance is 5 1/4 miles.

Question 2.
Sumaya is reading a book with 288 pages. She has already read 90 pages. She plans to read 20 more pages each day until she finishes the book.
a. Sumaya writes the equation 378 = -20d to find the number of days she will need to finish the book. Identify the errors that Sumaya made.
Type below:
__________

Answer:
She made the mistake of using -20 in the equation instead of a positive 20. The negative can’t be used since she is not reading a negative number of pages per day.
She also made the mistake of adding 90 to 288 instead of subtracting.
Since she has already read 90 pages she has less than 288 pages left to read, not more.
288 – 90 = 198
The correct equation is 198 = 20d

Question 2.
b. Write and solve an equation to determine how many days Sumaya will need to finish the book. In your answer, count part of a day as a full day. Show that your answer is reasonable.
______ days

Answer:

198 = 20d is dividing both sides by 20 gives d = 198/20 = 9.9
Rounding this up gives 10 days.
This answer is reasonable since the book is about 300 pages and she has read about 100 pages of the book leaving about 200 pages left to read.
She is reading 20 pages per day and 20 × 10 = 200
So it would take 10 days to read about 200 pages.

Question 2.
c. Estimate how many days you would need to read a book about the same length as Sumaya’s book. What information did you use to find the estimate?
Type below:
__________

Answer:
Sumaya’s book is about 300 pages. Reading 20 pages a day would mean it would take about 300/20 = 15 days to read the book.

Unit 1 Performance Tasks – Page No. 108

Question 3.
Jackson works as a veterinary technician and earns $12.20 per hour.
a. Jackson normally works 40 hours a week. In a normal week, what is his total pay before taxes and other deductions?
$ ______

Answer: $488

Explanation:
Jackson works as a veterinary technician and earns $12.20 per hour.
Jackson normally works 40 hours a week.
40 × $12.20 = $488
Thus the total pay before taxes and other deductions is $488.

Question 3.
b. Last week, Jackson was ill and missed some work. His total pay before deductions was $372.10. Write and solve an equation to find the number of hours Jackson worked.
______ hours

Answer: 30.5 hours

Explanation:
Jackson works as a veterinary technician and earns $12.20 per hour.
His total pay before deductions was $372.10.
$12.20h = $372.10
h = 372.10/12.20
h = 30.5 hours

Question 3.
c. Jackson records his hours each day on a time sheet. Last week when he was ill, his time sheet was incomplete. How many hours are missing? Show your work. Then show that your answer is reasonable.

______ hours

Answer: 6.75 hours

Explanation:
8 + 7.25 + 8.5 = 23.75
30.5 – 23.75 = 6.75 hours

Question 3.
d. When Jackson works more than 40 hours in a week, he earns 1.5 times his normal hourly rate for each of the extra hours. Jackson worked 43 hours one week. What was his total pay before deductions? Justify your answer.
$ __________________

Answer: $542.90

Explanation:
When Jackson works more than 40 hours in a week, he earns 1.5 times his normal hourly rate for each of the extra hours.
Jackson worked 43 hours one week.
40 × 12.20 + 3 × 1.5 × 12.20 = $488 + $54.90 = $542.90

Question 3.
e. What is a reasonable range for Jackson’s expected yearly pay before deductions? Describe any assumptions you made in finding your answer.
$ __________________

Answer:

Assuming he works between 40 and 45 hours per week, his weekly pay range is between 40 × 12.20 = $488
40 × 12.20 + 5 × 1.5 × 12.20 = 488 + 91.50 = $579.50
Since there are 52 weeks in a year, his yearly pay is between 52 × 488 ≈ $25,000
and 52 × $579.50 ≈ $30,000.

Unit 1 MIXED REVIEW – Selected Response – Page No. 109

Question 1.
What is −6 \(\frac{9}{16}\) written as a decimal?
Options:
a. -6.625
b. -6.5625
c. -6.4375
d. -6.125

Answer: -6.5625

Explanation:
−6 \(\frac{9}{16}\)
Divide 9 by 16 to get 9/16 = 0.5625.
6 \(\frac{9}{16}\) = 6 + 0.5625 = 6.5625
−6 \(\frac{9}{16}\) = -6.5625
Thus the correct answer is option B.

Question 2.
Working together, 6 friends pick 14 \(\frac{2}{5}\) pounds of pecans at a pecan farm. They divide the pecans equally among themselves. How many pounds does each friend get?
Options:
a. 20 \(\frac{2}{5}\) pounds
b. 8 \(\frac{2}{5}\) pounds
c. 2 \(\frac{3}{5}\) pounds
d. 2 \(\frac{2}{5}\) pounds

Answer: \(\frac{2}{5}\) pounds

Explanation:
Divide the number of pounds by the number of friends to get the number of pounds each friend gets.
14 \(\frac{2}{5}\)/6 = 14.4/6 = 2.4 pounds.
2.4 = 2 \(\frac{2}{5}\) pounds
Thus the correct answer is option D.

Question 3.
What is the value of (−3.25)(−1.56)?
Options:
a. -5.85
b. -5.07
c. 5.07
d. 5.85

Answer: 5.07

Explanation:
Multiply two negatives make a positive.
So the answer is positive.
(−3.25)(−1.56) = 5.07
The answer is option C.

Question 4.
Mrs. Rodriguez is going to use 6 \(\frac{1}{3}\) yards of material to make two dresses. The larger dress requires 3 \(\frac{2}{3}\) yards of material. How much material will Mrs. Rodriguez have left to use on the smaller dress?
Options:
a. 1 \(\frac{2}{3}\) yards
b. 2 \(\frac{1}{3}\) yards
c. 2 \(\frac{2}{3}\) yards
d. 3 \(\frac{1}{3}\) yards

Answer: 2 \(\frac{2}{3}\) yards

Explanation:
Subtract the yards of material for the larger dress from the total yards of material.
6 \(\frac{1}{3}\) yards – 3 \(\frac{2}{3}\) yards = 2 \(\frac{2}{3}\) yards
Thus the correct answer is option C.

Question 5.
Jaime had $37 in his bank account on Sunday. The table shows his account activity for the next four days. What was the balance in Jaime’s account after his deposit on Thursday?

Options:
a. $57.49
b. $59.65
c. $94.49
d. $138.93

Answer: $94.49

Explanation:
Add up all the deposits and withdrawals to his original balance make sure deposits are represented by positive numbers and withdrawals are represented by negative numbers.
37 + 17.42 – 12.60 – 9.62 + 62.29 = 94.49
Thus the correct answer is option C.

Question 6.
A used motorcycle is on sale for $3,600. Erik makes an offer equal to \(\frac{3}{4}\) of this price. How much does Erik offer for the motorcycle?
Options:
a. $4800
b. $2700
c. $2400
d. $900

Answer: $2700

Explanation:
Given that,
A used motorcycle is on sale for $3,600. Erik makes an offer equal to \(\frac{3}{4}\) of this price.
\(\frac{3}{4}\) × 3600 = 2700
Thus the correct answer is option B.

Question 7.
Ruby ate \(\frac{1}{3}\) of a pizza, and Angie ate \(\frac{1}{5}\) of the pizza. How much of the pizza did they eat in all?
Options:
a. 1 \(\frac{1}{5}\) of the pizza
b. \(\frac{1}{8}\) of the pizza
c. \(\frac{3}{8}\) of the pizza
d. \(\frac{8}{15}\) of the pizza

Answer: \(\frac{8}{15}\) of the pizza

Explanation:
Ruby ate \(\frac{1}{3}\) of a pizza, and Angie ate \(\frac{1}{5}\) of the pizza.
\(\frac{1}{3}\) = \(\frac{1}{5}\) = \(\frac{5}{15}\) + \(\frac{3}{15}\) = \(\frac{8}{15}\)
Thus the correct answer is option D.

Unit 1 MIXED REVIEW – Page No. 110

Question 8.
Winslow buys 1.2 pounds of bananas. The bananas cost $1.29 per pound. To the nearest cent, how much does Winslow pay for the bananas?
Options:
a. $1.08
b. $1.20
c. $1.55
d. $2.49

Answer: $1.55

Explanation:
Winslow buys 1.2 pounds of bananas. The bananas cost $1.29 per pound.
1.2 × $1.29 = $1.548 ≈ $1.55
Thus the correct answer is option C.

Question 9.
The temperature was -10 °F and dropped by 16 °F. Which statement represents the resulting temperature in degrees Fahrenheit?
Options:
a. -10 – (-16) = -6
b. -10 – 16 = -26
c. 10 – (-16) = 26
d. -10 + 16 = 6

Answer: -10 – 16 = -26

Explanation:
The temperature was -10 °F and dropped by 16 °F.
-10 + (-16) = -26°F.
So, the correct answer is option B.

Question 10.
A scuba diver at a depth of -12 ft (12 ft below sea level), dives down to a coral reef that is 3.5 times the diver’s original depth. What is the diver’s new depth?
Options:
a. -420 ft
b. -42 ft
c. 42 ft
d. about 3.4 ft

Answer: -42 ft

A scuba diver at a depth of -12 ft, dives down to a coral reef that is 3.5 times the diver’s original depth.
-12 × 3.5 = -42 ft
So, the correct answer is option B.

Question 11.
The school Spirit Club spent $320.82 on food and took in 643.59 selling the food. How much did the Spirit Club make?
Options:
a. -$322.77
b. -$964.41
c. $322.77
d. $964.41

Answer: $322.77

Explanation:
The school Spirit Club spent $320.82 on food and took in 643.59 selling the food.
$643.59 – $320.82 = $322.77
So, the answer is option C.

Question 12.
Lila graphed the points -2 and 2 on a number line. What does the distance between these two points represent?
Options:
a. the sum of -2 and 2
b. the difference of 2 and -2
c. the difference of -2 and 2
d. the product of -2 and 2

Answer: the difference of 2 and -2

Explanation:
Distance is found by subtracting the larger number and the smaller number so it is the difference of 2 and -2.
Thus the correct answer is option B.

Question 13.
What is a reasonable estimate of −3 \(\frac{4}{5}\) + (−5.25) and the actual value?
Options:
a. -4 + (-5) = -9; −9 \(\frac{1}{20}\)
b. -3 + (-5) = -8; −8 \(\frac{1}{20}\)
c. -4 + (-5) = -1; −8 \(\frac{9}{20}\)
d. -3 + (-5) = 8; 8 \(\frac{1}{20}\)

Answer: -4 + (-5) = -9; −9 \(\frac{1}{20}\)

Explanation:
−3 \(\frac{4}{5}\) + (−5.25)
−3 \(\frac{4}{5}\) ≈ -4
−5.25 ≈ -5
So the sum is about -4 + -5 = -9.
The estimated answer is -9.

Mini-Task

Question 14.
Juanita is watering her lawn using the water stored in her rainwater tank. The water level in the tank drops \(\frac{1}{3}\) inch every 10 minutes she waters.
a. What is the change in the tank’s water level after 1 hour?
______ inches

Answer: -2 inches

Explanation:
Juanita is watering her lawn using the water stored in her rainwater tank.
There are six 10 minute intervals in 1 hour so change is
6 × –\(\frac{1}{3}\) = -2 inches
Therefore, the tank’s water level after 1 hour is -2 inches.

Question 14.
b. What is the expected change in the tank’s water level after 2.25 hours?
______ inches

Answer: -4.5 inches

Explanation:
Since the water level drops 2 inches every hour, in 2.25 hours the water level change will be -2 × 2.25 = -4.5 inches
Thus the expected change in the tank’s water level after 2.25 hours is -4.5 inches.

Question 14.
c. If the tank’s water level is 4 feet, how many days can Juanita water if she waters for 15 minutes each day?
______ days

Answer: 96 days

Explanation:
15 minutes is 1/4 of an hour so in 15 minutes the water level will have dropped by 2 × 1/4 = 1/2 inches.
Since the water level is initially 4 feet = 48 inches
She can water for 48/1/2 = 48 × 2 = 96 days
It takes 96 days if she waters for 15 minutes.

Conclusion:

Hope the info prevailed in this article is beneficial for all the students. Keep in touch with us to get the latest updates regarding Go Math Grade 7 Answer Key Chapter 3 Rational Numbers. Also, the students of 4th grade can get the solutions for all the chapters on Go Math Grade 7 Answer Key page.

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Go Math Grade 7 Answer Key Chapter 2 Multiplying and Dividing Integers

Go through the topics covered in Multiplying and Dividing Integers before you start your preparation. Just tap the link and start preparing now. In addition to the exercise problems, you can also get a detailed explanation for all the questions in Module Review. So, first, practice the exercise problems and then try to solve the questions given in the module review. With the help of this Go Math 7th Grade Answer Key Chapter 2 Multiplying and Dividing Integers, you can improve the knowledge in the following topics.

Chapter 2 – Multiplying Integers

• Multiplying Integers – Guided Practice – Page No. 40
• Multiplying Integers – Independent Practice – Page No. 41
• Multiplying Integers – Page No. 42

Chapter 2 – Dividing Integers

• Dividing Integers – Guided Practice – Page No. 46
• Dividing Integers – Independent Practice – Page No. 47
• Dividing Integers – Page No. 48

Chapter 2- Applying Integer Operations

• Applying Integer Operations – Guided Practice – Page No. 52
• Applying Integer Operations – Independent Practice – Page No. 53
• Applying Integer Operations – Page No. 54

Chapter 2 – Module Review

• MODULE QUIZ – 2.1 Multiplying Integers – Page No. 55
• MIXED REVIEW – Selected Response – Page No. 56
• Module 2 Review – Multiplying and Dividing Integers – Page No. 104

Multiplying Integers – Guided Practice – Page No. 40

Find each product.

Question 1.
-1(9)
_______

Answer: -9

Explanation:
We have to find the product:
-1(9)
1. Determine the sign of the product
-1 < 0 and 9 > 0
Since the numbers have opposite signs, the product will be negative.
2. We find the absolute value of the numbers and multiply them:
|-1| = 1
|9| = 9
1 × 9 =9
3. We assign the correct sign to the product:
-1 × 9 = -9

Question 2.
14(-2)
_______

Answer: -28

Explanation:
We have to find the product:
14(-2)
1. Determine the sign of the product
-2 < 0 and 14 > 0
Since the numbers have opposite signs, the product will be negative.
2. We find the absolute value of the numbers and multiply them:
|-2| = 2
|14| = 14
2 × 14 = 28
3. We assign the correct sign to the product:
-2 × 14 = -28

Question 5.
(-4)(15)
_______

Answer: -60

Explanation:
We have to find the product:
(-4)(15)
1. Determine the sign of the product
-4 < 0 and 15 > 0
Since the numbers have opposite signs, the product will be negative.
2. We find the absolute value of the numbers and multiply them:
|-4| = 4
|15| = 15
3. We assign the correct sign to the product:
-4 × 15 = -60

Question 6.
-18(0)
_______

Answer: 0

Explanation:
We have to find the product:
-18(0)
Since one of the factors is zero, the product is zero.
-18 × 0 = 0

Question 7.
(-7)(-7)
_______

Answer: 49

Explanation:
We have to find the product:
(-7)(-7)
1. Determine the sign of the product
-7 < 0
Since the numbers have the same signs, the product will be positive.
2. We find the absolute value of the numbers and multiply them:
|-7| = 7
|-7| = 7
3. We assign the correct sign to the product:
-7 × -7 = 49

Question 8.
-15(9)
_______

Answer: -125

Explanation:
We have to find the product:
-15(9)
1. Determine the sign of the product
-15 < 0  and 9 > 0
Since the numbers have opposite signs, the product will be negative.
2. We find the absolute value of the numbers and multiply them:
|-15| = 15
|9| = 9
3. We assign the correct sign to the product:
-15 × 9 = -125

Question 9.
(8)(-12)
_______

Answer: -96

Explanation:
We have to find the product:
(8)(-12)
1. Determine the sign of the product
8 > 0 and -12 < 0
Since the numbers have opposite signs, the product will be negative.
2. We find the absolute value of the numbers and multiply them:
|8| = 8
|-15| = 15
3. We assign the correct sign to the product:
8 × -15 = -96

Question 10.
-3(-100)
_______

Answer: 300

Explanation:
We have to find the product:
-3 < 0 and -100 < 0
Since the numbers have the same signs, the product will be positive.
2. We find the absolute value of the numbers and multiply them:
|-3| = 3
|-100| = 100
3. We assign the correct sign to the product:
-3 × -100 = 300

Question 11.
0(-153)
_______

Answer: 0

Explanation:
We have to find the product:
0(-153)
Since one of the factors is zero, the product is zero.
0 × -153 = 0

Question 12.
-6(32)
_______

Answer: -192

Explanation:
We have to find the product:
-6(32)
1. Determine the sign of the product
-6 < 0  and 32 > 0
Since the numbers have opposite signs, the product will be negative.
2. We find the absolute value of the numbers and multiply them:
|-6| = 6
|32| = 32
3. We assign the correct sign to the product:
-6 × 32 = -192

Question 13.
Flora made 7 withdrawals of $75 each from her bank account. What was the overall change in her account?
$ _______

Answer: -525

Explanation:
Flora made 7 withdrawals of $75 each from her bank account.
Since she withdrew 7 amounts of money, the overall change in the account will be:
(-7)(75)
-7 < 0 and 75 > 0
The numbers have opposite signs, so the product will be negative.
7 × 75 = 525
We assign the correct sign to the product:
-7 × 75 = -525

Question 17.
A mountain climber climbed down a cliff 50 feet at a time. He did this 5 times in one day. What was the overall change in his elevation?
_______ feet

Answer: -250 feet

Explanation:
A mountain climber climbed down a cliff 50 feet at a time. He did this 5 times in one day.
5(-50)
5(-50) = (-50) + (-50) + (-50) + (-50) + (-50)
To graph 5(-50) we start at 0 and move 50 units to the left 5 times:
5 × -50 = -250 feet
Thus the overall change in his elevation is -250 feet.

Question 18.
Explain the process for finding the product of two integers.
Type below:
_____________

Answer: Determine the sign of the product, multiply the absolute value of the numbers then attach the sign to the product. If one of the numbers is zero then the product is zero.

Multiplying Integers – Independent Practice – Page No. 41

Question 19.
Critique Reasoning Lisa used a number line to model –2(3). Does her number line make sense? Explain why or why not.

_______

Answer:
Lisa used the number line incorrectly since in her number line she multiplied -3 twice.
The number line should show 2 movements to the left three times.

Question 20.
Represent Real-World Problems Mike got on an elevator and went down 3 floors. He meant to go to a lower level, so he stayed on the elevator and went down 3 more floors. How many floors did Mike go down altogether?
_______ floors

Answer: 6 floors

Explanation:
Mike got on an elevator and went down 3 floors. He meant to go to a lower level, so he stayed on the elevator and went down 3 more floors.
2(-3) = -6
This means he went down 6 floors.

Solve. Show your work.

Question 21.
When Brooke buys lunch at the cafeteria, money is withdrawn from a lunch account. The table shows amounts withdrawn in one week. By how much did the amount in Brooke’s lunch account change by the end of that week?

$ _______ decrease

Answer: $-20 decrease

Explanation:
He withdrew $4 each day, which means 5 times $4 each time. By the end of the week, his account will mark a decrease of: -$20

Question 24.
Casey uses some of his savings on batting practice. The cost of renting a batting cage for 1 hour is $6. He rents a cage for 9 hours in each of two months. What is the change in Casey’s savings after two months?
$ _______

Answer: $108

Explanation:
Given,
Casey uses some of his savings on batting practice.
The cost of renting a batting cage for 1 hour is $6.
He rents a cage for 9 hours in each of two months.
9(-6) = -54
For 2 months = -54 × 2 = -108

Multiplying Integers – Page No. 42

Question 26.
Communicate Mathematical Ideas Describe a real-world situation that can be represented by the product 8(–20). Then find the product and explain what the product means in terms of the real-world situation.
_______ points

Answer: -160

Example:
Irene has some savings in an account. Each of 8 months she uses $20 to pay a rate for a bicycle. What is the change in her account after the 8 months?
8(-20) = -160
The change in the account consists of 8 times $20:
Her account diminished by $160.

Question 27.
What If? The rules for multiplying two integers can be extended to a product of 3 or more integers. Find the following products by using the Associative Property to multiply 2 numbers at a time.
a. 3(3)(–3)
_______

Answer: -27

Explanation:
3 × 3 × (-3) = 9(-3) = -27

Question 27.
b. 3(–3)(–3)
_______

Answer: 27
3((-3)(-3))
3 × 9 = 27

Question 27.
c. –3(–3)(–3)
_______

Answer: -27
(-3)(–3)(–3)
(-3)(9)
9 × -3 = -27

Question 27.
d. 3(3)(3)(–3)
_______

Answer: -81
(3 × 3)(3 × -3)
9 × -9 = -81

Question 27.
e. 3(3)(–3)(–3)
_______

Answer: 81
(3 × 3)(-3 × -3)
9 × 9 = 81

Question 27.
f. 3(–3)(–3)(–3)
_______

Answer: -81
3(-3)(-3 × -3)
-9 × 9 = -81

Question 27.
g. Make a Conjecture Based on your results, complete the following statements:
When a product of integers has an odd number of negative factors, then the sign of the product is ____________.
____________

Answer: Negative

Explanation:
When a product of integers has an odd number of negative factors, then the sign of the product is negative.

Question 27.
When a product of integers has an even number of negative factors, then the sign of the product is ___________ .
____________

Answer: Positive

Explanation:
When a product of integers has an even number of negative factors, then the sign of the product is positive.

FOCUS ON HIGHER ORDER THINKING

Question 28.
Multiple Representations The product of three integers is –3. Determine all of the possible values for the three factors.
Type below:
_______________

Answer:
We are given the product:
a . b . c = -3
a, b, c integers
a = -1, b = 1, c = 3
a = -3, b = c = 1
a = -1, b = -1, c = -3
The elements of the product can be 1, -1, 3, -3. Since the result is negative, the number of negative factors is odd. Thus we can have either one negative number or three. The possibilities are:
-1, 1, 3
-3, 1 , 1
-1, -1, -3

Question 29.
Analyze Relationships When is the product of two nonzero integers less than or equal to both of the two factors?
Type below:
_______________

Answer:
Let a, b be the two integers.
We are given the data:
a.b ≤ a
a.b ≤ b
a > 0, b> 0
Case 1: both are positive numbers
a = 1, b = 1
= a.b = 1.1 = a = b
The product is greater than any of the two numbers except for the case in which both numbers are 1:
Case 2: both numbers are negative.
a < 0, b< 0
a.b > 0
a . b > a
a . b > b
The product is positive, thus it cannot be smaller than any of the numbers.
Case 3: the numbers have different signs
a . b < 0
a. b < a
a . b ≤ b
The product is negative, therefore smaller than the positive number. The product is also smaller than the negative number as it represents a times the number b:
a = b = 1
a > 0, b < 0

Dividing Integers – Guided Practice – Page No. 46

Find each quotient.

Question 1.
\(\frac{-14}{2}\) = _______

Answer: -7

Explanation:
We have to find the quotient:
-14 < 0
2 > 0
Since the numbers have opposite signs, the quotient will be negative.
-14/2 = -7

Question 2.
21 ÷ (−3) = _______

Answer: -7

Explanation:
We have to find the quotient:
21 ÷ (−3)
21 > 0
-3 < 0
Since the numbers have opposite signs, the quotient will be negative.
21/(-3) = -7

Question 4.
0 ÷ (−4) = _______

Answer: 0

Explanation:
We have to find the quotient:
0 ÷ (−4)
If one of the numbers is zero answers will be zero.
0 ÷ (−4) = 0

Question 5.
\(\frac{-45}{-5}\) = _______

Answer: 9

Explanation:
We have to find the quotient:
\(\frac{-45}{-5}\)
Since the numbers have the same sign, the quotient will be positive.
\(\frac{-45}{-5}\) = 9

Question 7.
\(\frac{-11}{-1}\) = _______

Answer: 11

Explanation:
We have to find the quotient:
\(\frac{-11}{-1}\)
Since the numbers have the same sign, the quotient will be positive.
\(\frac{-11}{-1}\) = 11

Question 8.
-31 ÷ (-31) = _______

Answer: 1

Explanation:
We have to find the quotient:
-31 ÷ (-31)
Since the numbers have the same sign, the quotient will be positive.
-31 ÷ (-31) = 1

Question 9.
\(\frac{0}{-7}\) = _______

Answer: 0

Explanation:
We have to find the quotient:
\(\frac{0}{-7}\)
If one of the numbers is zero answers will be zero.
0/-7 = 0

Question 11.
84 ÷ (-7) = _______

Answer: -12

Explanation:
We have to find the quotient:
84 ÷ (-7)
Since the numbers have opposite signs, the quotient will be negative.
84 ÷ (-7)
7 divides 84 twelve times.
So, the answer is -12.

Question 12.
\(\frac{500}{-25}\) = _______

Answer: -20

Explanation:
We have to find the quotient:
\(\frac{500}{-25}\)
Since the numbers have opposite signs, the quotient will be negative.
\(\frac{500}{-25}\) = -20

Question 13.
-6 ÷ 0 =
__________

Answer: undefined

Explanation:
We have to find the quotient:
-6 ÷ 0
Any number divided by 0 is undefined.

Write a division expression for each problem. Then find the value of the expression.

Question 15.
Clark made four of his truck payments late and was fined four late fees. The total change to his savings from late fees was -$40. How much was one late fee?
$ _______

Answer: 10

Explanation:
Clark made four of his truck payments late and was fined four late fees.
The total change to his savings from late fees was -$40.
We determine one late fee by dividing the total change in his savings by the number of late fees|:
-10 ÷ 4 = -10
One late fee was $10.

Question 16.
Jan received -22 points on her exam. She got 11 questions wrong out of 50 questions. How much was Jan penalized for each wrong answer?
_______ points

Answer: 2 points

Explanation:
Jan received -22 points on her exam. She got 11 questions wrong out of 50 questions.
We determine the number of points on the exam to the number of wrong questions:
-22 ÷ 11 = -2
Thus a wrong answer was penalized by 2 points.

Question 17.
Allen’s score in a video game was changed by -75 points because he missed some targets. He got -15 points for each missed target. How many targets did he miss?
_______ targets

Answer: 5 targets

Explanation:
Allen’s score in a video game was changed by -75 points because he missed some targets.
He got -15 points for each missed target.
We divide the change in the score by the number of points for a missed target:
-75 ÷ -15 = 5
Thus he missed 5 targets.

ESSENTIAL QUESTION CHECK-IN

Question 19.
How is the process of dividing integers similar to the process of multiplying integers?
Type below:
____________

Answer: The process of dividing integers is similar to the process of multiplying integers by the sign of the result which is positive in case both numbers have the same sign and negative when they have different signs.

Dividing Integers – Independent Practice – Page No. 47

Question 20.
Walter buys a bus pass for $30. Every time he rides the bus, money is deducted from the value of the pass. He rode 12 times and $24 was deducted from the value of the pass. How much does each bus ride cost?
$ _______

Answer: 2

Explanation:
We divide the total amount deducted from the value of the pass by the number of times he rode the bus:
-24 ÷ 12 = -2
The price of a bus ride is $2.

Question 22.
Multistep At 7 p.m. last night, the temperature was 10 °F. At 7 a.m. the next morning, the temperature was -2 °F.
a. By how much did the temperature change from 7 p.m. to 7 a.m.?
_______ degrees

Answer: 12 degrees

Explanation:
We are given the data:
7 p.m: 10°F.
7 a.m: -2 °F.
We determine by how much the temperature changed from 7 p.m to 7 a.m by subtraction the initial temperature from the final temperature:
-2 – 10 = -12
Thus the temperature decreased by 12 degrees.

Question 22.
b. The temperature changed by a steady amount overnight. By how much did it change each hour?
_______ degrees each hour

Answer: 1

Explanation:
We divide the total change of temperature by the number of hours to determine by how much the temperature changed each hour:
-12 ÷ 12 = -1
The temperature decreased by 1°F each hour.

Question 23.
Analyze Relationships Nola hiked down a trail at a steady rate for 10 minutes. Her change in elevation was -200 feet. Then she continued to hike down for another 20 minutes at a different rate. Her change in elevation for this part of the hike was -300 feet. During which portion of the hike did she walk down at a faster rate? Explain your reasoning.
___________ was faster

Answer:
First trail: -200 feet in 10 minutes
Second trail -300 feet in 20 minutes
we determine the rate she walked down on the first trail by dividing the elevation by the time she walked on that trail:
-200 ÷ 10 = -20
The rate was 20 feet/minute.
we determine the rate she walked down on the second trail by dividing the elevation by the time she walked on that trail:
-300 ÷ 20 = -15
The rate was 15 feet/minute.

Dividing Integers – Page No. 48

Question 25.
Communicate Mathematical Ideas Two integers, a and b, have different signs. The absolute value of integer a is divisible by the absolute value of integer b. Find two integers that fit this description. Then decide if the product of the integers is greater than or less than the quotient of the integers. Show your work.
product ___________ quotient

Answer:
Let’s consider two positive numbers to represent |a| and |b|
Case 1: a = 12, b = -4
a . b = 12 . (-4) = -48
a ÷ b = 12 ÷ (-4) = 3
or
Case 2: a = -12, b = 4
a . b = -12 . 4 = -48
a ÷ b = -12 ÷ 4 = -3
In both cases the product is smaller than the quotient and this happens because one number is positive and the other negative and because |a| is divisible by |b|
a . b ≤ a ÷ b

Determine if each statement is true or false. Justify your answer.

Question 26.
For any two nonzero integers, the product and quotient have the same sign.
___________

Answer: True
The statement is true because both division and multiplication operate in the same way about signs, the difference being that the absolute values are either multiplied or divided which doesn’t make any difference regarding the signs.

Question 27.
Any nonzero integer divided by 0 equals 0.
___________

Answer: False
The statement is false because the divisor cannot be zero, division is undefined in this case no matter the dividend.

FOCUS ON HIGHER ORDER THINKING

Question 28.
Multi-step A perfect score on a test with 25 questions is 100. Each question is worth the same number of points.
a. How many points is each question on the test worth?
_______ points

Answer: 4

Explanation:
We determine the number of points each question worth by dividing the perfect score to the number of questions:
100 ÷ 25 = 4

Question 28.
b. Fred got a score of 84 on the test. Write a division sentence using negative numbers where the quotient represents the number of questions Fred answered incorrectly.
_______ questions

Answer: 4

Explanation:
The number of questions Fred answered incorrectly is:
(84 – 100) ÷ (-4) = -16 ÷ -4 = 4
The number of questions Fred answered incorrectly is 4.

Question 30.
Justify Reasoning The quotient of two negative integers results in an integer. How does the value of the quotient compare to the value of the original two integers? Explain.
Type below:
___________

Answer: Since the quotient of two negative numbers is positive, it will always be greater than the original two integers.

Applying Integer Operations – Guided Practice – Page No. 52

Evaluate each expression.

Question 1.
−6(−5) + 12 =
_______

Answer: 42

Explanation:
We are given the expression:
−6(−5) + 12
First, multiply -6 and -5
−6(−5) + 12 = 30 + 12 = 42

Question 6.
−3(−5) − 16 =
_______

Answer: -1

Explanation:
We are given the expression:
−3(−5) − 16 = 15 – 16 = -1

Write an expression to represent the situation. Evaluate the expression and answer the question.

Question 7.
Bella pays 7 payments of $5 each to a game store. She returns one game and receives 20 dollars back. What is the change to the amount of money she has?
$ _______

Answer: 15 less

Explanation:
Given that,
Bella pays 7 payments of $5 each to a game store. She returns one game and receives 20 dollars back.
7(-5) + 20 = -35 + 20 = -15
Thus she will have $15 less.

Question 8.
Ron lost 10 points seven times playing a video game. He then lost an additional 100 points for going over the time limit. What was the total change in his score?
_______ points

Answer: 170 points

Explanation:
We use negative numbers for the number of points he losses.
7(-10) + (-100) = -70 – 100 = -170
Thus he will have 170 points less.

Question 9.
Ned took a test with 25 questions. He lost 4 points for each of the 6 questions he got wrong and earned an additional 10 points for answering a bonus question correctly. How many points did Ned receive or lose overall?
_______ points

Answer: He lost 14 points

Explanation:
Given,
Ned took a test with 25 questions. He lost 4 points for each of the 6 questions he got wrong and earned an additional 10 points for answering a bonus question correctly.
6(-4) + 10 = -24 + 10 = -14
Since he lost the same number of points for each of the 6 questions he answered incorrectly, we use multiplication to determine the number of points he lost, then we add the number of points he received as a bonus.
Thus he lost 14 points.

Question 10.
Mr. Harris has some money in his wallet. He pays the babysitter $12 an hour for 4 hours of babysitting. His wife gives him 10, and he puts the money in his wallet. By how much does the amount in his wallet change?
$ _______

Answer: $38 less

Explanation:
Given,
Mr. Harris has some money in his wallet. He pays the babysitter $12 an hour for 4 hours of babysitting.
His wife gives him 10, and he puts the money in his wallet.
Since she paid 4 times the amount of $12, we use multiplication to determine the money he spent paying the babysitter, then we add the money received from his wife.
The change to the amount of money he has is:
4(12) + 10 = -48 + 10 = -38
Thus he will have $38 less.

Compare the values of the two expressions using <, =, or >.

Question 11.
-3(-2) + 3 _______ 3(-4) + 9

Answer: -3(-2) + 3 > 3(-4) + 9

Explanation:
-3(-2) + 3 = 6 + 3 = 9
3(-4) + 9 = -12 + 9 = -3
9 is greater than -3
So, -3(-2) + 3 > 3(-4) + 9

Question 14.
-16(0) – 3 _______ -8(-2) – 3

Answer: -16(0) – 3 < -8(-2) – 3

Explanation:
-16(0) – 3 = 0 – 3 = -3
-8(-2) – 3 = 16 – 3 = 13
-3 is less than 13.
Thus -16(0) – 3 < -8(-2) – 3

Essential Question Check-In

Question 15.
When you solve a problem involving money, what can a negative answer represent?
Type below:
___________

Answer:
A negative answer to a problem involving money can represent:

• an amount of money spent on something
• a stolen amount of money
• a lent amount of money
• a donated amount of money
• an amount of money given for fines, fees

Applying Integer Operations – Independent Practice – Page No. 53

Evaluate each expression.

Question 16.
−12(−3) + 7
_______

Answer: 43

Explanation:
We are given the expression:
−12(−3) + 7
We perform multiplication first, then addition:
-12(-3) + 7 = 36 + 7 = 43

Question 17.
(−42) ÷ (−6) + 5 − 8
_______

Answer: 4

Explanation:
We are given the expression:
(−42) ÷ (−6) + 5 − 8
((−42) ÷ (−6)) + 5 − 8 = 7 + 5 – 8
12 – 8 = 4

Question 20.
35 ÷ (−7) + 6
_______

Answer: 1

Explanation:
We are given the expression:
35 ÷ (−7) + 6
We perform division first, then addition:
35 ÷ (−7) + 6 = -5 + 6 = 1
35 ÷ (−7) + 6 = 1

Question 21.
−13(−2) − 16 − 8
_______

Answer: 2

Explanation:
We are given the expression:
−13(−2) − 16 − 8
We perform multiplication first, then subtraction:
26 – 16 – 8
10 – 8 = 2
−13(−2) − 16 − 8 = 2

Question 22.
Multistep
Lily and Rose are playing a game. In the game, each player starts with 0 points and the player with the most points at the end wins. Lily gains 5 points two times, loses 12 points, and then gains 3 points. Rose loses 3 points two times, loses 1 point, gains 6 points, and then gains 7 points.
a. Write and evaluate an expression to find Lily’s score
_______ point(s)

Answer: 1 point

Explanation:
We write and evaluate an expression to find Lily’s score:
2(5) – 12 + 3 = 10 – 12 + 3 = -2 + 3 = 1

Question 22.
b. Write and evaluate an expression to find Rose’s score.
_______ point(s)

Answer: 6 points

Explanation:
We write and evaluate an expression to find Rose’s score:
2(-3) – 1 + 6 + 7 = -6 – 1 + 6 + 7 = -7 + 6 + 7
= 0 + 6 = 6

Question 22.
c. Who won the game?
___________

Answer: Rose

Explanation:
6 > 1
So, Rose won the game because her score is greater than Lily’s score.

Write an expression from the description. Then evaluate the expression.

Question 25.
Multistep
Arleen has a gift card for a local lawn and garden store. She uses the gift card to rent a tiller for 4 days. It costs 35 dollars per day to rent the tiller. She also buys a rake for $9.
a. Find the change to the value on her gift card.
$ _______

Answer: -149

Explanation:
We determine the change to the value of her gift card:
4(-35) + (-9) = -140 – 9 = – 149

Question 25.
b. The original amount on the gift card was $200. Does Arleen have enough left on the card to buy a wheelbarrow for $50? Explain.
________________

Answer: yes

Explanation:
We determine the amount of money she has left on the gift card after renting the tiler and buying the rake.
200 – 149 = 51
Since she has got $51 on the gift card and a wheelbarrow is $50 she is able to buy it.
51 > 50

Applying Integer Operations – Page No. 54

Question 26.
Carlos made up a game where, in a deck of cards, the red cards (hearts and diamonds) are negative and the black cards (spades and clubs) are positive. All face cards are worth 10 points, and number cards are worth their value.
a. Samantha has a king of hearts, a jack of diamonds, and a 3 of spades. Write an expression to find the value of her cards.
_______

Answer: -17

Explanation:
We use negative values for hearts and diamonds and positive values for spades and clubs
1(-10) + 1(-10) + 1(3)
-10 – 10 + 3 = -17

Question 26.
b. Warren has a 7 of clubs, a 2 of spades, and a 7 of hearts. Write an expression to find the value of his cards.
_______

Answer: 2

Explanation:
We use negative values for hearts and diamonds and positive values for spades and clubs
1(7) + 1(2) + 1(-7)
7 + 2 – 7 = 0 + 2 = 2

Question 26.
c. If the greater score wins, who won?
___________

Answer: Warren
2 > -17

Question 26.
d. If a player always gets three cards, describe two different ways to receive a score of 7.
Type below:
___________

Answer:
10 – 2 – 1 = 7 (a queen of spades, a 2 of hearts and an ace of diamonds)
1 + 2 + 4 = 7 (an ace of clubs, a 2 of spades and a 4 of clubs)
-10 + 10 + 7 = 7 (a king of diamonds, a jack of spades and a 7 of clubs)

H.O.T.

Focus On Higher Order Thinking

Question 27.
Represent Real-World Problems
Write a problem that the expression 3(-7) – 10 + 25 = -6 could represent.
Type below:
___________

Answer: -6

Explanation:
We are given the expression:
3(-7) – 10 + 25 = -6
Example:
Adrian has some savings from which he buys 3 books $7 each and a video game for which he pays $10. His sister gives him $5. What is the total change in his savings?
3(-7) – 10 + 25 = -21 – 10 + 25 = -31 + 25 = -6

Question 29.
Persevere in Problem-Solving
Lisa is standing on a dock beside a lake. She drops a rock from her hand into the lake. After the rock hits the surface of the lake, the rock’s distance from the lake’s surface changes at a rate of -5 inches per second. If Lisa holds her hand 5 feet above the lake’s surface, how far from Lisa’s hand is the rock 4 seconds after it hits the surface?
________ inches

Answer: 80 inches

Explanation:
We use negative values for the distances the rock gets into the water and the distance from Lisa’s hand until the water’s surface as both go down.
Convert from feet to inches.
1 feet = 12 inches
5 feet = 5 × 12 = 60 inches
After 4 seconds the distance from Lisa’s hand will be given by the sum of the distance from Lisa’s hand to the water’s surface and the distance traveled by the rock below the water’s surface.
4(-5) + (-60) = -20 – 60 = -80
Thus the rock will be 80 inches from Lisa’s hand.

MODULE QUIZ – 2.1 Multiplying Integers – Page No. 55

Find each product.

Question 1.
(−2)(3)
______

Answer: -6

Explanation:
We have to determine the product
(−2)(3)
The numbers have different signs, thus the result will be negative.
We multiply the absolute values of the numbers and assign the negative sign.
-2 × 3 = -6

Question 4.
(−3)2(−2)
______

Answer: 12

Explanation:
We have to determine the product
(−3)2(−2)
The numbers have the same signs, thus the result will be positive.
We multiply the absolute values of the numbers and assign a positive sign.
(−3)2(−2) = -6 × -2 = 12

2.2 Dividing Integers

Find each quotient.

Question 7.
\(\frac{-63}{7}\)
______

Answer: -9

Explanation:
We have to determine the quotient:
\(\frac{-63}{7}\)
The numbers have different signs, thus the result will be negative.
We divide the absolute values of the numbers and assign the negative sign.
\(\frac{-63}{7}\) = -9
Thus the quotient of \(\frac{-63}{7}\) is -9

Question 8.
0 ÷ (−15)
______

Answer: 0

Explanation:
We have to determine the quotient:
0 ÷ (−15)
If one of the numbers is zero then the quotient will be 0.
0 ÷ (−15) = 0

Question 9.
96 ÷ (−12)
______

Answer: -8

Explanation:
We have to determine the quotient:
96 ÷ (−12)
The numbers have different signs, thus the result will be negative.
We divide the absolute values of the numbers and assign the negative sign.
12 divides 96 eight times
So, 96 ÷ (−12) = -8

Question 10.
An elephant at the zoo lost 24 pounds over 6 months. The elephant lost the same amount of weight each month. Write an integer that represents the change in the elephant’s weight each month.
______ pounds

Answer: – 4 pounds

Explanation:
Given that,
An elephant at the zoo lost 24 pounds over 6 months.
The elephant lost the same amount of weight each month.
We use the negative numbers for the drop in weight.
Since the elephant’s weight decreased each month with the same amount, the change in the elephant’s weight each month will be represented by the result of the division:
-24 ÷ 6 = – 4
Thus the change in the elephant’s weight each month is -4 pounds.

2.3 Applying Integer Operations

Evaluate each expression.

Question 11.
(−4)(5) + 8
______

Answer: -12

Explanation:
Given the expression
(−4)(5) + 8
We have to perform multiplication first and then addition
-20 + 8 = -12
So, (−4)(5) + 8 is -12.

Essential Question

Question 15.
Write and solve a real-world problem that can be represented by the expression (–3)(5) + 10.
Type below:
___________

Answer: $5

Example:
Lily bought 5 DVDs $3 each and was given a prize of $ 10 for winning a competition. What is the change in her account after these events?
Answer:
5(-3) + 10 = -15 + 10 = -5
Thus she has $5 less in her account.

MIXED REVIEW – Selected Response – Page No. 56

Question 1.
A diver is at an elevation of -18 feet relative to sea level. The diver descends to an undersea cave that is 4 times as far from the surface. What is the elevation of the cave?
Options:
a. -72 feet
b. -22 feet
c. -18 feet
d. -14 feet

Answer: -72 feet

Explanation:
A diver is at an elevation of -18 feet relative to sea level.
The diver descends to an undersea cave that is 4 times as far from the surface.
We determine the elevation of the cave using multiplication as the diver descends 4 times the distance of -18 feet:
4 × -18 = -72
Thus the correct answer is option A.

Question 2.
The football team lost 4 yards on 2 plays in a row. Which of the following could represent the change in field position?
Options:
a. -12 yards
b. -8 yards
c. -6 yards
d. -2 yards

Answer: -8 yards

Explanation:
We determine the change in field position using multiplication as the team lost twice the distance of 4 yards:
2 × -4 = -8 yards
Thus the correct answer is option B.

Question 3.
Clayton climbed down 50 meters. He climbed down in 10-meter intervals. In how many intervals did Clayton make his climb?
Options:
a. 5
b. 10
c. 40
d. 500

Answer: 5

Explanation:
We determine the number of intervals using division as Clayton climbed down the total distance in equal 10 meter intervals
-50 ÷ -10 = 5
Thus the correct answer is option A.

Question 4.
Which expression results in a negative answer?
Options:
a. a negative number divided by a negative number
b. a positive number divided by a negative number
c. a negative number multiplied by a negative number
d. a positive number multiplied by a positive number

Answer:
a. a negative number divided by a negative number gives a positive result.
b. a positive number divided by a negative number gives a negative result.
c. a negative number multiplied by a negative number gives a positive result.
d. a positive number multiplied by a positive number gives a positive result.
The only situation in which we get a negative result is in case B, thus the correct answer is option B.

Question 5.
Clara played a video game before she left the house to go on a walk. She started with 0 points, lost 6 points 3 times, won 4 points, and then lost 2 points. How many points did she have when she left the house to go on the walk?
Options:
a. -20
b. -16
c. 12
d. 20

Answer: -16

Explanation:
Clara played a video game before she left the house to go on a walk.
She started with 0 points, lost 6 points 3 times, won 4 points, and then lost 2 points.
3(-6) + 4 – 2 = -18 + 2 = -16
Thus the correct answer is option B.

Question 6.
Which expression is equal to 0?
Options:
a. \(\frac{-24}{6}\) − 4
b. \(\frac{-24}{-6}\) + 4
c. \(\frac{24}{6}\) + 4
d. \(\frac{-24}{-6}\) − 4

Answer: \(\frac{-24}{-6}\) − 4

Explanation:
a. \(\frac{-24}{6}\) − 4
– 4 – 4 = -8
b. \(\frac{-24}{-6}\) + 4
4 + 4 = 8
c. \(\frac{24}{6}\) + 4
4 + 4 = 8
d. \(\frac{-24}{-6}\) − 4
4 – 4 = 0
Thus the correct answer is option D.

Mini-Task

Question 7.
Rochelle and Denae started with the same amount of money in their bank accounts. Rochelle made three withdrawals of $25 and then wrote a $100 check. Denae deposited $5 and then wrote a $200 check.
a. Find the total change in the amount of money in Rochelle’s account.
$ _______

Answer: -175

Explanation:
We use positive values for deposited money and negative values for withdrawals and written checks.
3(-25) – 100 = -75 – 100 = -175

Question 7.
b. Find the total change in the amount of money in Denae’s account.
$ _______

Answer: -195

Explanation:
We find the total change in the amount of money in Rochelle’s account:
5 + (-200) = -195

Question 7.
c. Compare the amounts of money the two women have in their accounts now.
Type below:
___________

Answer: 20

Explanation:
Since they started with the same amount of money and Rochelle’s account decreased by $175, while Denae’s account decreased by $195, it means Rochelle has an account greater than Denae’s by the sum of
195 – 175 = 20

Module 2 Review – Multiplying and Dividing Integers – Page No. 104

EXERCISES

Question 1.
−9 × (−5) =
________

Answer: 45

Explanation:
Given,
−9 × (−5)
Since the two integers have the same sign, the answer will be positive.
Multiply both numbers.
−9 × (−5) = 45

Question 2.
0 × (−10) =
________

Answer: 0

Explanation:
Given,
Any number multiplied by zero will be zero.
0 × (−10) = 0

Question 5.
−9 ÷ (−1) =
________

Answer: 9

Explanation:
Given,
Since the two integers have the same sign, the answer will be positive.
Divide both numbers.
−9 ÷ (−1) = 9

Question 6.
−56 ÷ 8 =
________

Answer: -7

Explanation:
Given,
Since the two integers have different signs, the answer will be negative.
Divide both numbers.
−56 ÷ 8 = -7

Question 7.
−14 ÷ 2 − 3 =
________

Answer: -10

Explanation:
Given,
−14 ÷ 2 − 3 = (−14 ÷ 2) − 3
– 7 – 3 = -10

Question 8.
8 + (−20) × 3 =
________

Answer: -52

Explanation:
Given,
8 + (−20) × 3 = 8 – 60 = -52

Question 10.
Tony bought 3 packs of pencils for 4 each and a pencil box for 7. Mario bought 4 binders for 6 each and used a coupon for 6 off. Write and evaluate expressions to find who spent more money.
_____________

Answer: Tony

Explanation:
Tony bought 3 packs of pencils for 4 each and a pencil box for 7.
Mario bought 4 binders for 6 each and used a coupon for 6 off.
Find the total amount that Tony spent
3 × 4 + 7 = 12 + 7 = $19
Find the total amount that Mario spent this is less than Tony’s amount so Tony spent more.
4 × 6 – 6 = 24 – 6 = $18
Compare the amount that Tony and Mario spent
Tony spent more.

Conclusion:

The solutions provided in the Go Math Grade 7 Answer Key Chapter 2 Multiplying and Dividing Integers pdf are prepared by the math experts. This Go Math Answer Key Grade 7 Chapter 2 helps the students to score the highest marks in the exams. It also helps the teachers and parents to help their children in solving the problems in Go Math Grade 7 Key Chapter 2 Multiplying and Dividing Integers.

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Go Math Grade 7 Answer Key Chapter 1 Adding and Subtracting Integers

We suggest the students check out the topics of Grade 7 chapter 1 before you start your preparation for exams. The chapter Adding and Subtracting Integers contains topics such as Adding Integers with the Same Sign, Different sign, Subtracting Integers, Applying Addition and Subtraction of Integers, and so on. Tap the below links and try to solve the questions provided in the Go Math Grade 7 Solution Key Chapter 1 Adding and Subtracting Integers.

Chapter 1 – Adding Integers with the Same Sign

• Adding Integers with the Same Sign – Guided Practice – Page No. 10
• Adding Integers with the Same Sign – Independent Practice – Page No. 11
• Adding Integers with the Same Sign – Page No. 12

Chapter 1 – Adding Integers with Different Signs

• Adding Integers with Different Signs – Guided Practice – Page No. 16
• Adding Integers with Different Signs – Independent Practice – Page No. 17
• Adding Integers with Different Signs – Page No. 18

Chapter 1 – Subtracting Integers

• Subtracting Integers – Guided Practice – Page No. 22
• Subtracting Integers – Independent Practice – Page No. 23
• Subtracting Integers – Page No. 24

Chapter 1 – Applying Addition and Subtraction of Integers

• Applying Addition and Subtraction of Integers – Guided Practice – Page No. 28
• Applying Addition and Subtraction of Integers – Independent Practice – Page No. 29

Chapter 1 – MODULE 1

• Module Quiz – Ready to Go On – Page No. 31
• Module Quiz – MODULE 1 MIXED REVIEW – Page No. 32
• MODULE 1 MIXED REVIEW
• Module 1 Review – Adding and Subtracting Integers – Page No. 103

Adding Integers with the Same Sign – Guided Practice – Page No. 10

Find each sum.

Question 1.
-5 + (-1)

a. How many counters are there?
_______ counters

Answer: 6

Explanation:
By seeing the above pictures we can say that there are 6 counters.

Question 1.
b. Do the counters represent positive or negative numbers?
____________

Answer: negative numbers

Explanation:
The counters are red so they represent negative numbers.

Question 1.
c. -5 + (-1) =
_______

Answer: -6

Explanation:
There are 6 counters so -5 + (-1) = – 6

Question 2.
-2 + (-7)

a. How many counters are there?
_______ counters

Answer: 9

Explanation:
The above figure shows that there are 9 counters.

Question 2.
b. Do the counters represent positive or negative numbers?
____________

Answer: negative numbers

Explanation:
The counters are red so they represent the negative numbers.

Question 2.
c. -2 + (-7) =
_______

Answer: -9

Explanation:
There are 9 counters so -2 + (-7) = -9
The answer is -9.

Model each addition problem on the number line to find each sum.

Question 3.
-5 + (-2) =

_______

Answer: -7

Explanation:
Remember if the number being added is positive more units go to the right and if the number being added is negative more units to the left.
Since we are adding the negative number starting from -5, we move 2 units to the left. This results in -7.

Question 4.
-1 + (-3) =

_______

Answer: -4

Explanation:
Remember if the number being added is positive more units go to the right and if the number being added is negative more units to the left.
Since we are adding a negative number starting from -1, we move 3 units to left. This results in -4.

Question 5.
-3 + (-7) =

_______

Answer: -10

Explanation:
Remember if the number being added is positive more units go to the right and if the number being added is negative more units to the left.
Since we are adding a negative number starting from -3, we move 7 units to left. This results in -10.

Question 6.
-4 + (-1) =

_______

Answer: -5

Explanation:
Remember if the number being added is positive more units go to the right and if the number being added is negative more units to the left.
Since we are adding a negative number starting from -4, we move 1 unit to left. This results in -5.

Question 7.
-2 + (-2) =

_______

Answer: -4

Explanation:
Remember if the number being added is positive more units go to the right and if the number being added is negative more units to the left.
Since we are adding the negative number starting at -2, we move 2 units to the left which gives the result -4.

Question 8.
-6 + (-8) =

_______

Answer: -14

Explanation:
Remember if the number being added is positive more units go to the right and if the number being added is negative more units to the left.
Since we are adding the negative number starting from -6 we have to move 8 units to the left which shows the result -14.

Find each sum.

Question 9.
-5 + (-4) =
_______

Answer: -9

Explanation:
In adding two integers with the same signs you add both integers and keep the common sign.
Since -5 + (-4) has the same sign we add their absolute value and keep the same sign.
-5 + (-4) = -(5 + 4) = -9

Question 13.
-52 + (-48) =
_______

Answer: -100

Explanation:
In adding two integers with the same signs you add both integers and keep the common sign.
Since -52 + (-48) has the same sign we add their absolute value and keep the same sign.
-52 + (-48) = -(52 + 48)
= -100
The answer is -100.

Question 14.
5 + 198 =
_______

Answer: 203

Explanation:
In adding two integers with the same signs you add both integers and keep the common sign.
Since 5 + 198 has the same sign we add their absolute value and keep the same sign.
5 + 198 = 203
The answer is 203.

Question 15.
-4 + (-5) + (-6) =
_______

Answer: -15

Explanation:
In adding two integers with the same signs you add both integers and keep the common sign.
Since -4 + (-5) + (-6) has the same sign we add their absolute value and keep the same sign.
-4 + (-5) + (-6) = -(4 + 5 + 6)
= -15
The answer is -15.

Adding Integers with the Same Sign – Independent Practice – Page No. 11

Question 18.
Represent Real-World Problems Jane and Sarah both dive down from the surface of a pool. Jane first dives down 5 feet and then dives down 3 more feet. Sarah first dives down 3 feet and then dives down 5 more feet.
a. Multiple Representations Use the number line to model the equation -5 + (-3) = -3 + (-5).

Type below:
______________

Answer: -8

Explanation:
Start at -3 and move 5 units down for one number line. Next, start at -5 and move down 3 units for another number line.
Both have a final answer of -8.
So, -5 + (-3) = -3 + (-5) = -8.

Question 18.
b. Does the order in which you add two integers with the same sign affect the sum? Explain.
_______

Answer: no

Explanation:

Based on the results of part a, the order doesn’t matter. Since the commutative properties of addition hold for the sum of two negative numbers.

Question 19.
A golfer has the following scores for a 4-day tournament.

What was the golfer’s total score for the tournament?
_______

Answer: -11

Explanation:
The total score is the sum of each day’s score
= -3 + (-1) + (-5) + (-2)
= -(3 + 1 + 5 + 2)
= -11
Thus the total score for 4 days tournament is -11.

Question 21.
When the quarterback is sacked, the team loses yards. In one game, the quarterback was sacked four times. What was the total sack yardage?

_______

Answer: -54

Explanation:
The total sack yardage = -14 + (-5) + (-12) + (-23)
= -(14 + 5 + 12 + 23)
= -54
Therefore the total sack yardage is -54.

Question 22.
Multistep The temperature in Jonestown and Cooperville was the same at 1:00. By 2:00, the temperature in Jonestown dropped 10 degrees, and the temperature in Cooperville dropped 6 degrees. By 3:00, the temperature in Jonestown dropped 8 more degrees, and the temperature in Cooperville dropped 2 more degrees.
a. Write an equation that models the change to the temperature in Jonestown since 1:00.
Type below:
______________

Answer: J = T – 18

Explanation:
Let J be the final temperature and T be the initial temperature. Then the equation is J = T + (-10) + (-8)
J = T – 18

Question 22.
b. Write an equation that models the change to the temperature in Cooperville since 1:00.
Type below:
______________

Answer: C = T – 8

Explanation:
Let C be the final temperature and T be the initial temperature. Then the equation is C = T + (-6) + (-2)
C = T – 8

Question 22.
c. Where was it colder at 3:00, in Jonestown or Cooperville?
__________

Answer: Jonestown

Explanation:
Since they both started at the same temperature and Jonestown dropped a total of 18 degrees while Cooperville dropped a total of 8 degrees, Jonestown is colder.

Adding Integers with the Same Sign – Page No. 12

H.O.T. FOCUS ON HIGHER ORDER THINKING

Question 24.
Multistep On Monday, Jan made withdrawals of $25, $45, and $75 from her savings account. On the same day, her twin sister Julie made withdrawals of $35, $55, and $65 from her savings account.
a. Write a sum of negative integers to show Jan’s withdrawals on Monday. Find the total amount Jan withdrew.
Jan withdrew $ _______

Answer: 145

Explanation:
Each withdrawal is represented by a negative integer so find the sum of those negative integers = -25 + (-45) + (-75)
= -(25 + 45 + 75)
= -145
Thus Jan withdrew $145.

Question 24.
b. Write a sum of negative integers to show Julie’s withdrawals on Monday. Find the total amount Julie withdrew.
Julie withdrew $ _______

Answer: 155

Explanation:
Each withdrawal is represented by a negative integer so find the sum of those negative integers
= -35 + (-55) + (-65)
= – (35 + 55 + 65)
= -155
The total amount Julie withdrew is -$155.

Question 24.
c. Julie and Jan’s brother also withdrew money from his savings account on Monday. He made three withdrawals and withdrew $10 more than Julie did. What are three possible amounts he could have withdrawn?
Type below:
______________

Answer:

If he withdrew $10 more than Julie then he withdrew $165 in total. The possible amounts could then be $35, $55, $75.

Question 26.
Critique Reasoning The absolute value of the sum of two different integers with the same sign is 8. Pat says there are three pairs of integers that match this description. Do you agree? Explain.
__________

Answer: Disagree

Explanation:
Pat is saying that x + y = 8 is true for only three pairs of numbers with the same sign.
This is not true though. The pairs could be 1, 7, 2 and 6, 3, 5, 4 and -4, -1 and -7, -2 and -6, -3 and -5 and -4, -4.

Adding Integers with Different Signs – Guided Practice – Page No. 16

Use a number line to find each sum.

Question 1.
9 + (-3) =

_______

Answer: 6

Explanation:
Remember if the number being added is positive more units go to the right and if the number being added is negative more units to the left.
Since we are adding a negative number starting from 9, move 3 units to the left. This results in 6.

Question 2.
-2 + 7 =

_______

Answer: 5

Explanation:
Remember if the number being added is positive more units go to the right and if the number being added is negative more units to the left.
Since we are adding a positive number starting from -2 we move 7 units to the right. This results in 5.

Question 3.
-15 + 4 =

_______

Answer: -11

Explanation:
Remember if the number being added is positive more number of units going to the right and if the number being added is negative more number of units to the left.
Since we are adding a positive number starting from -15, we move 4 units to the right. This results in -11

Question 4.
1 + (-4) =

_______

Answer: -3

Explanation:
Remember if the number being added is positive more number of units going to the right and if the number being added is negative more number of units to the left.
Since we are adding the negative number starting from 1, we move 4 units to the left. This results in -3.

Circle the zero pairs in each model. Find the sum.

Question 5.
-4 + 5 =

_______

Answer: 1

Explanation:
In adding two integers with the same sign, add their absolute value, and keep the common sign.
When adding two integers with opposite signs, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value.
Above is an illustration of which are the zero pairs and what remains. In this item 1 yellow counter remains which means the sum is 1.

Question 6.
-6 + 6 =

_______

Answer: 0

Explanation:
In adding two integers with the same sign, add their absolute value, and keep the common sign.
When adding two integers with opposite signs, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value.
Above is an illustration of which are the zero pairs and what remains. In this item, there are no counters so the sum is 0.

Question 7.
2 + (-5) =

_______

Answer: -3

Explanation:
In adding two integers with the same sign, add their absolute value, and keep the common sign.
When adding two integers with opposite signs, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value.
Above is an illustration of which are the zero pairs and what remains. In this item, 3 red counters are remaining so the sum is -3.

Question 8.
-3 + 7 =

_______

Answer: 4

Explanation:
In adding two integers with the same sign, add their absolute value, and keep the common sign.
When adding two integers with opposite signs, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value.
Above is an illustration of which are the zero pairs and what remains. In this item, 4 yellow counters remain so the sum is 4.

Find each sum.

Question 9.
-8 + 14 =
_______

Answer: 6

Explanation:
In adding two integers with the same sign, add their absolute value, and keep the common sign.
When adding two integers with opposite signs, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value.
Here we are the opposite number with the negative number.
-8 + 14 = 6
The more significant number is having a positive sign so the sum is 6.

Question 12.
14 + (-14) =
_______

Answer: 0

Explanation:
In adding two integers with the same sign, add their absolute value, and keep the common sign.
When adding two integers with opposite signs, subtract the smaller absolute value from the larger one and keep the sign of the number with the larger absolute value.
14 + (-14) =14 – 14 = 0

Question 13.
0 + (-5) =

Answer: -5

Explanation:
In adding two integers with the same sign, add their absolute value, and keep the common sign.
When adding two integers with opposite signs, subtract the smaller absolute value from the larger one and keep the sign of the number with the larger absolute value.
0 + (-5) = 0 – 5 = -5
The larger is having a negative sign so the sum is -5.

Adding Integers with Different Signs – Independent Practice – Page No. 17

Find each sum.

Question 16.
-15 + 71 =
_______

Answer: 56

Explanation:
In adding two integers with the same sign, add their absolute value, and keep the common sign.
When adding two integers with opposite signs, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value.
-15 + 71 = |71| – |-15|
= 71 – 15
= 56

Question 17.
-53 + 45 =
_______

Answer: -8

Explanation:
In adding two integers with the same sign, add their absolute value, and keep the common sign.
When adding two integers with opposite signs, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value.
-53 + 45 = |-53| – |45|
53 – 45 = 8
The larger number is having the negative symbol so the answer is -8.

Question 18.
-79 + 79 =
_______

Answer: 0

Explanation:
In adding two integers with the same sign, add their absolute value, and keep the common sign.
When adding two integers with opposite signs, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value.
79 + (-79) = |79| – |-79|
79 – 79 = 0

Question 19.
-25 + 50 =
_______

Answer: 25

Explanation:
In adding two integers with the same sign, add their absolute value, and keep the common sign.
When adding two integers with opposite signs, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value.
-25 + 50 = |50| – |-25|
50 – 25 = 25

Question 20.
18 + (-32) =
_______

Answer: -14

Explanation:
In adding two integers with the same sign, add their absolute value, and keep the common sign.
When adding two integers with opposite signs, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value.
18 + (-32) = |-32| – |18|
32 – 18 = 14
The larger number is having a negative sign so the answer is -14.

Question 21.
5 + (-100) =
_______

Answer: -95

Explanation:
In adding two integers with the same sign, add their absolute value, and keep the common sign.
When adding two integers with opposite signs, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value.
5 + (-100) = |-100| – |5|
100 – 5 = 95
The larger number is having a negative sign so the answer is -95.

Question 22.
-12 + 8 + 7 =
_______

Answer: 3

Explanation:
In adding two integers with the same sign, add their absolute value, and keep the common sign.
When adding two integers with opposite signs, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value.
-12 + 8 + 7 = -12 + (8 + 7)
For the terms have different signs, we subtract the lesser absolute value from the greater absolute value and use the sign of the integer with the greater absolute value for the sum: 3
-12 + 15 = 3

Question 23.
-8 + (-2) + 3 =
_______

Answer: -7

Explanation:
In adding two integers with the same sign, add their absolute value, and keep the common sign.
When adding two integers with opposite signs, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value.
-(8 + 2) + 3
For the terms have different signs, we subtract the lesser absolute value from the greater absolute value and use the sign of the integer with the greater absolute value for the sum: -7
-10 + 3 = -7

Question 24.
15 + (-15) + 200 =
_______

Answer: 200

Explanation:
We are given the expression:
15 + (-15) + 200 = 0 + 200
The sum of the opposite number is 0.
0 + 200 = 200

Question 27.
A soccer team is having a car wash. The team spent $55 on supplies. They earned $275, including tips. The team’s profit is the amount the team made after paying for supplies. Write a sum of integers that represents the team’s profit.
Type below:
____________

Answer: 220

Explanation:
(-55) + (+275)
The money spent on supplies diminishes the profit, so they contribute to the profit with -55, while the earned money increases the profit, so they contribute to the profit with +275.
The sum of integers that represents the team’s profit is:
(-55) + (+275) = (275 -55) = 220

Question 28.
As shown in the illustration, Alexa had a negative balance in her checking account before depositing a $47.00 check. What is the new balance of Alexa’s checking account?

$ _______

Answer: 0

Explanation:
(-47) + 47 = 0
The new balance consists of the sum between the old balance and the amount she deposits: 0

Question 29.
The sum of two integers with different signs is 8. Give two possible integers that fit this description.
Type below:
____________

Answer: 10 and -2

Explanation:
10 and 2
10 – 2 = 8
Because the sum of the two numbers is positive and the two numbers have different signs, it means the absolute value of the positive number is 8 units greater than the absolute value of the negative number. First, we find two positive numbers which are different by 8, which will be the positive values of our numbers.
10 and -2
Our positive number will be a greater one while our negative number will be the smaller one (-2). So the desired numbers are:
10 + (-2) = 8
12 + (-4) = 8
15 + (-7) = 8

Question 30.
Multistep Bart and Sam played a game in which each player earns or loses points in each turn. A player’s total score after two turns is the sum of his points earned or lost. The player with the greater score after two turns wins. Bart earned 123 points and lost 180 points. Sam earned 185 points and lost 255 points. Which person won the game? Explain.
____________

Answer: Bart

Explanation:
123 + (-180) = -(180 – 123) = -57
The person who has the greatest number of points after 2 turns wins.
We find the number of points Bart has, by adding the number of points from the two turns:
185 + (-255) = -(255 – 185) – 70
We find the number of points Sam has, by adding the number of points from the two turns:
The winner is Bart because -57 is greater than -70.

Adding Integers with Different Signs – Page No. 18

H.O.T. FOCUS ON HIGHER ORDER THINKING

Question 31.
Critical Thinking Explain how you could use a number line to show that -4 + 3 and 3 + (-4) have the same value. Which property of addition states that these sums are equivalent?
____________ Property of Addition

Answer: Commutative property of addition

Explanation:
In order to prove that -4 + 3 and 3 + (-4) have the same value we use the number line twice: -1 we start from -4 and we move 3 units in the positive direction to the right we get the sum -1.
We start from 3 and we move 4 units in the negative direction to the left where we find again -1.
The property of addition which states that the sum is the same no matter the order in which we add the terms is called commutative property.

Question 32.
Represent Real-World Problems Jim is standing beside a pool. He drops weight from 4 feet above the surface of the water in the pool. The weight travels a total distance of 12 feet down before landing on the bottom of the pool. Explain how you can write a sum of integers to find the depth of the water.
Type below:
____________

Answer: 12 + (-4) = 8

Explanation:
Given that,
Jim is standing beside a pool.
He drops weight from 4 feet above the surface of the water in the pool.
The weight travels a total distance of 12 feet down before landing on the bottom of the pool.
12 + (-4) = 12 – 4 = 8
The depth of the water can be calculated by adding to the total distance of 12 feet the negative distance of -4 feet.

Question 33.
Communicate Mathematical Ideas Use counters to model two integers with different signs whose sum is positive. Explain how you know the sum is positive.
Type below:
____________

Answer: The result is positive because there are more positive counters than negative counters.

Explanation:
○○○○○○○
●●●
Let’s model the sum 7 + (-3) using counters we use 7 white counters for the positive numbers and 3 black counters for the negative numbers.
We pair each white counter with a black counter their sum is being 0.
The result is +4 as we are left with 4 white counters.
The result is positive because there are more positive counters than negative counters.

Question 34.
Analyze Relationships You know that the sum of -5 and another integer is a positive integer. What can you conclude about the sign of the other integer? What can you conclude about the value of the other integer? Explain.
Type below:
____________

Answer:
We know that the sum is -5 and another integer is a positive integer. This means that the absolute value of the positive number is greater than the absolute value of -5.
The absolute value of -5 is 5, so the absolute value of the positive integer must be greater than 5. But because the number is positive, its absolute value is the number itself, so the positive number must be greater than 5.
-5 + 7 = 7 – 5 = 2

Subtracting Integers – Guided Practice – Page No. 22

Explain how to find each difference using counters.

Question 1.
5 – 8 =
_______

Answer: -3

Explanation:
5 – 8
We start with 5 black counters.
Since we have to subtract more black counters than we have (5 instead of 8), we add 3 zero pairs:
We subtract the 8 black counters: -3
We are left with 3 white counters, which means the result is -3.

Use a number line to find each difference.

Question 3.
− 4 − 5 = − 4 + ( _______ ) = _______

Answer: -9

Explanation:
-4 – 5
We have to compute the difference:
-4 – 5 = -(4 + 5)
On a number line, we start from -4 and we go to the left by 5 units:
-4 -5 = -9

Question 4.
1 − 4 = 1 + ( _______ ) = _______

Answer: -3

Explanation:
1 – 4
We have to compute the difference:
1 – 4 = 1 + (-4)
We replace the subtraction by addition with the opposite:
On a number line, we start from 1 and we go to the left by 4 units:
1 – 4 = – 3
The result is -3.

Solve.

Question 8.
-17 – 1 =
_______

Answer: -18

Explanation:
We have to perform the subtraction:
-17 – 1 = -17 + (-1)
We replace subtraction by addition with the opposite number:
-17 + (-1) = -18
We use the rule for adding integers: -18

Question 9.
0 – (-5) =
_______

Answer: 5

Explanation:
We have to perform the subtraction:
0 – (-5) = 0 + 5
We replace subtraction b addition with the opposite number:
0 + 5 = 5
We use the rule for adding integers: 5

Question 10.
1 – (-18) =
_______

Answer: 19

Explanation:
We have to perform the subtraction:
1 – (-18) = 1 + 18
We replace subtraction by addition with the opposite number:
1 + 18 = 19
We use the rule for adding integers: 19

Question 11.
15 – 1 =
_______

Answer: 14

Explanation:
We have to perform the subtraction:
15 – 1 = 14
We subtract the numbers directly as in this case it is simpler than to replace subtraction by addition with the opposite: 14

Question 12.
-3 – (-45) =
_______

Answer: 42

Explanation:
We have to perform the subtraction:
-3 – (-45) = -3 + 45
We replace subtraction by addition with the opposite number:
-3 + 45 = 42
We use the rule for adding integers: 42

Subtracting Integers – Independent Practice – Page No. 23

Question 16.
Theo had a balance of -$4 in his savings account. After making a deposit, he has $25 in his account. What is the overall change to his account?
$ _______

Answer: $29

Explanation:
Theo had a balance of -$4 in his savings account.
After making a deposit, he has $25 in his account.
25 – (-4)
The overall change to the account is the difference between the amount in the account after making the deposit and the amount before it, so we have to perform the subtraction.
25 – (-4) = 25 + 4
We change subtraction to addition with the opposite number:
25 + 4 = 29
We apply the rules for adding integers: $29

Question 17.
As shown, Suzi starts her hike at an elevation below sea level. When she reaches the end of the hike, she is still below sea level at -127 feet. What was the change in elevation from the beginning of Suzi’s hike to the end of the hike?

_______ feet

Answer: 98 feet

Explanation:
127 – (-225)
The change in the elevation from the beginning of Suzi’s hike to the end of the hike is the difference between the elevation at the end of the hike and the elevation at the beginning of it, so we have to perform the subtraction:
-127 – (-225) = -127 + 225
We change subtraction to addition with the opposite number:
-127 + 225 = 98
We apply the rules for adding integers: 98 feet

Question 20.
A scientist conducts three experiments in which she records the temperature of some gases that are being heated. The table shows the initial temperature and the
final temperature for each gas.

a. Write a difference of integers to find the overall temperature change for each gas.
Gas A: __________ °C increase
Gas B: __________ °C increase
Gas C: __________ °C increase

Answer:
We determine the overall change of temperature for each gas by subtracting the initial temperature from the final temperature.
Gas A:
-8 – (-21) = -8 + 21 = 13
Gas B:
12 – (-12) = 12 + 12 = 24
Gas C:
-15 – (-19) = -15 + 19 = 4

Question 20.
What If? Suppose the scientist performs an experiment in which she cools the three gases. Will the changes in temperature be positive or negative for this experiment? Why?
__________

Answer: Negative

Explanation:
Cooling the gases means diminishing their temperature, thus their final temperature will be lower than the initial temperature, so the change in temperature will be negative.

Subtracting Integers – Page No. 24

Question 21.
Analyze Relationships For two months, Nell feeds her cat Diet Chow brand cat food. Then for the next two months, she feeds her cat Kitty Diet brand cat food. The table shows the cat’s change in weight over 4 months.

Which brand of cat food resulted in the greatest weight loss for Nell’s cat? Explain.
__________

Answer: Diet Chow

Explanation:
(-8) + (-18) = -26
We count the total change of weight resulted after using the diet chow for two months.
We count the total change of weight resulted after using the Kitty Diet for two months:
3 + (-19) = -16
This means that by using the Diet Chow the cat lost 26 oz, while using the Kitty Diet she lost 16 oz, thus the greatest loss of weight resulted in using the Diet Chow food.

FOCUS ON HIGHER ORDER THINKING

Question 22.
Represent Real-World Problems Write and solve a word problem that can be modeled by the difference -4 – 10.
Type below:
____________

Answer:
We have to write and solve a problem using the difference:
-4 – 10
For example:
Yesterday the temperature was -4 degrees. Today the temperature decreased by 10 degrees. What is the temperature today?
– 4 – 10 =- + (-10) = -14

Question 23.
Explain the Error When Tom found the difference -11 – (-4), he got -15. What might Tom have done wrong?
Type below:
____________

Answer:
We have to find the error in computing the difference:
-11 – (-4) = -15
In order to perform subtraction, Tom replaced it with addition, but he was wrong in adding -4 instead of adding its opposite 4.
The correct form is -11 – (-4) = -11 + 4 = -7

Applying Addition and Subtraction of Integers – Guided Practice – Page No. 28

Write an expression. Then find the value of the expression.

Question 1.
Tomas works as an underwater photographer. He starts at a position that is 15 feet below sea level. He rises 9 feet, then descends 12 feet to take a photo of a coral reef. Write and evaluate an expression to find his position relative to sea level when he took the photo.
_______ feet below sea level

Answer: 18 feet

Explanation:
When he rises, we add the distance. When he descends, we subtract the distance.
The initial position is -15. We write an expression to find his position relative to sea level when he took the photo:
-15 + 9 – 12 = (-15) + 9 + (-12)
(-15) + (-12) + 9
-(15 + 12) + 9
-27 + 9 = -18
Thus he was 18 feet below sea level when he took the photo.

Question 2.
The temperature on a winter night was -23 °F. The temperature rose by 5 °F when the sun came up. When the sun set again, the temperature dropped by 7 °F. Write and evaluate an expression to find the temperature after the sun set.
_______ °F

Answer: -25

Explanation:
When the temperature rises, we add the temperature. When the temperature drops, we subtract the temperature. The initial temperature is -23.
We write an expression to find the temperature after sunset:
-23 + 5 – 7 = -(23 + 7) + 5
-30 + 5 = -25
Thus the temperature is -25°F after sunset.

Find the value of each expression.

Question 4.
-6 + 15 + 15 =
_______

Answer: 24

Explanation:
We have to find the value of the expression:
-6 + 15 + 15 = – 6 + 30 = 24
-6 + 15 + 15 = 24

Question 5.
9 – 4 – 17 =
_______

Answer: -12

Explanation:
We have to find the value of the expression:
9 – 4 – 17 = 9 – (4 + 17)
= 9 – 21 = -12

Question 6.
50 – 42 + 10 =
_______

Answer: 18

Explanation:
We have to find the value of the expression:
50 + (-42) + 10 = 60 – 42
We use the commutative property:
60 – 42 = 18

Determine which expression has a greater value.

Question 10.
-12 + 6 – 4 or -34 – 3 + 39
___________

Answer:
We have to compare the expressions:
-12 + 6 – 4 or -34 – 3 + 39
We compute the first expression:
-12 + 6 – 4
-(12 + 4) + 6
-16 + 6 = -10
We compute the second expression:
-34 – 3 + 39
-(34 + 3) + 39
-37 + 39 = 2
2 > -10
Since 2 is greater than -10, the second expression is greater than the first expression.

Question 11.
21 – 3 + 8 or -14 + 31 – 6
___________

Answer:
We have to compare the expressions:
21 – 3 + 8 or -14 + 31 – 6
We compute the first expression:
21 – 3 + 8
21 + 8 – 3
21 + 5 = 26
We compute the second expression:
-14 + 31 – 6
31 – (14 + 6)
31 – 20 = 11
26 > 11
Since 26 is greater than 11, the first expression is greater than the second.

Question 12.
Explain how you can find the value of the expression -5 + 12 + 10 – 7.
Type below:
___________

Answer: 10

Explanation:
We have to find the value of the expression:
-5 + 12 + 10 – 7 = 12 + 10 – (5 + 7)
22 – 12 = 10

Applying Addition and Subtraction of Integers – Independent Practice – Page No. 29

Question 13.
Sports Cameron is playing 9 holes of golf. He needs to score a total of at most 15 over par on the last four holes to beat his best golf score. On the last four holes, he scores 5 over par, 1 under par, 6 over par, and 1 under par.
a. Write and find the value of an expression that gives Cameron’s score for 4 holes of golf.
Type below:
___________

Answer:
We write the expression that gives Cameron’s score for 4 holes:
5 – 1 + 6 – 1
5 + 6 – (1 + 1)
11 – 2 = 9

Question 13.
b. Is Cameron’s score on the last four holes over or under par?
Type below:
___________

Answer: The result shows that Cameron’s score is over par.

Question 13.
c. Did Cameron beat his best golf score?
_______

Answer:
Since his score of 9 is beaten his best score of 9 > 15.

Question 16.
Lee and Barry play a trivia game in which questions are worth different numbers of points. If a question is answered correctly, a player earns points. If a question is answered incorrectly, the player loses points. Lee currently has -350 points.

a. Before the game ends, Lee answers a 275-point question correctly, a 70-point question correctly, and a 50-point question incorrectly. Write and find the value of an expression to find Lee’s final score.
_______ points

Answer: -55 points

Explanation:
The initial score is -350 points. We write and find the value of an expression to find Lee’s final score:
-350 + 275 + 70 – 50
-(350 + 50) + 275 + 70
-400 + 345 = -55

Question 17.
b. Barry’s final score is 45. Which player had the greater final score?
___________

Answer: Since -55 < 45, it means Barry has a greater final score.

Question 17.
Multistep Rob collects data about how many customers enter and leave a store every hour. He records a positive number for customers entering the store each hour and a negative number for customers leaving the store each hour.

a. During which hour did more customers leave than arrive?
___________

Answer: 3:00 – 4:00

Explanation:
since in the last column, the only positive value is in the last position, the hour in which more customers leave than arrive is 3:00 – 4:00

Question 17.
b. There were 75 customers in the store at 1:00. The store must be emptied of customers when it closes at 5:00. How many customers must leave the store between 4:00 and 5:00?
_______ customers

Answer: 87

Explanation:
75 + 30 – 12 + 14 – 8 + 18 – 30
75 + 30 + 14 + 18 – (12 + 8 + 30)
137 – 50 = 87
Since there are 87 customers in the store at 4:00 and the store must be emptied at 5:00, the number of clients who must leave is 87.

Applying Addition and Subtraction of Integers – Page No. 30

The table shows the changes in the values of two friends’ savings accounts since the previous month.

Question 18.
Carla had $100 in her account in May. How much money does she have in her account in August?
$ _______

Answer: $51

Explanation:
We are given the data:
100 – 18 + 22 – 53
100 + 22 -(18 + 53)
122 – 71 = 51
Thus Carla saved $51 in her account in August.

FOCUS ON HIGHER ORDER THINKING

Question 21.
Represent Real-World Problems Write and solve a word problem that matches the diagram shown.

Type below:
___________

A diver leaves from a point situated 1 meter below the sea level. First, he dives 6 meters, then he rises 3 meters and stops. At which level under the sea level does he stop?
We start from the initial point -1, we add distance if he rises and we subtract distance when he dives. We determine the final level under the sea level where he stops:
-1 – 6 + 3
-(1 + 6) + 3
-7 + 3 = -4
-4 or 4 meters below sea level.

Question 23.
Draw Conclusions An expression involves subtracting two numbers from a positive number. Under what circumstances will the value of the expression be negative? Give an example.
Type below:
___________

Answer:
The sum of the two numbers to be subtracted from the positive number is a number, we will study this first. Since we subtract this number from the positive number and we get a negative number, it means that the number is greater than the positive number, therefore mandatory positive. This means the two numbers cannot be both negative.
Example:
10 – (7 + 5) = 10 – 12 = -2
-2 < 0

Module Quiz – Ready to Go On – Page No. 31

Adding Integers with the Same Sign

Add

Question 1.
−8 + (−6) = _______

Answer: -14

Explanation:
In adding two integers with the same signs you add both integers and keep the common sign.
−8 + (−6) = -(8 + 6) = -14

Question 2.
−4 + (−7) = _______

Answer: -11

Explanation:
In adding two integers with the same signs you add both integers and keep the common sign.
−4 + (−7) = – 4 – 7
-(4 + 7) = -11
−4 + (−7) = -11

Question 3.
−9 + (−12) = _______

Answer: -21

Explanation:
In adding two integers with the same signs you add both integers and keep the common sign.
−9 + (−12) = -9 – 12
-(9 + 12) = – 21
Thus −9 + (−12) = -21

Adding Integers with Different Signs

Add

Question 4.
5 + (−2) = _______

Answer: 3

Explanation:
In adding two integers with the same sign, add their absolute value, and keep the common sign.
When adding two integers with opposite signs, subtract the smaller absolute value from the larger one and keep the sign of the number with the larger absolute value.
5 + (−2) = 5 – 2 = 3
The larger number is having a positive sign thus the sum is 3

Question 5.
−8 + 4 = _______

Answer: -4

Explanation:
In adding two integers with the same sign, add their absolute value, and keep the common sign.
When adding two integers with opposite signs, subtract the smaller absolute value from the larger one and keep the sign of the number with the larger absolute value.
−8 + 4 = (-8) + 4 = -4
The larger number is having a negative sign thus the sum is -4.

Question 6.
15 + (−8) = _______

Answer: 7

Explanation:
In adding two integers with the same sign, add their absolute value, and keep the common sign.
When adding two integers with opposite signs, subtract the smaller absolute value from the larger one and keep the sign of the number with the larger absolute value.
15 + (−8) = 15 – 8 = 7
The larger number is having a positive sign thus the sum is 7.

Subtracting Integers

Subtract.

Applying Addition and Subtraction of Integers

Question 10.
A bus makes a stop at 2:30, letting off 15 people and letting on 9. The bus makes another stop ten minutes later to let off 4 more people. How many more or fewer people are on the bus after the second stop compared to the number of people on the bus before the 2:30 stop?
_______ people

Answer: 10

Explanation:
Assume that the total number of passengers on the bus before 2:30 was x
15 passengers got off and 9 got on.
number of passengers = x – 15 + 9
number of passengers = x -6
4 passengers got off the bus
number of passengers = (x-6) – 4
number of passengers = x – 10
The original number of passengers on the bus decreased by 10 after the second stop.

Question 11.
Cate and Elena were playing a card game. The stack of cards in the middle had 24 cards in it to begin with. Cate added 8 cards to the stack. Elena then took 12 cards from the stack. Finally, Cate took 9 cards from the stack. How many cards were left in the stack?
_______ cards

Answer: 11 cards

Explanation:
When cards are put to the stack, we perform addition.
When cards are taken from the stack we perform subtraction.
24 + 8 – 12 – 9
32 – (12 + 9)
32 – 21 = 11
Thus in the end the stack has 11 cards.

ESSENTIAL QUESTION

Question 12.
Write and solve a word problem that can be modeled by the addition of two negative integers.
Type below:
_____________

Answer: -25

Explanation:
A football team played two games. During the first game, the team lost 15 points and during the second game, it lost another 10 points. What is the change in the team’s score after these two games?
(-15) + (-10) = -25

Module Quiz – MODULE 1 MIXED REVIEW – Page No. 32

Assessment Readiness

Selected Response

Question 1.
Which expression has the same value as -3 + (-5):
Options:
a. -3 – (-5)
b. -3 + 5
c. -5 + (-3)
d. -5 – (-3)

Answer: -5 + (-3)

Explanation:
a. -3 – (-5)
-3 + 5 = 2
b. -3 + 5
5 – 3 = 2
c. -5 + (-3)
– 5 – 3 = -8
d. -5 – (-3)
-5 + 3 = -2
Thus the correct answer is option C.

Question 2.
A diver’s elevation is -30 feet relative to sea level. She dives down 12 feet. What is her elevation after the dive?
Options:
a. 12 feet
b. 18 feet
c. -30 feet
d. -42 feet

Answer: -42 feet

Explanation:
A diver’s elevation is -30 feet relative to sea level. She dives down 12 feet.
-30 -12 = (-30) + (-12) = -42 feet
Thus the correct answer is option D.

Question 3.
Which number line models the expression -3 + 5?
Options:
a.
b.
c.
d.

Answer:

Explanation:
-3 + 5
On the numeric line, his is modeled by starting at -3 and going right by 5 units. The number which models this is is:

Thus the correct answer is option B.

Question 4.
Which number can you add to 5 to get a sum of 0?
Options:
a. -10
b. -5
c. 0
d. 5

Answer: -5

Explanation:
The number we can add to 5 to get a sum of 0 is its opposite:
5 + (-5) = 0
The correct answer is option B.

Question 5.
The temperature in the morning was -3 °F. The temperature dropped 11 degrees by night. What was the temperature at night?
Options:
a. -14 °F
b. -8 °F
c. 8 °F
d. 14 °F

Answer: -14 °F

Explanation:
The temperature in the morning was -3 °F. The temperature dropped 11 degrees by night.
-3 + (-11) = -3 – 11 = -14°F
Therefore the correct answer is option A.

Question 6.
Which of the following expressions has the greatest value?
Options:
a. 3 – 7 + (-10)
b. 3 + 7 – (-10)
c. 3 – 7 – (-10)
d. 3 + 7 + (-10)

Answer: 3 + 7 – (-10)

Explanation:
a. 3 – 7 + (-10)
3 – 7 – 10 = 3 -(7 + 10) = 3 – 17 = -14
b. 3 + 7 – (-10)
3 + 7 + 10 = 20
c. 3 – 7 – (-10)
3 – 7 + 10 = 13 – 7 = 6
d. 3 + 7 + (-10)
10 – 10 = 0
Thus the correct answer is option B.

Mini-Task

Question 7.
At the end of one day, the value of a share of a certain stock was $12. Over the next three days, the change in the value of the share was -$1, then, -$1, and then $3.
a. Write an expression that describes the situation.
Type below:
____________

Answer:
We write an expression that describes the changes in the value of the share:
12 – 1 – 1 + 3

Question 7.
b. Evaluate the expression.
______

Answer: 13

Explanation:
12 – 1 – 1 + 3
12 + 3 – (1 + 1)
15 – 2 = 13

Question 7.
c. What does your answer to part b mean in the context of the problem?
Type below:
____________

Answer: After 3 days, the value of the share changed from $12 to $13.

MODULE 1

MIXED REVIEW

Assessment Readiness

Look at each expression. Does it have the same value as -6 – 4?

Select Yes or No for expressions A–C.

Question 8.
A. -6 + (-4)
______

Answer: Yes

Explanation:
-6 + (-4) = – 6 – 4
-6 + (-4) has the same value as – 6 – 4

Question 8.
B. -4 + (-6)
______

Answer: Yes

Explanation:
-4 + (-6) = -4 – 6
-4 + (-6) has the same value as – 6 – 4

Question 8.
C. 6 + (-4)
______

Answer: No

Explanation:
6 + (-4) = 6 – 4
6 – 4 ≠ – 6 – 4
So, 6 – 4 does not have the same value as – 6 – 4

Choose True or False for A–C.

Question 9.
A. x = 4 is the solution for x + 4 = 0.
i. True
ii. False

Answer: False

Explanation:
x + 4 = 0
x = 4
4 + 4 = 0
8 ≠ 0
So, the statement is false.

Question 9.
B. x = 24 is the solution for \(\frac{x}{3}\) = 8.
i. True
ii. False

Answer: True

Explanation:
\(\frac{x}{3}\) = 8
x = 24
24/3 = 8
8 = 8
Thus the statement is true.

Question 9.
C. x = 6 is the solution for 6x = 1
i. True
ii. False

Answer: False

Explanation:
6x = 1
x = 6
6(6) = 1
36 ≠ 1
Thus the statement is false.

Module 1 Review – Adding and Subtracting Integers – Page No. 103

EXERCISES

Question 1.
−10 + (−5) =
________

Answer: -15

Explanation:
In adding two integers with the same sign, add their absolute value, and keep the common sign.
When adding two integers with opposite signs, subtract the smaller absolute value from the larger one and keep the sign of the number with the larger absolute value.
-10 – 5 = -(10 + 5) = -15

Question 5.
25 − (−4) =
________

Answer: 29

Explanation:
In adding two integers with the same sign, add their absolute value, and keep the common sign.
When adding two integers with opposite signs, subtract the smaller absolute value from the larger one and keep the sign of the number with the larger absolute value.
25 − (−4) = 25 + 4 = 29

Question 6.
−3 − (−40) =
________

Answer: 37

Explanation:
In adding two integers with the same sign, add their absolute value, and keep the common sign.
When adding two integers with opposite signs, subtract the smaller absolute value from the larger one and keep the sign of the number with the larger absolute value.
-3 – (-40) = -3 + 40 = 37

Question 7.
Antoine has $13 in his checking account. He buys some school supplies and ends up with $5 in his account. What was the overall change in Antoine’s account?
$ ________

Answer: $8

Explanation:
The overall change in his account is given by the difference between the final amount of money and the initial amount of money
5 – 13 = 5 + (-13) = -8
The amount in his account is decreased by $8.

Conclusion:
We believe that the solutions provided in Go Math Answer Key Grade 7 Chapter 1 Adding and Subtracting Integers are helpful for you. Also, share the Go Math Grade 7 Key Chapter 1 Adding and Subtracting Integers with your dear ones to help to overcome the issues in solving the integer problems. It helps to learn simple techniques to solve adding and subtracting integer problems. If you have any doubts you can clarify them by posting the comments in the below section.