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This handy Spectrum Math Grade 7 Answer Key Chapter 4 Lesson 4.6 Problem Solving provides detailed answers for the workbook questions.

Spectrum Math Grade 7 Chapter 4 Lesson 4.6 Problem Solving Answers Key

Proportional relationships can be used to solve ratio and percent problems.
Mika’s lunch costs $ 12.50. She wants to leave an 18% tip. How much should she leave?
Set up a proportion. \(\frac{x}{12.50}\) = \(\frac{18}{100}\)
Solve for the variable. 225 = 100x
So, Mika should leave a $2.25 tip. 2.25 = x

Solve each problem.

Question 1.
A store is having a 25% off sale. If an item originally cost $ 19.36, how much should be taken off the price?
_____ should be taken off the original price.
Answer: $ 4.84 should be taken off the original price.

A store is having a 25% off sale.
An item originally cost $ 19.36
Proportional relationships can be used to solve ratio and percent problems.
Now, write a proportion using the above information
\(\frac{x}{19.36}\) = \(\frac{25}{100}\)
By solving the above equation,
x = \(\frac{25}{100}\) × 19.36
x  = $ 4.84
Therefore, $ 4.84 should be taken off the original price.

Question 2.
Dario bought a new bike for $90.00. Sales tax is 5\(\frac{1}{2}\)%. How much tax does he have to pay? How much is his total bill?
Dario’s tax is _____.
Dario’s total bill is ______
Answer: Dario’s tax is $ 4.95
Dario’s total bill is $ 94.95

Dario bought a new bike for $90.00.
Sales tax is 5\(\frac{1}{2}\)%.
Proportional relationships can be used to solve ratio and percent problems.
Now, write a proportion using the above information
\(\frac{x}{90}\) = 5\(\frac{1}{2}\)%
\(\frac{x}{90}\) = \(\frac{11}{2}\)%
\(\frac{x}{90}\) = \(\frac{11}{2}\) × \(\frac{1}{100}\)
\(\frac{x}{90}\) = \(\frac{11}{200}\)
By solving the above equation,
x = \(\frac{11}{200}\) × 90
x  = $ 4.95
Dario’s tax is $ 4.95
Therefore, Dario’s total bill is 90 + 4.95 = $ 94.95

Question 3.
A flower arrangement has 8 carnations for every 4 roses. There are 14 carnations. How many roses are in the arrangement?
There are ____________ roses in the arrangement.

Answer: There are 7 roses in the arrangement.

A flower arrangement has 8 carnations for every 4 roses.
There are 14 carnations.
Proportional relationships can be used to solve ratio and percent problems.
Now, write a proportion using the above information
\(\frac{x}{14}\) = \(\frac{4}{8}\)
By solving the above equation,
x = \(\frac{4}{8}\) × 14 = 7 roses
Therefore, There are 7 roses in the arrangement.

Question 4.
There are 18 girls in the school choir. The ratio of girls to boys is 1 to 2. How many boys are in the choir?
There are ____________ boys in the chair.
Answer: There are 9 boys in the chair.

There are 18 girls in the school choir.
The ratio of girls to boys is 1 to 2.
Proportional relationships can be used to solve ratio and percent problems.
Now, write a proportion using the above information
\(\frac{x}{18}\) = \(\frac{1}{2}\)
By solving the above equation,
x = \(\frac{1}{2}\) × 18 = 9 boys
Therefore, There are 9 boys in the chair.

Question 5.
A baseball player strikes out 3 times for every 2 hits he gets. If the player strikes out 15 times, how many hits does he get? If the player gets 46 hits, how many times does he strike out?
The player gets ____ hits for every 15 minutes he strikes out.
If the player gets 46 hits, he strikes out ____ times.
Answer: The player gets 10 hits for every 15 minutes he strikes out.
If the player gets 46 hits, he strikes out 69 times.

A baseball player strikes out 3 times for every 2 hits he gets.
If the player strikes out 15 times:
Proportional relationships can be used to solve ratio and percent problems.
Now, write a proportion using the above information
\(\frac{x}{15}\) = \(\frac{2}{3}\)
By solving the above equation,
x = \(\frac{2}{3}\) × 15 = 10 hits
Therefore, The player gets 10 hits for every 15 minutes he strikes out.
If the player gets 46 hits:
Proportional relationships can be used to solve ratio and percent problems.
Now, write a proportion using the above information
\(\frac{x}{46}\) = \(\frac{3}{2}\)
By solving the above equation,
x = \(\frac{3}{2}\) × 46 = 69 times
Therefore, If the player gets 46 hits, he strikes out 69 times.

Solve each problem.

Question 1.
Mr. Johnson borrowed $750 for 1 year. He has to pay 6% simple interest. How much interest will he pay?
Mr. Johnson will pay _____________ in interest.
Answer:
Mr. Johnson borrowed $750 for 1 year.
He has to pay 6% simple interest.
Proportional relationships can be used to solve ratio and percent problems.
Now, write a proportion using the above information
\(\frac{x}{750}\) = \(\frac{6}{100}\) × 1
By solving the above equation,
x = \(\frac{6}{100}\) × 750 = $45
Therefore, Mr. Johnson will pay $45 in interest.

Question 2.
Mrs. Soto invested in a certificate of deposit that pays 8% interest. Her investment was $325. How much interest will she receive in year?
Mrs. Soto will receive _____________ in interest.
Answer: Mrs. Soto will receive $26 in interest.
Mrs. Soto invested in a certificate of deposit that pays 8% interest.
Her investment was $325 in a year
Proportional relationships can be used to solve ratio and percent problems.
Now, write a proportion using the above information
x = \(\frac{8}{100}\) × 325 × 1
x = $26
Therefore, Mrs. Soto will receive $26 in interest.

Question 3.
Andrea put $52 in a savings account that pays 4% interest. How much interest will she earn in 1 year?
Andrea will earn _____________ in interest.
Answer: Andrea will earn $2.08 in interest.
Andrea put $52 in a savings account that pays 4% interest for 1 year
Proportional relationships can be used to solve ratio and percent problems.
Now, write a proportion using the above information
x = \(\frac{4}{100}\) × 52 × 1
x = $2.08
Therefore, Andrea will earn $2.08 in interest.

Question 4.
Jonas purchased a 2-month (3\(\frac{1}{2}\) year) certificate of deposit. It cost $600 and pays 7% interest each year. How much interest will he get? How much will the certificate be worth when he cashes it in?
Jonas will get _____________ in interest.
The certificate will be worth _____.
Answer: Jonas will get $147 in interest.
And the certificate will be worth = $747

Jonas purchased a 2-month (3\(\frac{1}{2}\) year) certificate of deposit.
It cost $600 and pays 7% interest each year.
Proportional relationships can be used to solve ratio and percent problems.
Now, write a proportion using the above information
x = \(\frac{7}{100}\) × 600 × 3\(\frac{1}{2}\)
x = \(\frac{7}{100}\) × 600 × \(\frac{7}{2}\)
x = \(\frac{7 × 600 × 7}{2 × 100}\)
by simplification,
x = $147
Therefore, Jonas will get $147 in interest.
And the certificate will be worth = 600 + 147 = $747

Question 5.
Rick borrowed $50 from his sister for 3 months (\(\frac{1}{4}\) year). She charged him 14% interest. How much does Rick have to pay to his sister?
Rick must pay his sister a total of _____.
Answer: Rick must pay his sister a total of $51.75

Rick borrowed $50 from his sister for 3 months (\(\frac{1}{4}\) year).
She charged him 14% interest.
Proportional relationships can be used to solve ratio and percent problems.
Now, write a proportion using the above information
x = \(\frac{14}{100}\) × 50 × \(\frac{1}{4}\)
x = \(\frac{14 × 50 × 1}{4 × 100}\)
by simplification,
x = $1.75
Therefore, Rick must pay his sister a total of 50 + 1.75 = $51.75

Question 6.
The grocery store borrowed $ 15,000 to remodel. The term is 7 years and the yearly interest rate is 4\(\frac{1}{4}\)%.
How much interest will the store pay? What is the total amount to be repaid?

The store will pay _____________ in interest.
The total amount to be repaid is _____.
Answer: The store will pay $4462.5 in interest.
The total amount to be repaid is  $19462.5

The grocery store borrowed $ 15,000 to remodel. The term is 7 years and the yearly interest rate is 4\(\frac{1}{4}\)%.
Proportional relationships can be used to solve ratio and percent problems.
Now, write a proportion using the above information
x = 4\(\frac{1}{4}\) %× 15000 × 7
x = \(\frac{17}{4}\) %× 15000 × 7
x = \(\frac{17}{4}\) × \(\frac{1}{100}\) × 15000 × 7
x = \(\frac{17}{400}\) × 15000 × 7
By simplification,
x = $4462.5
Therefore, The store will pay $4462.5 in interest.
The total amount to be repaid is 15000 + 4462.5 = $19462.5

Go through the Spectrum Math Grade 6 Answer Key Chapter 3 Lesson 3.3 Solving Ratio Problems and get the proper assistance needed during your homework.

Spectrum Math Grade 6 Chapter 3 Lesson 3.3 Solving Ratio Problems Answers Key

Tables can be used to help find missing values in real-life ratio problems.
A car can drive 60 miles on two gallons of gas. Create a table to find out how many miles the car can travel on 10 gallons of gas.

Complete the tables to solve the ratio problems. Circle your answer in the table.

Question 1.
You can buy 4 cans of green beans at the market for $2.25. How much will it cost to buy 12 cans of beans?

Answer:

Explanation:
Number of cans of green beans = 4.
Cost of 4 cans of green beans at the market = $2.25
Cost of each can of green beans = Cost of 4 cans of green beans at the market ÷ Number of cans of green beans
= $2.25 ÷ 4
= $0.5625.
Cost of 8 cans of green beans = Cost of each can of green beans × 8
= $0.5625 × 8
= $4.5.
Cost of 12 cans of green beans = Cost of each can of green beans × 12
= $0.5625 × 12
= $6.75.

Question 2.
An ice-cream factory makes 180 quarts of ice cream in 2 hours. How many quarts could be made in 12 hours?

Answer:

Explanation:
Number of quarts of ice cream an ice- cream factory makes = 180.
Number of hours an ice- cream factory makes 180 quarts of ice cream = 2.
Number of quarts of ice cream an ice- cream factory makes in an hour = Number of quarts of ice cream an ice- cream factory makes  ÷ Number of hours an ice- cream factory makes 180 quarts of ice cream
= 180 ÷ 2
= 90.
Number of quarts of ice cream an ice- cream factory makes in 4 hours = 4 × Number of quarts of ice cream an ice- cream factory makes in an hour
= 4 × 90
= 360.
Number of quarts of ice cream an ice- cream factory makes in 6 hours = 6 × Number of quarts of ice cream an ice- cream factory makes in an hour
= 6 × 90
= 540.
Number of quarts of ice cream an ice- cream factory makes in 8 hours = 8 × Number of quarts of ice cream an ice- cream factory makes in an hour
= 8 × 90
= 720.
Number of quarts of ice cream an ice- cream factory makes in 10 hours = 10 × Number of quarts of ice cream an ice- cream factory makes in an hour
= 10 × 90
= 900.
Number of quarts of ice cream an ice- cream factory makes in 12 hours = 12 × Number of quarts of ice cream an ice- cream factory makes in an hour
= 12 × 90
= 1080.

Question 3.
A jet travels 650 miles in 3 hours. At this rate, how far could the jet fly in 9 hours?

Answer:

Explanation:
Number of miles a jet travels = 650.
Number of hours a jet travels = 3.
Number of miles each hour jet travels = Number of miles a jet travels ÷ Number of hours a jet travels
= 650 ÷ 3
= 216.67.
Number of miles a jet travels in 6 hours  = 6 × Number of miles each hour jet travels
= 6 × 216.67
= 1,300.
Number of miles a jet travels in 9 hours  = 9 × Number of miles each hour jet travels
= 9 × 216.67
= 1950.03.

Question 4.’
A bakery can make 640 bagels in 4 hours. How many can they bake in 16 hours?

Answer:

Explanation:
Number of bagels a bakery can make in 4 hours = 640.
Number of hours a bakery can make 640 bagles = 4.
Number of bagels a bakery can make in an hour = Number of bagels a bakery can make in 4 hours ÷ Number of hours a bakery can make 640 bagles
= 640 ÷ 4
= 160.
Number of bagels a bakery can make in 8 hours = 8 × Number of bagels a bakery can make in an hour
= 8 × 160
= 1,280.
Number of bagels a bakery can make in 12 hours = 12 × Number of bagels a bakery can make in an hour
= 12 × 160
= 1,920.
Number of bagels a bakery can make in 16 hours = 16 × Number of bagels a bakery can make in an hour
= 16 × 160
= 2,560.

Go through the Spectrum Math Grade 6 Answer Key Chapter 3 Lesson 3.1 Understanding Ratios and get the proper assistance needed during your homework.

Spectrum Math Grade 6 Chapter 3 Lesson 3.1 Understanding Ratios Answers Key

A ratio compares 2 numbers. When written out, several phrases can show how the ratio should be written.
4 to 2 4:2 \(\frac{4}{2}\) or \(\frac{2}{1}\)
6 out of 8 6:8 \(\frac{6}{8}\) or \(\frac{3}{4}\)

Express each ratio as a fraction in simplest form.

Question 1.
a. 15 feet out of 36 feet ___________
Answer:
Fraction in simplest form of 15 feet out of 36 feet is \(\frac{5}{12}\).

Explanation:
15 feet out of 36 feet
= 15 ÷ 36
= \(\frac{5}{12}\).

b. 5 pounds to 35 pounds ______
Answer:
Fraction in simplest form of 5 pounds to 35 pounds is \(\frac{1}{7}\).

Explanation:
5 pounds to 35 pounds
= 5 ÷ 35
= \(\frac{1}{7}\).

Question 2.
a. 8 rainy days out of 60 days ___________

Answer:
Fraction in simplest form of 8 rainy days out of 60 days is \(\frac{2}{15}\).

Explanation:
8 rainy days out of 60 days
= 8 ÷ 60
= 4 ÷ 30
= 2 ÷ 15 or \(\frac{2}{15}\).

b. 28 snow days out of 49 days ______
Answer:
Fraction in simplest form of 28 snow days out of 49 days is \(\frac{4}{7}\).

Explanation:
28 snow days out of 49 days
= 28 ÷ 49
= 4 ÷ 7 or \(\frac{4}{7}\).

Question 3.
a. 10 pints to 20 pints __________
Answer:
Fraction in simplest form of 10 pints to 20 pints is

Explanation:
10 pints to 20 pints
= 10 ÷ 20
= 1 ÷ 2 or \(\frac{1}{2}\).

b. 40 cups to 55 cups ____
Answer:
Fraction in simplest form of 40 cups to 55 cups is \(\frac{8}{11}\).

Explanation:
40 cups to 55 cups
= 40 ÷ 55
= 8 ÷ 11 or \(\frac{8}{11}\).

Question 4.
a. 10 miles out of 12 miles ____________
Answer:
Fraction in simplest form of 10 miles out of 12 miles is \(\frac{5}{6}\).

Explanation:
10 miles out of 12 miles
= 10 ÷ 12
= 5 ÷ 6 or \(\frac{5}{6}\).

b. 28 red bikes out of 40 bikes ____

Answer:
Fraction in simplest form of 28 red bikes out of 40 bikes is \(\frac{7}{10}\).

Explanation:
28 red bikes out of 40 bikes
= 28 ÷ 40
= 14 ÷ 20
= 7 ÷ 10 or \(\frac{7}{10}\).

Question 5.
a. 18 beetles out of 72 insects ___________
Answer:
Fraction in simplest form of 18 beetles out of 72 insects is \(\frac{1}{4}\).

Explanation:
18 beetles out of 72 insects
= 18 ÷ 72
= 9 ÷ 36
= 1 ÷ 4 or \(\frac{1}{4}\).

b. 63 gallons to 8tf gallons ______
Answer:
Fraction in simplest form of 63 gallons to 8tf gallons is

Explanation:
63 gallons to 8tf gallons
Conversion:

Question 6.
a. 49 dimes out of 77 coins ___________
Answer:
Fraction in simplest form of 49 dimes out of 77 coins is 490 dimes.

Explanation:
49 dimes out of 77 coins
Conversion:
1 dollar = 100 coins/cents.
= 49 dollars × 100 coins
= 4900 coins.
1 dime = 10 coins.
=> 4900 coins÷ 10
=> 490 dimes.

b. 12 cakes out of 36 cakes ______
Answer:
Fraction in simplest form of 12 cakes out of 36 cakes is \(\frac{1}{3}\).

Explanation:
12 cakes out of 36 cakes
= 12 ÷ 36
= 1 ÷ 3 or \(\frac{1}{3}\).

Question 7.
a. 15 students out of 30 students ____________
Answer:
Fraction in simplest form of 15 students out of 30 students is \(\frac{1}{2}\).

Explanation:
15 students out of 30 students
= 15 ÷ 30
= 1 ÷ 2 or \(\frac{1}{2}\).

b. 3 floors out of 18 floors ____
Answer:
Fraction in simplest form of 3 floors out of 18 floors is \(\frac{1}{6}\).

Explanation:
3 floors out of 18 floors
= 3 ÷ 18
= 1 ÷ 6 or \(\frac{1}{6}\).

Question 8.
a. 36 meters out of 100 meters ______
Answer:
Fraction in simplest form of 36 meters out of 100 meters is \(\frac{18}{25}\).

Explanation:
36 meters out of 100 meters
= 36 ÷ 100
= 18 ÷ 25 or \(\frac{18}{25}\).

b. 14 hats out of 20 accessories ____
Answer:
Fraction in simplest form of 14 hats out of 20 accessories is \(\frac{7}{10}\).

Explanation:
14 hats out of 20 accessories
= 14 ÷ 20
= 7 ÷ 10 or \(\frac{7}{10}\).

Question 9.
a. 80 scores out of 90 scores ____________
Answer:
Fraction in simplest form of 80 scores out of 90 scores is \(\frac{8}{9}\).

Explanation:
80 scores out of 90 scores
= 80 ÷ 90
= 8 ÷ 9 or \(\frac{8}{9}\).

b. 2 sports out of 9 sports ______
Answer:
Fraction in simplest form of 2 sports out of 9 sports is \(\frac{2}{9}\).

Explanation:
2 sports out of 9 sports
= 2 ÷ 9 or \(\frac{2}{9}\).

Question 10.
a. 42 cars out of 124 cars ___________

Answer:
Fraction in simplest form of 42 cars out of 124 cars is \(\frac{21}{62}\).

Explanation:
42 cars out of 124 cars
= 42 ÷ 124
= 21 ÷ 62 or \(\frac{21}{62}\).

b. 7 messages out of 84 messages __________
Answer:
Fraction in simplest form of 7 messages out of 84 messages is \(\frac{1}{12}\).

Explanation:
7 messages out of 84 messages
= 7 ÷ 84
= 1 ÷ 12 or \(\frac{1}{12}\).

Ratios can be written based on the number of objects in a set.
There are 2 bottles of soda and 5 bottles of water in the refrigerator. Write the ratio of sodas to waters.
Answer:
The ratio of sodas to waters = \(\frac{2}{5}\).

Explanation:
Number of bottles of Soda = 2.
Number of bottles of water = 5.
Ratio of sodas to waters = Number of bottles of Soda ÷ Number of bottles of water
= 2 ÷ 5 or \(\frac{2}{5}\).

Express each ratio as a fraction in simplest form.

Question 1.
a. There are 2 cubes and 15 spheres in a geometry box. Write the ratio of spheres to cubes.
_____________
Answer:
Ratio of spheres to cubes = \(\frac{15}{2}\).

Explanation:
Number of cubes in a geometry box = 2.
Number of spheres in a geometry box = 15.
Ratio of spheres to cubes = Number of spheres in a geometry box ÷ Number of cubes in a geometry box
= 15 ÷ 2 or \(\frac{15}{2}\).

b. There are 5 cars and 4 vans in a parking lot. Write the ratio of vans to cars.
_____________
Answer:
Ratio of vans to cars = \(\frac{4}{5}\).

Explanation:
Number of cars in a parking lot = 5.
Number of vans in a parking lot = 4.
Ratio of vans to cars = Number of vans in a parking lot ÷ Number of cars in a parking lot
= 4 ÷ 5 or \(\frac{4}{5}\).

Question 2.
a. There are 5 horses and 15 elephants in a circus. Write the ratio of elephants to horses.
_____________
Answer:
Ratio of elephants to horses = 3.

Explanation:
Number of horses in a circus = 5.
Number of elephants in a circus = 15.
Ratio of elephants to horses = Number of elephants in a circus ÷ Number of horses in a circus
= 15 ÷ 5
= 3 ÷ 1
= 3.

b. There are 16 horses and 14 elephants in a circus. Write the ratio of horses to elephants.
_____________

Answer:
Ratio of horses to elephants = \(\frac{8}{7}\).

Explanation:
Number of horses in a circus = 16.
Number of elephants in a circus = 14.
Ratio of horses to elephants = Number of horses in a circus ÷ Number of elephants in a circus
= 16 ÷ 14
= 8 ÷ 7 or \(\frac{8}{7}\).

Question 3.
a. There are 11 blue marbles and 7 red marbles in a box. Write the ratio of red marbles to blue marbles.
_____________
Answer:
Ratio of red marbles to blue marbles = \(\frac{7}{11}\).

Explanation:
Number of blue marbles in a box = 11.
Number of red marbles in a box = 7.
Ratio of red marbles to blue marbles = Number of red marbles in a box ÷ Number of blue marbles in a box
= 7 ÷ 11 or \(\frac{7}{11}\).

b. There are 12 apples and 15 oranges in a fruit basket. Write the ratio of apples to oranges.
_____________
Answer:
Ratio of apples to oranges = \(\frac{4}{5}\).

Explanation:
Number of apples in a fruit basket = 12.
Number of oranges in a fruit basket = 15.
Ratio of apples to oranges = Number of apples in a fruit basket ÷ Number of oranges in a fruit basket
= 12 ÷ 15
= 4 ÷ 5 or \(\frac{4}{5}\).

Question 4.
a. There are 5 blue marbles and 16 red marbles in a box. Write the ratio of blue marbles to red marbles.
_____________
Answer:
Ratio of blue marbles to red marbles = \(\frac{5}{16}\).

Explanation:
Number of blue marbles in a box = 5.
Number of red marbles in a box = 16.
Ratio of blue marbles to red marbles = Number of blue marbles in a box ÷ Number of red marbles in a box
= 5 ÷ 16 or \(\frac{5}{16}\).

b. There are 12 dogs and 7 cats in a park. Write the ratio of cats to dogs.
_____________
Answer:
Ratio of cats to dogs = \(\frac{7}{12}\).

Explanation:
Number of dogs in a park = 12.
Number of cats in a park = 7.
Ratio of cats to dogs = Number of cats in a park ÷ Number of dogs in a park
= 7 ÷ 12 or \(\frac{7}{12}\).

Question 5.
a. There are 14 cars and 7 vans in a parking lot. Write the ratio of cars to vans.
_____________
Answer:
Ratio of cars to vans = 2.

Explanation:
Number of cars in a parking lot = 14.
Number of vans in a parking lot = 7.
Ratio of cars to vans = Number of cars in a parking lot ÷ Number of vans in a parking lot
= 14 ÷ 7
= 2 ÷ 1
= 2.

b. There are 7 blue marbles and 8 red marbles in a bag. Write the ratio of red marbles to blue marbles.

_____________
Answer:
Ratio of red marbles to blue marbles = \(\frac{8}{7}\) or 1\(\frac{1}{7}\).

Explanation:
Number of blue marbles in a bag = 7.
Number of red marbles in a bag = 8.
Ratio of red marbles to blue marbles = Number of red marbles in a bag ÷ Number of blue marbles in a bag
= 8 ÷ 7
= \(\frac{8}{7}\) or 1\(\frac{1}{7}\).

Question 6.
a. There are 6 pennies and 10 dimes in a jar. Write the ratio of pennies to dimes.
_____________
Answer:
Ratio of pennies to dimes = \(\frac{3}{5}\).

Explanation:
Number of pennies in a jar = 6.
Number of dimes in a jar = 10.
Ratio of pennies to dimes = Number of pennies in a jar ÷ Number of dimes in a jar
= 6 ÷ 10
= 3 ÷ 5 or \(\frac{3}{5}\).

b. There are 24 butterflies and 16 snails on the ground. Write the ratio of butterflies to snails.
_____________
Answer:
Ratio of butterflies to snails = \(\frac{3}{2}\) or 1\(\frac{1}{2}\).

Explanation:
Number of butteries on the ground = 24.
Number of snails on the ground = 16.
Ratio of butterflies to snails = Number of butteries on the ground ÷ Number of snails on the ground
= 24 ÷ 16
= 12 ÷ 8
= 3 ÷ 2
= \(\frac{3}{2}\) or 1\(\frac{1}{2}\).

Go through the Spectrum Math Grade 6 Answer Key Chapter 2 Lesson 2.3 Dividing Fractions and get the proper assistance needed during your homework.

Spectrum Math Grade 6 Chapter 2 Lesson 2.3 Dividing Fractions Answers Key

To divide, multiply by the reciprocal of the divisor.
\(\frac{4}{5}\) ÷ \(\frac{8}{9}\) = \(\frac{4}{5}\) × \(\frac{9}{8}\) = \(\frac{36}{40}\) = \(\frac{9}{10}\)

Divide. Write answers in simplest form.

Question 1.
a. \(\frac{1}{2}\) ÷ \(\frac{3}{5}\)
Answer:
Simplest form of \(\frac{1}{2}\) ÷ \(\frac{3}{5}\) is \(\frac{5}{6}\).

Explanation:
\(\frac{1}{2}\) ÷ \(\frac{3}{5}\)
= \(\frac{1}{2}\) × \(\frac{5}{3}\)
= \(\frac{5}{6}\)

b. \(\frac{3}{8}\) ÷ \(\frac{2}{3}\)
Answer:
Simplest form of \(\frac{3}{8}\) ÷ \(\frac{2}{3}\) is \(\frac{9}{16}\).

Explanation:
\(\frac{3}{8}\) ÷ \(\frac{2}{3}\)
= \(\frac{3}{8}\) × \(\frac{3}{2}\)
= \(\frac{9}{16}\)

c. \(\frac{5}{8}\) ÷ \(\frac{3}{4}\)
Answer:
Simplest form of \(\frac{5}{8}\) ÷ \(\frac{3}{4}\) is \(\frac{5}{6}\).

Explanation:
\(\frac{5}{8}\) ÷ \(\frac{3}{4}\)
= \(\frac{5}{8}\) ×  \(\frac{4}{3}\)
= \(\frac{5}{2}\) × \(\frac{1}{3}\)
= \(\frac{5}{6}\)

d. \(\frac{2}{5}\) ÷ \(\frac{3}{8}\)
Answer:
Simplest form of \(\frac{2}{5}\) ÷ \(\frac{3}{8}\) is \(\frac{16}{15}\) or 1\(\frac{1}{15}\).

Explanation:
\(\frac{2}{5}\) ÷ \(\frac{3}{8}\)
= \(\frac{2}{5}\) × \(\frac{8}{3}\)
= \(\frac{16}{15}\) or 1\(\frac{1}{15}\)

Question 2.
a. \(\frac{1}{2}\) ÷ \(\frac{7}{8}\)
Answer:
Simplest form of \(\frac{1}{2}\) ÷ \(\frac{7}{8}\) is \(\frac{4}{7}\)

Explanation:
\(\frac{1}{2}\) ÷ \(\frac{7}{8}\)
= \(\frac{1}{2}\) ×  \(\frac{8}{7}\)
= \(\frac{1}{1}\) ×  \(\frac{4}{7}\)
= \(\frac{4}{7}\)

b. \(\frac{4}{5}\) ÷ \(\frac{3}{4}\)
Answer:
Simplest form of \(\frac{4}{5}\) ÷ \(\frac{3}{4}\) is \(\frac{16}{15}\) or 1\(\frac{1}{15}\).

Explanation:
\(\frac{4}{5}\) ÷ \(\frac{3}{4}\)
= \(\frac{4}{5}\) × \(\frac{4}{3}\)
= \(\frac{16}{15}\) or 1\(\frac{1}{15}\)

c. \(\frac{5}{6}\) ÷ \(\frac{3}{8}\)
Answer:
Simplest form of \(\frac{5}{6}\) ÷ \(\frac{3}{8}\) is \(\frac{20}{9}\) or 2\(\frac{2}{9}\).

Explanation:
\(\frac{5}{6}\) ÷ \(\frac{3}{8}\)
= \(\frac{5}{6}\) × \(\frac{8}{3}\)
= \(\frac{5}{3}\) × \(\frac{4}{3}\)
= \(\frac{20}{9}\) or 2\(\frac{2}{9}\)

d. \(\frac{2}{3}\) ÷ \(\frac{4}{5}\)
Answer:
Simplest form of \(\frac{2}{3}\) ÷ \(\frac{4}{5}\) is \(\frac{5}{6}\).

Explanation:
\(\frac{2}{3}\) ÷ \(\frac{4}{5}\)
= \(\frac{2}{3}\) × \(\frac{5}{4}\)
= \(\frac{1}{3}\) × \(\frac{5}{2}\)
= \(\frac{5}{6}\)

Question 3.
a. \(\frac{7}{8}\) ÷ \(\frac{1}{3}\)
Answer:
Simplest form of \(\frac{7}{8}\) ÷ \(\frac{1}{3}\) is \(\frac{21}{8}\) or 2\(\frac{7}{8}\).

Explanation:
\(\frac{7}{8}\) ÷ \(\frac{1}{3}\)
= \(\frac{7}{8}\) × \(\frac{3}{1}\)
= \(\frac{21}{8}\) or 2\(\frac{7}{8}\)

b. \(\frac{7}{9}\) ÷ \(\frac{2}{3}\)
Answer:
Simplest form of \(\frac{7}{9}\) ÷ \(\frac{2}{3}\) is \(\frac{7}{6}\) or 1\(\frac{1}{6}\).

Explanation:
\(\frac{7}{9}\) ÷ \(\frac{2}{3}\)
= \(\frac{7}{9}\) × \(\frac{3}{2}\)
= \(\frac{7}{3}\) × \(\frac{1}{2}\)
= \(\frac{7}{6}\) or 1\(\frac{1}{6}\)

c. \(\frac{1}{3}\) ÷ \(\frac{2}{3}\)
Answer:
Simplest form of \(\frac{1}{3}\) ÷ \(\frac{2}{3}\) is \(\frac{1}{2}\).

Explanation:
\(\frac{1}{3}\) ÷ \(\frac{2}{3}\)
= \(\frac{1}{3}\) × \(\frac{3}{2}\)
= \(\frac{1}{1}\) × \(\frac{1}{2}\)
= \(\frac{1}{2}\)

d. \(\frac{5}{6}\) ÷ \(\frac{1}{3}\)
Answer:
Simplest form of \(\frac{5}{6}\) ÷ \(\frac{1}{3}\) is \(\frac{5}{2}\) or 2\(\frac{1}{2}\).

Explanation:
\(\frac{5}{6}\) ÷ \(\frac{1}{3}\)
= \(\frac{5}{6}\) × \(\frac{3}{1}\)
= \(\frac{5}{2}\) × \(\frac{1}{1}\)
= \(\frac{5}{2}\) or 2\(\frac{1}{2}\)

Question 4.
a. \(\frac{3}{5}\) ÷ \(\frac{2}{3}\)
Answer:
Simplest form of \(\frac{3}{5}\) ÷ \(\frac{2}{3}\) is \(\frac{9}{10}\).

Explanation:
\(\frac{3}{5}\) ÷ \(\frac{2}{3}\)
= \(\frac{3}{5}\) × \(\frac{3}{2}\)
= \(\frac{9}{10}\)

b. \(\frac{4}{9}\) ÷ \(\frac{3}{7}\)
Answer:
Simplest form of \(\frac{4}{9}\) ÷ \(\frac{3}{7}\) is \(\frac{28}{27}\) or 1\(\frac{1}{27}\).

Explanation:
\(\frac{4}{9}\) ÷ \(\frac{3}{7}\)
= \(\frac{4}{9}\) × \(\frac{7}{3}\)
= \(\frac{28}{27}\) or 1\(\frac{1}{27}\)

c. \(\frac{1}{2}\) ÷ \(\frac{5}{8}\)
Answer:
Simplest form of \(\frac{1}{2}\) ÷ \(\frac{5}{8}\) is \(\frac{4}{5}\).

Explanation:
\(\frac{1}{2}\) ÷ \(\frac{5}{8}\)
= \(\frac{1}{2}\) × \(\frac{8}{5}\)
= \(\frac{1}{1}\) × \(\frac{4}{5}\)
= \(\frac{4}{5}\)

d. \(\frac{2}{3}\) ÷ \(\frac{7}{9}\)
Answer:
Simplest form of \(\frac{2}{3}\) ÷ \(\frac{7}{9}\) is \(\frac{6}{7}\).

Explanation:
\(\frac{2}{3}\) ÷ \(\frac{7}{9}\)
= \(\frac{2}{3}\) × \(\frac{9}{7}\)
= \(\frac{2}{1}\) × \(\frac{3}{7}\)
= \(\frac{6}{7}\)

Divide. Write answers in simplest form.

Question 1.
a. \(\frac{3}{5}\) ÷ \(\frac{2}{7}\) =
Answer:
Simplest form of \(\frac{3}{5}\) ÷ \(\frac{2}{7}\) is \(\frac{21}{10}\) or 2\(\frac{1}{10}\).

Explanation:
\(\frac{3}{5}\) ÷ \(\frac{2}{7}\)
= \(\frac{3}{5}\) × \(\frac{7}{2}\)
= \(\frac{21}{10}\) or 2\(\frac{1}{10}\)

b. \(\frac{3}{4}\) ÷ \(\frac{1}{2}\) =
Answer:
Simplest form of \(\frac{3}{4}\) ÷ \(\frac{1}{2}\) is \(\frac{3}{2}\) or 1\(\frac{1}{2}\).

Explanation:
\(\frac{3}{4}\) ÷ \(\frac{1}{2}\)
= \(\frac{3}{4}\) × \(\frac{2}{1}\)
= \(\frac{3}{2}\) × \(\frac{1}{1}\)
= \(\frac{3}{2}\) or 1\(\frac{1}{2}\)

c. \(\frac{5}{8}\) ÷ \(\frac{3}{5}\) =
Answer:
Simplest form of \(\frac{5}{8}\) ÷ \(\frac{3}{5}\) is \(\frac{25}{24}\) or 1\(\frac{1}{24}\).

Explanation:
\(\frac{5}{8}\) ÷ \(\frac{3}{5}\)
= \(\frac{5}{8}\) × \(\frac{5}{3}\)
= \(\frac{25}{24}\) or 1\(\frac{1}{24}\)

d. \(\frac{5}{6}\) ÷ \(\frac{1}{10}\) =
Answer:
Simplest form of \(\frac{5}{6}\) ÷ \(\frac{1}{10}\) is \(\frac{25}{3}\) or 8\(\frac{1}{3}\).

Explanation:
\(\frac{5}{6}\) ÷ \(\frac{1}{10}\)
= \(\frac{5}{6}\) × \(\frac{10}{1}\)
= \(\frac{5}{3}\) × \(\frac{5}{1}\)
= \(\frac{25}{3}\) or 8\(\frac{1}{3}\)

Question 2.
a. \(\frac{1}{5}\) ÷ \(\frac{1}{4}\) =
Answer:
Simplest form of \(\frac{1}{5}\) ÷ \(\frac{1}{4}\) is \(\frac{4}{5}\).

Explanation:
\(\frac{1}{5}\) ÷ \(\frac{1}{4}\)
= \(\frac{1}{5}\) × \(\frac{4}{1}\)
= \(\frac{4}{5}\)

b. \(\frac{1}{2}\) ÷ \(\frac{2}{3}\) =
Answer:
Simplest form of \(\frac{1}{2}\) ÷ \(\frac{2}{3}\) is \(\frac{3}{4}\).

Explanation:
\(\frac{1}{2}\) ÷ \(\frac{2}{3}\)
= \(\frac{1}{2}\) × \(\frac{3}{2}\)
= \(\frac{3}{4}\)

c. \(\frac{6}{7}\) ÷ \(\frac{1}{8}\) =
Answer:
Simplest form of \(\frac{6}{7}\) ÷ \(\frac{1}{8}\) is \(\frac{48}{7}\) or 6\(\frac{6}{7}\).

Explanation:
\(\frac{6}{7}\) ÷ \(\frac{1}{8}\)
= \(\frac{6}{7}\) × \(\frac{8}{1}\)
= \(\frac{48}{7}\) or 6\(\frac{6}{7}\)

d. \(\frac{1}{4}\) ÷ \(\frac{1}{2}\) =
Answer:
Simplest form of \(\frac{1}{4}\) ÷ \(\frac{1}{2}\) is \(\frac{1}{2}\).

Explanation:
\(\frac{1}{4}\) ÷ \(\frac{1}{2}\)
= \(\frac{1}{4}\) × \(\frac{2}{1}\)
= \(\frac{1}{2}\) × \(\frac{1}{1}\)
= \(\frac{1}{2}\)

Question 3.
a. \(\frac{7}{10}\) ÷ \(\frac{1}{4}\) =
Answer:
Simplest form of \(\frac{7}{10}\) ÷ \(\frac{1}{4}\) is \(\frac{14}{5}\) or 2\(\frac{4}{5}\).

Explanation:
\(\frac{7}{10}\) ÷ \(\frac{1}{4}\)
= \(\frac{7}{10}\) × \(\frac{4}{1}\)
= \(\frac{7}{5}\) × \(\frac{2}{1}\)
= \(\frac{14}{5}\) or 2\(\frac{4}{5}\)

b. \(\frac{1}{2}\) ÷ \(\frac{6}{11}\) =
Answer:
Simplest form of \(\frac{1}{2}\) ÷ \(\frac{6}{11}\) is \(\frac{11}{12}\).

Explanation:
\(\frac{1}{2}\) ÷ \(\frac{6}{11}\)
= \(\frac{1}{2}\) × \(\frac{11}{6}\)
= \(\frac{1}{2}\) × \(\frac{11}{6}\)
= \(\frac{11}{12}\)

c. \(\frac{3}{5}\) ÷ \(\frac{1}{3}\) =
Answer:
Simplest form of \(\frac{3}{5}\) ÷ \(\frac{1}{3}\) is \(\frac{9}{5}\) or 1\(\frac{4}{5}\).

Explanation:
\(\frac{3}{5}\) ÷ \(\frac{1}{3}\)
= \(\frac{3}{5}\) × \(\frac{3}{1}\)
= \(\frac{9}{5}\) or 1\(\frac{4}{5}\)

d. \(\frac{1}{4}\) ÷ \(\frac{3}{8}\) =
Answer:
Simplest form of \(\frac{1}{4}\) ÷ \(\frac{3}{8}\) is \(\frac{2}{3}\).

Explanation:
\(\frac{1}{4}\) ÷ \(\frac{3}{8}\)
= \(\frac{1}{4}\) × \(\frac{8}{3}\)
= \(\frac{1}{1}\) × \(\frac{2}{3}\)
= \(\frac{2}{3}\)

Question 4.
a. \(\frac{10}{12}\) ÷ \(\frac{2}{7}\) =
Answer:
Simplest form of \(\frac{10}{12}\) ÷ \(\frac{2}{7}\) is \(\frac{35}{12}\) or 2\(\frac{11}{12}\).

Explanation:
\(\frac{10}{12}\) ÷ \(\frac{2}{7}\)
= \(\frac{10}{12}\) × \(\frac{7}{2}\)
= \(\frac{5}{12}\) × \(\frac{7}{1}\)
= \(\frac{35}{12}\) or 2\(\frac{11}{12}\)

b. \(\frac{1}{15}\) ÷ \(\frac{4}{5}\) =
Answer:
Simplest form of \(\frac{1}{15}\) ÷ \(\frac{4}{5}\) is \(\frac{1}{12}\).

Explanation:
\(\frac{1}{15}\) ÷ \(\frac{4}{5}\)
= \(\frac{1}{15}\) × \(\frac{5}{4}\)
= \(\frac{1}{3}\) × \(\frac{1}{4}\)
= \(\frac{1}{12}\)

c. \(\frac{12}{15}\) ÷ \(\frac{1}{4}\) =
Answer:
Simplest form of \(\frac{12}{15}\) ÷ \(\frac{1}{4}\) is \(\frac{48}{15}\) or 3\(\frac{3}{15}\).

Explanation:
\(\frac{12}{15}\) ÷ \(\frac{1}{4}\)
= \(\frac{12}{15}\) × \(\frac{4}{1}\)
= \(\frac{48}{15}\) or 3\(\frac{3}{15}\)

d. \(\frac{4}{5}\) ÷ \(\frac{9}{10}\) =
Answer:
Simplest form of \(\frac{4}{5}\) ÷ \(\frac{9}{10}\) is \(\frac{8}{9}\).

Explanation:
\(\frac{4}{5}\) ÷ \(\frac{9}{10}\)
= \(\frac{4}{5}\) × \(\frac{10}{9}\)
= \(\frac{4}{1}\) × \(\frac{2}{9}\)
= \(\frac{8}{9}\)

Question 5.
a. \(\frac{9}{10}\) ÷ \(\frac{2}{6}\) =
Answer:
Simplest form of \(\frac{9}{10}\) ÷ \(\frac{2}{6}\) is \(\frac{27}{10}\) or 2\(\frac{7}{10}\).

Explanation:
\(\frac{9}{10}\) ÷ \(\frac{2}{6}\)
= \(\frac{9}{10}\) × \(\frac{6}{2}\)
= \(\frac{9}{5}\) × \(\frac{3}{2}\)
= \(\frac{27}{10}\) or 2\(\frac{7}{10}\)

b. \(\frac{7}{15}\) ÷ \(\frac{8}{10}\) =
Answer:
Simplest form of \(\frac{7}{15}\) ÷ \(\frac{8}{10}\) is \(\frac{7}{12}\).

Explanation:
\(\frac{7}{15}\) ÷ \(\frac{8}{10}\)
= \(\frac{7}{15}\) × \(\frac{10}{8}\)
= \(\frac{7}{3}\) × \(\frac{2}{8}\)
= \(\frac{14}{24}\)
= \(\frac{7}{12}\)

c. \(\frac{2}{12}\) ÷ \(\frac{3}{4}\) =
Answer:
Simplest form of \(\frac{2}{12}\) ÷ \(\frac{3}{4}\) is

Explanation:
\(\frac{2}{12}\) ÷ \(\frac{3}{4}\)
= \(\frac{2}{12}\) × \(\frac{4}{3}\)
= \(\frac{2}{3}\) × \(\frac{1}{3}\)
= \(\frac{2}{9}\)

d.
\(\frac{7}{15}\) ÷ \(\frac{7}{9}\) =
Answer:
Simplest form of \(\frac{7}{15}\) ÷ \(\frac{7}{9}\) is \(\frac{3}{5}\).

Explanation:
\(\frac{7}{15}\) ÷ \(\frac{7}{9}\)
= \(\frac{7}{15}\) × \(\frac{9}{7}\)
= \(\frac{1}{15}\) × \(\frac{9}{1}\)
= \(\frac{1}{5}\) × \(\frac{3}{1}\)
= \(\frac{3}{5}\)

Divide. Write answers in simplest form.

Question 1.
a.
\(\frac{4}{9}\) ÷ \(\frac{1}{3}\) =
Answer:
Simplest form of \(\frac{4}{9}\) ÷ \(\frac{1}{3}\) is \(\frac{4}{3}\) or 1\(\frac{1}{3}\).

Explanation:
\(\frac{4}{9}\) ÷ \(\frac{1}{3}\)
= \(\frac{4}{9}\) × \(\frac{3}{1}\)
= \(\frac{4}{3}\) × \(\frac{1}{1}\)
= \(\frac{4}{3}\) or 1\(\frac{1}{3}\)

b.
12 ÷ \(\frac{1}{5}\) =
Answer:
Simplest form of 12 ÷ \(\frac{1}{5}\) is 60.

Explanation:
12 ÷ \(\frac{1}{5}\)
= 12 × \(\frac{5}{1}\)
= 60.

c.
2 ÷ \(\frac{2}{3}\) =
Answer:
Simplest form of 2 ÷ \(\frac{2}{3}\) is 3.

Explanation:
2 ÷ \(\frac{2}{3}\)
= 2 × \(\frac{3}{2}\)
= 1 × \(\frac{3}{1}\)
= \(\frac{3}{1}\)
= 3.

d.
\(\frac{1}{5}\) ÷ \(\frac{1}{3}\) =
Answer:
Simplest form of \(\frac{1}{5}\) ÷ \(\frac{1}{3}\) is \(\frac{3}{5}\).

Explanation:
\(\frac{1}{5}\) ÷ \(\frac{1}{3}\)
= \(\frac{1}{5}\) × \(\frac{3}{1}\)
= \(\frac{3}{5}\)

Question 2.
a.
\(\frac{1}{7}\) ÷ \(\frac{3}{5}\) =
Answer:
Simplest form of \(\frac{1}{7}\) ÷ \(\frac{3}{5}\) is \(\frac{5}{21}\).

Explanation:
\(\frac{1}{7}\) ÷ \(\frac{3}{5}\)
= \(\frac{1}{7}\) × \(\frac{5}{3}\)
= \(\frac{5}{21}\)

b.
\(\frac{2}{3}\) ÷ \(\frac{3}{4}\) =
Answer:
Simplest form of \(\frac{2}{3}\) ÷ \(\frac{3}{4}\) is \(\frac{8}{9}\).

Explanation:
\(\frac{2}{3}\) ÷ \(\frac{3}{4}\)
= \(\frac{2}{3}\) × \(\frac{4}{3}\)
= \(\frac{8}{9}\)

c.
5 ÷ \(\frac{2}{3}\) =
Answer:
Simplest form of 5 ÷ \(\frac{2}{3}\) is \(\frac{15}{2}\) or 7\(\frac{1}{2}\).

Explanation:
5 ÷ \(\frac{2}{3}\)
= 5 × \(\frac{3}{2}\)
= \(\frac{15}{2}\) or 7\(\frac{1}{2}\)

d.
\(\frac{2}{3}\) ÷ \(\frac{1}{9}\) =
Answer:
Simplest form of \(\frac{2}{3}\) ÷ \(\frac{1}{9}\) is 6.

Explanation:
\(\frac{2}{3}\) ÷ \(\frac{1}{9}\)
= \(\frac{2}{3}\) × \(\frac{9}{1}\)
= \(\frac{2}{1}\) × \(\frac{3}{1}\)
= \(\frac{6}{1}\)
= 6.

Question 3.
a.
\(\frac{7}{8}\) ÷ 4 =
Answer:
Simplest form of \(\frac{7}{8}\) ÷ 4 is \(\frac{7}{32}\).

Explanation:
\(\frac{7}{8}\) ÷ 4
= \(\frac{7}{8}\) × \(\frac{1}{4}\)
= \(\frac{7}{32}\)

b.
\(\frac{2}{15}\) ÷ \(\frac{15}{17}\) =
Answer:
Simplest form of \(\frac{2}{15}\) ÷ \(\frac{15}{17}\) is \(\frac{34}{225}\).

Explanation:
\(\frac{2}{15}\) ÷ \(\frac{15}{17}\)
= \(\frac{2}{15}\) × \(\frac{17}{15}\)
= \(\frac{34}{225}\)

c.
\(\frac{3}{8}\) ÷ \(\frac{2}{3}\) =
Answer:
Simplest form of \(\frac{3}{8}\) ÷ \(\frac{2}{3}\) is \(\frac{9}{16}\).

Explanation:
\(\frac{3}{8}\) ÷ \(\frac{2}{3}\)
= \(\frac{3}{8}\) × \(\frac{3}{2}\)
= \(\frac{9}{16}\)

d.
\(\frac{3}{11}\) ÷ \(\frac{17}{23}\) =
Answer:
Simplest form of \(\frac{3}{11}\) ÷ \(\frac{17}{23}\) is \(\frac{69}{187}\).

Explanation:
\(\frac{3}{11}\) ÷ \(\frac{17}{23}\)
= \(\frac{3}{11}\) × \(\frac{23}{17}\)
= \(\frac{69}{187}\)

Question 4.
a.
\(\frac{4}{11}\) ÷ \(\frac{2}{3}\) =
Answer:
Simplest form of \(\frac{4}{11}\) ÷ \(\frac{2}{3}\) is \(\frac{6}{11}\).

Explanation:
\(\frac{4}{11}\) ÷ \(\frac{2}{3}\)
= \(\frac{4}{11}\) × \(\frac{3}{2}\)
= \(\frac{2}{11}\) × \(\frac{3}{1}\)
= \(\frac{6}{11}\)

b.
\(\frac{1}{11}\) ÷ \(\frac{5}{7}\) =
Answer:
Simplest form of \(\frac{1}{11}\) ÷ \(\frac{5}{7}\) is \(\frac{7}{55}\).

Explanation:
\(\frac{1}{11}\) ÷ \(\frac{5}{7}\)
= \(\frac{1}{11}\) × \(\frac{7}{5}\)
= \(\frac{7}{55}\)

c.
\(\frac{9}{20}\) ÷ \(\frac{9}{17}\) =
Answer:
Simplest form of \(\frac{9}{20}\) ÷ \(\frac{9}{17}\) is \(\frac{17}{20}\).

Explanation:
\(\frac{9}{20}\) ÷ \(\frac{9}{17}\)
= \(\frac{9}{20}\) × \(\frac{17}{9}\)
= \(\frac{1}{20}\) × \(\frac{17}{1}\)
= \(\frac{17}{20}\)

d.
\(\frac{3}{7}\) ÷ \(\frac{9}{20}\) =
Answer:
Simplest form of \(\frac{3}{7}\) ÷ \(\frac{9}{20}\) is \(\frac{20}{21}\).

Explanation:
\(\frac{3}{7}\) ÷ \(\frac{9}{20}\)
= \(\frac{3}{7}\) × \(\frac{20}{9}\)
= \(\frac{1}{7}\) × \(\frac{20}{3}\)
= \(\frac{20}{21}\)

Question 5.
a.
\(\frac{2}{3}\) ÷ \(\frac{10}{11}\) =
Answer:
Simplest form of \(\frac{2}{3}\) ÷ \(\frac{10}{11}\) is \(\frac{11}{15}\).

Explanation:
\(\frac{2}{3}\) ÷ \(\frac{10}{11}\)
= \(\frac{2}{3}\) × \(\frac{11}{10}\)
= \(\frac{1}{3}\) × \(\frac{11}{5}\)
= \(\frac{11}{15}\)

b.
\(\frac{1}{13}\) ÷ \(\frac{3}{13}\) =
Answer:
Simplest form of \(\frac{1}{13}\) ÷ \(\frac{3}{13}\) is \(\frac{1}{3}\).

Explanation:
\(\frac{1}{13}\) ÷ \(\frac{3}{13}\)
= \(\frac{1}{13}\) × \(\frac{13}{3}\)
= \(\frac{1}{1}\) × \(\frac{1}{3}\)
= \(\frac{1}{3}\)

c.
\(\frac{6}{11}\) ÷ \(\frac{5}{7}\) =
Answer:
Simplest form of \(\frac{6}{11}\) ÷ \(\frac{5}{7}\) is \(\frac{42}{55}\).

Explanation:
\(\frac{6}{11}\) ÷ \(\frac{5}{7}\)
= \(\frac{6}{11}\) × \(\frac{7}{5}\)
= \(\frac{42}{55}\)

d.
\(\frac{1}{4}\) ÷ \(\frac{6}{17}\) =
Answer:
Simplest form of \(\frac{1}{4}\) ÷ \(\frac{6}{17}\) is \(\frac{17}{24}\).

Explanation:
\(\frac{1}{4}\) ÷ \(\frac{6}{17}\)
= \(\frac{1}{4}\) × \(\frac{17}{6}\)
= \(\frac{17}{24}\)

This handy Spectrum Math Grade 7 Answer Key Chapter 2 Lesson 2.9 Problem Solving provides detailed answers for the workbook questions.

Spectrum Math Grade 7 Chapter 2 Lesson 2.9 Problem Solving Answers Key

Solve each problem. Write each answer in simplest form.

Question 1.
David worked 7\(\frac{1}{3}\) hours today and planted 11 trees. It takes him about the same amount of time to plant each tree. How long did it take him to plant each tree?
It took him _____________ hour to plant each tree.
Answer: \(\frac{2}{3}\)
Number of trees planted by David = 11
Number of hours worked by David to plant 11 trees = 7\(\frac{1}{3}\) hours
Convert the above numbers in improper fractions to make calculations easy.
7\(\frac{1}{3}\) = \(\frac{22}{3}\)
It takes him about the same amount of time to plant each tree.
Let time taken to plan 1 tree be x
then, time taken to plan 1 ×11 tree = 11 ×x
But, given that total time taken is \(\frac{22}{3}\) hours
thus,11x = 22/3 hours
x = \(\frac{22}{3 × 11}\) = \(\frac{2}{3}\)
Thus, it took \(\frac{2}{3}\) hours to plant each tree.
As 1 hour has 60 minutes
Time taken in minutes to plant 1 tree = \(\frac{2}{3}\)  × 60 mins = 40 mins
So, It took him \(\frac{2}{3}\) hour to plant each tree.

Question 2.
A car uses 3\(\frac{1}{8}\) gallons of gasoline per hour when driving on the highway. How many gallons will it use after 4\(\frac{2}{3}\) hours?
It will use ____ gallons.
Answer: 14\(\frac{7}{12}\)
A car uses 3\(\frac{1}{8}\) gallons of gasoline per hour when driving on the highway.
The trip is for 4\(\frac{2}{3}\) hours
Gallons per trip  = Gallons per hour × hours per trip
So, 3\(\frac{1}{8}\)  × 4\(\frac{2}{3}\)
Convert the above numbers in improper fractions to make calculations easy.
= \(\frac{25}{8}\)  × \(\frac{14}{3}\)
Reduce the above fractions into simplest form if possible. In this case, it is possible.
Then, multiply the numerators and denominators separately.
= \(\frac{25 × 14}{8 × 3}\)
Divide the 14 in numerator and 8 in denominator by 2, which is a common factor.
= \(\frac{25 × 7}{4 × 3}\)
= \(\frac{175}{12}\)
= 14\(\frac{7}{12}\)
It will use 14\(\frac{7}{12}\) gallons.

Question 3.
A board was 24\(\frac{3}{8}\) inches long. A worker cut it into pieces that were 4\(\frac{7}{8}\) inches long. The worker cut the board into how many pieces?
The worker cut the board into ____ pieces.
Answer: 5 pieces
A board was 24\(\frac{3}{8}\) inches long.
A worker cut it into pieces that were 4\(\frac{7}{8}\) inches long.
Number of pieces the worker  cut the board = 24\(\frac{3}{8}\) ÷ 4\(\frac{7}{8}\)
Convert the above numbers in improper fractions to make calculations easy.
= \(\frac{195}{8}\) ÷ \(\frac{39}{8}\)
To divide by a fraction, multiply by its reciprocal.
Here, the reciprocal of \(\frac{39}{8}\) is \(\frac{8}{39}\)
So, \(\frac{195}{8}\) × \(\frac{8}{39}\)
Reduce the above fractions into simplest form if possible. In this case, it is possible.
Then, multiply the numerators and denominators separately.
= \(\frac{195 × 8}{8 × 39}\)
Divide the 8 in numerator and 8 in denominator by 8, which is a common factor.
= \(\frac{195 × 1}{1 × 39}\)
now, Divide the 195 in numerator and 39 in denominator by 5, which is a common factor.
= \(\frac{5 × 1}{1 × 1}\)
= \(\frac{5}{1}\)
= 5
The worker cut the board into 5 pieces.

Question 4.
Susan must pour 6\(\frac{1}{2}\) bottles of juice into 26 drink glasses for her party. If each glass gets the same amount of juicer, what fraction of a bottle will each glass hold?

Each glass will hold ___________ bottles.
Answer: \(\frac{1}{4}\)
Susan must pour 6\(\frac{1}{2}\) bottles of juice into 26 drink glasses for her party.
6\(\frac{1}{2}\)  can be converted into improper fraction, which is \(\frac{13}{2}\)
The capacity of each glass = 6\(\frac{1}{2}\) ÷ 26
= \(\frac{13}{2}\)  ÷ 26
To divide by a fraction, multiply by its reciprocal.
Here, the reciprocal of \(\frac{26}{1}\) = \(\frac{1}{26}\)
= \(\frac{13}{2}\)  × \(\frac{1}{26}\)
Reduce the above fractions into simplest form if possible. In this case, it is possible.
Then, multiply the numerators and denominators separately.
= \(\frac{13 × 1}{2 × 26}\)
Divide the 13 in numerator and 26 in denominator by 13, which is a common factor.
= \(\frac{1 × 1}{1 × 2}\)
= \(\frac{1}{4}\)
Therefore, Each glass will hold \(\frac{1}{4}\) bottles.

Question 5.
The standard size of a certain bin holds 2\(\frac{2}{3}\) gallons. The large size of that bin is 1\(\frac{1}{4}\) if times larger. How many gallons does the large bin hold?
The large bin holds ____ gallons.
Answer: \(\frac{10}{3}\)=3\(\frac{1}{3}\)
The standard size of a certain bin holds 2\(\frac{2}{3}\) gallons. The large size of that bin is 1\(\frac{1}{4}\) if times larger.
The capacity of large bin = 2\(\frac{2}{3}\) × 1\(\frac{1}{4}\)
Convert the above numbers in improper fractions to make calculations easy.
= \(\frac{8}{3}\) × \(\frac{5}{4}\)
Reduce the above fractions into simplest form if possible. In this case, it is possible.
Then, multiply the numerators and denominators separately.
= \(\frac{8 × 5}{3 × 4}\)
Divide the 8 in numerator and 4 in denominator by 4, which is a common factor.
= \(\frac{2 × 5}{3 × 1}\)
= \(\frac{10}{3}\)
=3\(\frac{1}{3}\)
The large bin holds \(\frac{10}{3}\)=3\(\frac{1}{3}\) gallons.

Question 6.
Diana has 3\(\frac{1}{4}\) bags of nuts. Each bag holds 4\(\frac{1}{2}\) pounds. How many pounds of nuts does Diana have?
Diana has ____ pounds of nuts.
Answer: \(\frac{117}{8}\) = 14\(\frac{5}{8}\)
Diana has 3\(\frac{1}{4}\) bags of nuts.
Each bag holds × pounds.
Total number of pounds of nuts Diana has = 3\(\frac{1}{4}\) × 4\(\frac{1}{2}\)
Convert the above numbers in improper fractions to make calculations easy.
= \(\frac{13}{4}\) × \(\frac{9}{2}\)
Reduce the above fractions into simplest form if possible. In this case, it is not possible.
Then, multiply the numerators and denominators separately.
= \(\frac{13 × 9}{4 × 2}\)
= \(\frac{117}{8}\)
= 14\(\frac{5}{8}\)
Therefore, Diana has 14\(\frac{5}{8}\) pounds of nuts.

Question 7.
There is a stack of 7 crates. Each crate is 10\(\frac{2}{3}\) inches high. How many inches high is the stack of crates?
The stack of crates is ___________ inches high.
Answer: 74\(\frac{2}{3}\)
There is a stack of 7 crates. Each crate is 10\(\frac{2}{3}\) inches high.
The number of inches high is the stack of crates = 7 × 10\(\frac{2}{3}\)
Convert the above numbers in improper fractions to make calculations easy.
= \(\frac{7}{1}\) × \(\frac{32}{3}\)
Reduce the above fractions into simplest form if possible. In this case, it is not possible.
Then, multiply the numerators and denominators separately.
= \(\frac{7 × 32}{1 × 3}\)
= \(\frac{224}{3}\)
= 74\(\frac{2}{3}\)
The stack of crates is 74\(\frac{2}{3}\) inches high.

Solve each problem. Write each answer in simplest form.

Question 1.
Each month, Kelsey donates \(\frac{1}{5}\) of her allowance to her school for supplies. \(\frac{1}{2}\) of that amount goes to the chorus class. How much of her allowance goes to supplies for the chorus class?
____________ of her allowance goes to help the chorus classes.
Answer: \(\frac{1}{10}\)
Each month, Kelsey donates \(\frac{1}{5}\) of her allowance to her school for supplies. \(\frac{1}{2}\) of that amount goes to the chorus class.
The amount of her allowance that goes to supplies for the chorus class = \(\frac{1}{5}\) × \(\frac{1}{2}\)
Reduce the above fractions into simplest form if possible. In this case, it is not possible.
Then, multiply the numerators and denominators separately.
= \(\frac{1 × 1}{5 × 2}\)
= \(\frac{1}{10}\)
Therefore, \(\frac{1}{10}\) of her allowance goes to help the chorus classes.

Question 2.
Alvin cuts \(\frac{3}{4}\) of a piece of cheese. He gives \(\frac{1}{8}\) of it to Mall. How much of the cheese does Alvin give to Mall?
Alvin gives ____________ of the cheese to Mall.
Answer: \(\frac{3}{32}\)
Alvin cuts \(\frac{3}{4}\) of a piece of cheese. He gives \(\frac{1}{8}\) of it to Mall.
The amount of chees that Alvin gave to Mall = \(\frac{3}{4}\) × \(\frac{1}{8}\)
Reduce the above fractions into simplest form if possible.
Then, multiply the numerators and denominators separately.
= \(\frac{3 × 1}{4 × 8}\)
= \(\frac{3}{32}\)
Therefore, Alvin gives \(\frac{3}{32}\) of the cheese to Mall.

Question 3.
Kati has 16\(\frac{3}{4}\) hours to finish 3 school projects. How much time may she spend on each project, if she plans to spend the same amount of time on each?
Katie will spend ____________________ hours on each project.
Answer: 5\(\frac{7}{12}\)
Kati has 16\(\frac{3}{4}\) hours to finish 3 school projects.
If she plans to spend the same amount of time on each, the number of hours she spend on each project = 16\(\frac{3}{4}\) ÷  3
Convert the above number into improper fractions to make the calculations easy.
16\(\frac{3}{4}\) ÷ 3 = \(\frac{67}{4}\) ÷ \(\frac{3}{1}\)
To divide by a fraction, multiply by its reciprocal.
Here, the reciprocal of \(\frac{3}{1}\) is \(\frac{1}{3}\)
So, \(\frac{67}{4}\) × \(\frac{1}{3}\)
Reduce the above fractions into simplest form if possible.
Then, multiply the numerators and denominators separately.
= \(\frac{67 × 1}{4 × 3}\)
= \(\frac{67}{12}\)
= 5\(\frac{7}{12}\)
Therefore, Katie will spend 5\(\frac{7}{12}\) hours on each project.

Question 4.
Martha spent $2.90 on 3\(\frac{1}{2}\) pounds of bananas. How much did she spend on each pound of bananas?

She spent ____________________ on each pound.
Answer: $0.825
Martha spent $2.90 on 3\(\frac{1}{2}\) pounds of bananas.
The amount she spent on each pound of bananas = $2.90 ÷ 3\(\frac{1}{2}\)
Converting 3\(\frac{1}{2}\) into improper fraction to make the calculations easy.
3\(\frac{1}{2}\) = \(\frac{7}{2}\)
So, $2.90 ÷ \(\frac{7}{2}\)
To divide by a fraction, multiply by its reciprocal.
Here, the reciprocal of \(\frac{7}{2}\) is \(\frac{2}{7}\)
So, $2.90× \(\frac{2}{7}\)
Reduce the above fractions into simplest form if possible.
Then, multiply the numerators and denominators separately.
= \(\frac{2.90 × 2}{7}\)
= $0.825
Therefore, She spent $0.825 on each pound.

Question 5.
Monica has 5\(\frac{1}{2}\) cups of sugar to make pies. If each pie uses cup of sugar, how many pies can Monica make?
Monica can make ____ pies.
Answer: 5
Monica has 5\(\frac{1}{2}\) cups of sugar to make pies.
If each pie uses cup of sugar, the number of pies she can make  = 5\(\frac{1}{2}\) ÷ 1 = 5\(\frac{1}{2}\)
As we cannot make \(\frac{1}{2}\) pie, Monica can make 5 pies.

Question 6.
Vince has 12\(\frac{1}{2}\) hours to mow the lawn, do the laundry, make dinner, and finish his homework. How much time can Vince spend on each task, if he plans to spend the same amount of time on each?
Vince will spend ___________________ hours on éach project.
Answer: 3\(\frac{1}{8}\)
Vince has 12\(\frac{1}{2}\) hours to mow the lawn, do the laundry, make dinner, and finish his homework.
So, number of projects he has to do are 4
If he plans to spend the same amount of time on each, the number of hours he spends on each project = 12\(\frac{1}{2}\) ÷ 4
Convert the above number into improper fractions to make the calculations easy.
12\(\frac{1}{2}\) ÷ 4 = \(\frac{25}{2}\) ÷ \(\frac{4}{1}\)
To divide by a fraction, multiply by its reciprocal.
Here, the reciprocal of \(\frac{4}{1}\) is \(\frac{1}{4}\)
So, \(\frac{25}{2}\) × \(\frac{1}{4}\)
Reduce the above fractions into simplest form if possible.
Then, multiply the numerators and denominators separately.
= \(\frac{25  ×1}{2 × 4}\)
= \(\frac{25}{8}\)
= 3\(\frac{1}{8}\)
Therefore, Vince will spend 3\(\frac{1}{8}\) hours on éach project.

Question 7.
Drew spent $38.97 on 3\(\frac{1}{4}\) pounds of shrimp. How much did he spend on each pound of shrimp?
Drew spent _____ on each pound of shrimp.
Answer: $11.99
Drew spent $38.97 on 3\(\frac{1}{4}\) pounds of shrimp.
The amount Drew spent on each pound = $38.97 ÷ 3\(\frac{1}{4}\)
Converting 3\(\frac{1}{4}\) into improper fraction to make the calculations easy.
3\(\frac{1}{4}\) = \(\frac{13}{4}\)
So, $38.97 ÷ \(\frac{13}{4}\)
To divide by a fraction, multiply by its reciprocal.
Here, the reciprocal of \(\frac{13}{4}\) is \(\frac{4}{13}\)
So, $38.97× \(\frac{4}{13}\)
Reduce the above fractions into simplest form if possible.
Then, multiply the numerators and denominators separately.
= \(\frac{38.97 × 4}{13}\)
= $11.99
Therefore, Drew spent $11.99 on each pound of shrimp.

This handy Spectrum Math Grade 7 Answer Key Chapter 3 Posttest provides detailed answers for the workbook questions

Spectrum Math Grade 7 Chapter 3 Posttest Answers Key

Check What You Learned

Expressions, Equations, and Inequalities

Rewrite each expression using the property indicated.

Question 1.
a. commutative: 4 × 5
______________
Answer: 5 × 4
According to the commutative property of multiplication, changing the order of the numbers we are multiplying does not change the product. If there are two numbers, x and y, the commutative property of multiplication implies that x × y = y × x.
4 × 5 = 5 × 4
20 = 20

b. distributive: 6 × (8 – 5)
________________
Answer: (6 × 8) – (6 × 5)
The distributive property states that multiplying the sum of two or more addends by a number yields the same outcome as multiplying each addend separately by the number and combining the resulting products. a × (b – c) = (a × b) – (a × c) is the rule for distributive property.
6 × (8 – 5) = (6 × 8) – (6 × 5)
18 =18

Question 2.
a. associative: (12 × 7) × 8
_____________
Answer: 12 × (7 × 8)
According to the associative principle of addition, when adding three integers, the outcome will always be the same regardless of how the numbers are grouped. If there are three numbers, x, y and z, the associative property of addition implies that x × (y × z) = (x × y) × z. The grouping of addends does not change the sum.
(12 × 7) × 8 = 12 × (7 × 8)
672 = 672

b. associative: (3 + 4) + 5
_____________
Answer:
According to the associative principle of addition, when adding three integers, the outcome will always be the same regardless of how the numbers are grouped. If there are three numbers, x, y and z, the associative property of addition implies that x + (y + z) = (x + y) + z. The grouping of addends does not change the sum.
(3 + 4) + 5 = 3 + (4 + 5)
12 =12

Question 3.
a. identity: 32 × 1
___________
Answer:
According to the identity property of multiplication, if a number is multiplied by 1 (one), the result will be the original number. This property is applied when numbers are multiplied by 1. If there is a number, x then the identity property implies that x × 1 = x.
32 × 1 = 32

b. zero: 0 × 4
_______________
Answer: 0
According to the zero property of multiplication, if a number is multiplied by 0 (zero), the result will be zero. This property is applied when numbers are multiplied by 0. If there is a number, x then the identity property implies that x × 0 = 0.
0 × 4 = 0

Write each phrase as an expression or equation.

Question 4.
a. seven less than a number
______________
Answer: 7 < x
Let x be the number.
The expression for ‘seven less than a number’ can be given as 7 < x

b. eight more than a number
______________
Answer: 8 > x
Let x be the number.
The expression for ‘eight more than a number’ can be given as 8 > x

Question 5.
a. the product of six and a number
______________
Answer: 6 × x
Let x be the number.
The expression for ‘the product of six and a number’ can be given as 6 × x

b. a number divided by twelve
______________
Answer: x ÷ 12
Let x be the number.
The expression for ‘a number divided by twelve’ can be given as x ÷ 12

Question 6.
a. the product of 4 and a number is 16
______________
Answer:
Let x be the number.
The equation for ‘the product of 4 and a number is 16’ can be given as 4 ×  x = 16

b. nine more than a number is 11
______________
Answer: 9 > x = 11
Let x be the number.
The equation for ‘nine more than a number is 11’ can be given as 9 > x = 11

Question 7.
a. three less than a number is twenty
______________
Answer: 3 < x = 20
Let x be the number.
The equation for ‘three less than a number is twenty’ can be given as 3 < x = 20

b. twenty-five divided by a number is five
______________
Answer: 25 ÷ x = 5
Let x be the number.
The equation for ’twenty-five divided by a number is five′ can be given as 25 ÷ x = 5

Question 8.
a. a number divided by 10 is 11
______________
Answer: x ÷ 10 = 11
Let x be the number.
The equation for ’a number divided by 10 is 11′ can be given as x ÷ 10 = 11

b. the product of 5 and a number is 25
______________
Answer: 5 ×  x = 25
Let x be the number.
The equation for ‘the product of 5 and a number is 25’ can be given as 5 ×  x = 25

Question 9.
a. 1.2 more than a number
______________
Answer: 1.2 > x
Let x be the number.
The expression for ‘1.2 more than a number’ can be given as 1.2 > x

b. thirty-two divided by a number is 16
______________
Answer: 32 ÷ x = 16
Let x be the number.
The equation for ’thirty-two divided by a number is 16′ can be given as 32 ÷ x = 16

Question 10.
a. fifteen less than a number
______________
Answer: 15 < x
Let x be the number.
The expression for ‘fifteen less than a number’ can be given as 15 < x

b. 14 divided by a number is two
______________
Answer: 14 ÷ x = 2
Let x be the number.
The equation for ’14 divided by a number is two′ can be given as 14 ÷ x = 2

Solve each problem by creating an equation or inequality.

Question 11.
Yael bought two magazines for $5 and some erasers that cost $ 1.00 each. He could only spend $25. How many erasers could he buy?
Let e represent the number of erasers he was able to buy.

Equation or Inequality: ____________
Yael can buy _____ erasers.
Answer:
Inequality: 2 × $5 + e × $ 1.00 ≤ $25
Yael can buy 15 erasers.
Yael bought two magazines for $5 and some erasers that cost $ 1.00 each.
He could only spend $25.
Let e represent the number of erasers he was able to buy.
Inequality: 2 × $5 + e × $ 1.00 ≤ $25
10 + e ≤ 25
e ≤ 25 -10
e ≤ 15
Therefore, Yael can buy 15 erasers.

Question 12.
The sum of three consecutive numbers Is 75. What is the smallest of these numbers?
Let n represent the smallest number.
Equation or Inequality: ___________
_____ is the smallest number in the set.
Answer:
Equation: x + x+1+ x+2 = 75
24 is the smallest number in the set.
The sum of three consecutive numbers Is 75.
Let the numbers be x, x+1 and x+2.
Then the sum is : Equation: x + x+1+ x+2 = 75
3x + 3 = 75
3x = 75 – 3 = 72
x = 72 ÷ 3
x = 24
Therefore, The smallest of these numbers is 24

Question 13.
Summer won 40 super bouncy balls playing Skee Ball at her school’s fall festival. Later, she gave 3 to each of her friends. She only has 7 remaining. How many friends does she have?

Let f represent the number of friends.
Equation or Inequality: ______________
Summer shared with _____ friends.
Answer:
Equation: 40 – 3 × f = 7
Summer shared with 11 friends.
Summer won 40 super bouncy balls playing Skee Ball at her school’s fall festival.
Later, she gave 3 to each of her friends.
She only has 7 remaining.
Let f represent the number of friends.
Number of friends Summer has = 40 – 3 × f = 7
40 – 3f = 7
40 – 3f – 7 = 0
-3f + 33 = 0
-3f = -33
3f = 33
f = 33 ÷ 3
f = 11
Therefore, Summer shared with 11 friends.

Question 14.
Mrs. Watson had some candy to give to her students. She first took ten pieces for herself and then evenly divided the rest among her students. Each student received two pieces. If she started with 50 pieces of candy, how many students does
she teach?
Let s represent the number of students.
Equation or Inequality: ________________
Mrs. Watson teaches ______ students.
Answer:
Equation: 10 + 2 × s = 50
Mrs. Watson teaches 20 students.
Mrs. Watson had some candy to give to her students.
She first took ten pieces for herself and then evenly divided the rest among her students. Each student received two pieces.
She started with 50 pieces of candy.
Let s represent the number of students.
Therefore, number of students = 10 + 2 × s = 50
10 + 2s = 50
2s = 50 -10
2s = 40
s = 40 ÷  2
s = 20
so, Mrs. Watson teaches 20 students.

Question 15.
The Cooking Club made some cakes to sell at a baseball game to raise money for the school library. The cafeteria contributed 5 cakes to the sale. Each cake was then cut into 10 pieces and sold. There were a total of 80 pieces to sell. How many cakes did the club make?

Let c represent the number of cakes.
Equation or Inequality: ____________
The club made _____ cakes.
Answer:
Equation: 50 + c × 10 = 80
The club made 3 cakes.
The Cooking Club made some cakes to sell at a baseball game to raise money for the school library.
The cafeteria contributed 5 cakes to the sale.
Each cake was then cut into 10 pieces and sold.
Therefore, total pieces contributed by cafeteria = 5 × 10 = 50 pieces
There were a total of 80 pieces to sell.
Let c represent the number of cakes.
Number of cakes made by club = 50 + c × 10 = 80
50 + 10c = 80
10c = 80 – 50
10c = 30
c = 30 ÷ 10
c = 3
So, The cooking club made 3  cakes.

This handy Spectrum Math Grade 7 Answer Key Chapter 1 Lesson 1.5 Adding Integers provides detailed answers for the workbook questions.

Spectrum Math Grade 7 Chapter 1 Lesson 1.5 Adding Integers Answers Key

The sum of two positive integers is a positive integer.

The sum of two negative integers is a negative integer.

To find the sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.

Add.

Question 1.
a. 3 + 4 _____
Answer: 7
3 + 4 = 7
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 3 and 4 both are positive integers. Hence their sum is also positive.

b. -3 + (-4) _____
Answer: -7
-3 + (-4) = -3 – 4 =  -7
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -3 and -4 both are negative integers. Hence their sum is also negative.

c. 3 + (-4) _____
Answer: -1
3 + (-4) =3 – 4=  -1
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 3 is a positive integer and -4 is a negative integer. And 4 is greater than 3. Hence their sum is also negative.

d. -3 + 4 _____
Answer: 1
-3 + 4 = 1
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -3 is a negative integer and 4 is a positive integer. And 4 is greater than 3. Hence their sum is also positive.

Question 2.
a. -3 + (-3) ____
Answer: -6
-3 + (-3) = -3 – 3 =  -6
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -3 and -3 both are negative integers. Hence their sum is also negative.

b. 3 + (-3) _____
Answer: 0
3 + (-3) =3 – 3=  0
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 3 is a positive integer and -3 is a negative integer. Hence their sum is zero.

c. -3 + 3 _____
Answer: 0
-3 + 3=  0
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 3 is a positive integer and -3 is a negative integer. Hence their sum is zero.

d. 3 + 3 ____
Answer: 6
3 + 3 = 6
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 3 and 3 both are positive integers. Hence their sum is also positive.

Question 3.
a. 5 + (-1) ____
Answer: 4
5 + (-1) =5 – 1=  4
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 5 is a positive integer and -1 is a negative integer. And 5 is greater than 1. Hence their sum is also positive.

b. -5 + 1 ____
Answer: -4
-5 + 1 =  -4
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 1 is a positive integer and -5 is a negative integer. And 5 is greater than 1. Hence their sum is also negative.

c. -5 + (-1) ____
Answer: -6
-5 + (-1) = -5 – 1 =  -6
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -5 and -1 both are negative integers. Hence their sum is also negative.

d. 5 + 1 ____
Answer: 6
5 + 1 = 6
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 5 and 1 both are positive integers. Hence their sum is also positive.

Question 4.
a. -7 + 3 ____
Answer: -4
-7 + 3 =  -4
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 3 is a positive integer and -7 is a negative integer. And 7 is greater than 3. Hence their sum is also negative.

b. -7 + (-3) _____
Answer: -10
-7 + (-3) = -7- 3 =  -10
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -7 and -3 both are negative integers. Hence their sum is also negative.

c. 7 + (-3) _____
Answer: 4
7 + (-3) = 7 – 3 =  4
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 7 is a positive integer and -3 is a negative integer. And 7 is greater than 3. Hence their sum is also positive.

d. 7 + 3 _____
Answer: 10
7 + 3 = 10
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 7 and 3 both are positive integers. Hence their sum is also positive.

Question 5.
a. 4 + 7 ____
Answer: 11
4 + 7 = 11
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 4 and 7 both are positive integers. Hence their sum is also positive.

b. 4 + (-7) _____
Answer: -3
4 + (-7) = 4 – 7 = -3
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore,  4 is a positive integer and -7 is a negative integer. And 7 is greater than 4. Hence their sum is also negative.

c. -4 + (7) ____
Answer: 3
-4 + (7) = -4 + 7 = 3
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore,  7 is a positive integer and -4 is a negative integer. And 7 is greater than 4. Hence their sum is also positive.

d. -4 + (-7) _____
Answer: -11
-4 + (-7) = -4 – 7 =  -11
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -7 and -4 both are negative integers. Hence their sum is also negative.

Question 6.
a. 8 + (-8) _____
Answer: 0
8 + (-8) = 8 – 8=  0
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 8 is a positive integer and -8 is a negative integer. Hence their sum is zero.

b. -8 + (-8) ______
Answer: -16
-8 + (-8) = -8 – 8 =  -16
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -8 and -8 both are negative integers. Hence their sum is also negative.

c. 8 + 8 ____
Answer: 16
8 + 8 = 16
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 8 and 8 both are positive integers. Hence their sum is also positive.

d. -8 + 8 _____
Answer: 0
-8 + 8=  0
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 8 is a positive integer and -8 is a negative integer. Hence their sum is zero.

Question 7.
a. -3 + 0 _____
Answer: -3
-3 + 0=  -3
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -3 is a negative integer and 0 is neither a positive number nor a negative number. Hence their sum is also negative.

b. 3 + 0 ____
Answer: 3
3 + 0 = 3
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 3 is a positive integer and 0 is neither a positive number nor a negative number. Hence their sum is also positive.

c. -5 + (-6) _____
Answer: -11
-5 + (-6) =-5 – 6 =  -11
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -5 and -6 both are negative integers. Hence their sum is also negative.

d. -5 + 6 ____
Answer: 1
-5 + 6 =  1
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 6 is a positive integer and -5 is a negative integer. And 6 is greater than 5. Hence their sum is also positive.

Question 8.
a. 5 + (-6) _____
Answer: -1
5 + (-6) = 5 – 6 = -1
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore,  5 is a positive integer and -6 is a negative integer. And 6 is greater than 5. Hence their sum is also negative.

b. 5 + 6 ____
Answer: 11
5 + 6 = 11
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 5 and 6 both are positive integers. Hence their sum is also positive.

c. -8 + 0 ____
Answer: -8
-8 + 0  =  -8
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -8 is a negative integer and 0 is neither a positive number nor a negative number. Hence their sum is also negative.

d. 8 + 0 ____
Answer: 8
8 + 0 = 8
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 8 and 0 both are positive integers. Hence their sum is also positive.

Question 9.
a. -3 + 6 ____
Answer: 3
-3 + 6 =  3
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 6 is a positive integer and -3 is a negative integer. And 6 is greater than 3. Hence their sum is also positive.

b. -3 + (-6) ____
Answer: -9
-3 + (-6) = -3 – 6 =  -9
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -3 and -6 both are negative integers. Hence their sum is also negative.

c. 3 + 6 ____
Answer: 9
3 + 6 = 9
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 3 and 6 both are positive integers. Hence their sum is also positive.

d. 3 + (-6) _____
Answer: -3
3 + (-6) = 3 – 6 = -3
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore,  3 is a positive integer and -6 is a negative integer. And 6 is greater than 3. Hence their sum is also negative.

Question 10.
a. -6 + (-4) ______
Answer: -10
-6 + (-4) = -6 – 4 =  -10
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -4 and -6 both are negative integers. Hence their sum is also negative.

b. -6 + 4 ____
Answer: -2
-6 + 4 =  -2
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 4 is a positive integer and -6 is a negative integer. And 6 is greater than 4. Hence their sum is also negative.

c. 6 + (-4) _____
Answer: 2
6 + (-4) = 6 – 4 = 2
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore,  6 is a positive integer and -4 is a negative integer. And 6 is greater than 4. Hence their sum is also positive.

d. 6 + 4 ____
Answer: 10
6 + 4 = 10
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 6 and 4 both are positive integers. Hence their sum is also positive.

Adding Integers

To find the sum of two integers with different signs, find their absolute values. Remember, absolute value is the distance (in units) that a number is from 0, expressed as a positive quantity. Subtract the lesser number from the greater number. Absolute value is written as |n|.

4 > 3, so the sum is negative
The sum has the same sign as the integer with the larger absolute value.

Add.

Question 1.
a. 6 + 2 = ____
Answer: 8
6 + 2 = 8
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 6 and 2 both are positive integers. Hence their sum is also positive.

b. 9 + (-4) = ____
Answer: 5
9 + (-4) = 9 – 4 = 5
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore,  9 is a positive integer and -4 is a negative integer. And 9 is greater than 4. Hence their sum is also positive.

c. 7 + (-9) = ____
Answer: -2
7 + (-9) = 7 – 9 = -2
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore,  7 is a positive integer and -9 is a negative integer. And 9 is greater than 7. Hence their sum is also negative.

Question 2.
a. -4 + 7 = ____
Answer: 3
-4 + 7 =  3
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 7 is a positive integer and -4 is a negative integer. And 7 is greater than 4. Hence their sum is also positive.

b. -3 + (-6) = ____
Answer: -9
-3 + (-6) = -3 – 6 =  -9
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -3 and -6 both are negative integers. Hence their sum is also negative.

c. -12 + 11 = ____
Answer: -1
-12 + 11 =  -1
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 11 is a positive integer and -12 is a negative integer. And 12 is greater than 11. Hence their sum is also negative.

Question 3.
a. -16 + 0 = ____
Answer: -16
-16 + 0  =  -16
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -16 is a negative integer and 0 is neither a positive number nor a negative number. Hence their sum is also negative.

b. 13 + (-24) = ____
Answer: -11
13 + (-24) = 13 – 24 =  -11
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 13 is a positive integer and -24 is a negative integer. And 24 is greater than 13. Hence their sum is also negative.

c. -6 + 8 = ____
Answer: 2
-6 + 8 =  2
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 8 is a positive integer and -6 is a negative integer. And 8 is greater than 6. Hence their sum is also positive.

Question 4.
a. 0 + (-9) = ____
Answer: -9
0 + (-9) = 0 – 9=  -9
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -9 is a negative integer and 0 is neither a positive number nor a negative number. Hence their sum is also negative.

b. -1 + 2 = ____
Answer: 1
-1 + 2 =  1
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 2 is a positive integer and -1 is a negative integer. And 2 is greater than 1. Hence their sum is also positive.

c. 1 + (-2) = ____
Answer: -1
1 + (-2) =  -1
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 1 is a positive integer and -2 is a negative integer. And 2 is greater than 1. Hence their sum is also negative.

Question 5.
a. -4 + 4 = ____
Answer: 0
-4 + 4=  0
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 4 is a positive integer and -4 is a negative integer. Hence their sum is zero.

b. 3 + (-6) = ____
Answer: -3
3 + (-6) = 3 – 6=  -3
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -6 is a negative integer and 3 is a positive integer. And 6 is greater than 3. Hence their sum is also negative.

c. 7 + (-17) = ____
Answer: -10
7 + (-17) = 7 – 17=  -10
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -17 is a negative integer and 7 is a positive integer. And 17 is greater than 7. Hence their sum is also negative.

Question 6.
a. -45 + 21 = ____
Answer: -24
-45 + 21 =  -24
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -45 is a negative integer and 21 is a positive integer. And 45 is greater than 21. Hence their sum is also negative.

b. 41 + 44 = ____
Answer: 85
41 + 44 = 85
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 41 and 44 both are positive integers. Hence their sum is also positive.

c. 33 + 25 = ____
Answer: 58
33 + 25 = 58
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 33 and 25 both are positive integers. Hence their sum is also positive.

Question 7.
a. 27 + (-39) = ____
Answer: -12
27 + (-39) = 27 – 39 =  -12
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -39 is a negative integer and 27 is a positive integer. And 39 is greater than 27. Hence their sum is also negative.

b. 20 + 1 = ___
Answer: 21
20 + 1 = 21
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 20 and 1 both are positive integers. Hence their sum is also positive.

c. 3 + (-3) = ____
Answer: 0
3 + (-3) =3 – 3=  0
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 3 is a positive integer and -3 is a negative integer. Hence their sum is zero.

Question 8.
a. -12 + (-12) = ____
Answer: -24
-12 + (-12) = -12 – 12 =  -24
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -12 and -12 both are negative integers. Hence their sum is also negative.

b. 35 + (-26) = ____
Answer: 9
35 + (-26) = 35 – 26 =  9
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -26 is a negative integer and 35 is a positive integer. And 35 is greater than 26. Hence their sum is also positive.

c. -22 + 16 = ____
Answer: 6
-22 + 16 =  6
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -22 is a negative integer and 16 is a positive integer. And 22 is greater than 16. Hence their sum is also negative.

Question 9.
a. 31 + 17 = ___
Answer: 48
31 + 17 = 48
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 31 and 17 both are positive integers. Hence their sum is also positive.

b. -9 + (-6) = ____
Answer: -15
-9 + (-6) =  -15
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -9 and -6 both are negative integers. Hence their sum is also negative.

c. -47 + 36 = _____
Answer: 11
-47 + 36 =  11
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -47 is a negative integer and 36 is a positive integer. And 47 is greater than 36. Hence their sum is also negative.

Question 10.
a. 4 + 5 = ____
Answer: 9
4 + 5 = 9
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, 4 and 5 both are positive integers. Hence their sum is also positive.

b. -43 + 35 = ___
Answer: -8
-43 + 35 =  -8
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -43 is a negative integer and 35 is a positive integer. And 43 is greater than 35. Hence their sum is also negative.

c. 24 + (-33) = ____
Answer: -9
24 + (-33) = 24 – 33 =  -9
The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
The sum of two integers with opposite signs, subtract the digit of lesser value from the digit of greater value and keep the sign of the greater digit.
Therefore, -33 is a negative integer and 24 is a positive integer. And 33 is greater than 24. Hence their sum is also negative.