This handy **Spectrum Math Grade 7 Answer Key**** Chapter 4 Lesson 4.1 Unit Rates with Fractions** provides detailed answers for the workbook questions.

## Spectrum Math Grade 7 Chapter 4 Lesson 4.1 Unit Rates with Fractions Answers Key

A rate is a special ratio in which two terms are in different units. A unit rate is when one of those terms is expressed as a value of 1. Rates can be calculated with whole numbers or with fractions.

Emily ate \(\frac{1}{4}\) of an ice-cream cone in \(\frac{1}{2}\) of a minute. How long would it take her to eat one ice-cream cone?

1.Set up equivalent ratios using the information from the problem and l to represent the ice cream cone. Let t represent the time.

\(\frac{1}{4}\) × t = \(\frac{1}{2}\) × l

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

t = 2

2. Use cross multiplication.

3. Isolate the variable.

4. Solve.

**Find the unit rate in each problem.**

Question 1.

For Bill’s birthday his mom is bringing donuts to school. She has a coupon to get 2\(\frac{1}{2}\) dozen donuts for $8.00. How much would just one dozen donuts cost at this price?

Let c represent the cost of the donuts.

Equivalent ratios: _______________________

One dozen donuts would cost ____________________

Answer: Equivalent ratios: 2\(\frac{1}{2}\) ÷ 8 = \(\frac{1}{c}\)

One dozen donuts would cost $3.20

Bill’s mom has a coupon to get 2\(\frac{1}{2}\) dozen donuts for $8.00

Let c represent the cost of the donuts

2\(\frac{1}{2}\) dozen donuts = 2\(\frac{1}{2}\) × 12

= \(\frac{5}{2}\) × 12

= 30 donuts

Therefore, Bill’s mom has a coupon to get 30 donuts for $8.00

Equivalent ratios:

2\(\frac{1}{2}\) ÷ 8 = \(\frac{1}{c}\)

\(\frac{30}{8}\) = \(\frac{1}{c}\)

By cross multiplication,

30 × c = 1 × 8

c = \(\frac{8}{30}\)

so, c = 2.6667

the cost of each donut = 2.667, so the cost of One dozen donuts would be 2.6667 × 12 = $3.20

Question 2.

Jake ate 4\(\frac{1}{2}\) pounds of candy in one week. If he ate the same amount of candy every day, how much candy did he eat each day?

Let c represent the amount of candy.

Equivalent ratios: ______________________

He ate ___________________ pounds of candy each day.

Answer: Equivalent ratios: 4\(\frac{1}{2}\) ÷ 7 = \(\frac{1}{c}\)

He ate 1\(\frac{5}{9}\) pounds of candy each day.

Jake ate 4\(\frac{1}{2}\) pounds of candy in one week.

Converting the above mixed fraction into the improper fraction

4\(\frac{1}{2}\) = \(\frac{9}{2}\)

Therefore, Jake ate \(\frac{9}{2}\) pounds of candy in 7 days.

Let c represent the amount of candy.

Equivalent ratios:

4\(\frac{1}{2}\) ÷ 7 = \(\frac{1}{c}\)

\(\frac{9}{2}\) ÷ 7 = \(\frac{1}{c}\)

\(\frac{9}{14}\) = \(\frac{1}{c}\)

By cross multiplication,

9 × c = 1 × 14

c = \(\frac{14}{9}\)

c =1\(\frac{5}{9}\)

Therefore, He ate 1\(\frac{5}{9}\) pounds of candy each day.

Question 3.

A bakery used 6\(\frac{1}{4}\) cups of Hour this morning to make 5 batches of cookies. How much flour went into each batch of cookies?

Let f represent the amount of flour.

Equivalent ratios: ______________

Each batch of cookies used _______________________ cups of flour.

Answer: Equivalent ratios: 6\(\frac{1}{4}\) ÷ 5 = \(\frac{f}{1}\)

Each batch of cookies used 1\(\frac{1}{4}\) cups of flour.

A bakery used 6\(\frac{1}{4}\) cups of Hour this morning to make 5 batches of cookies.

Let f represent the amount of flour.

Equivalent ratios:

6\(\frac{1}{4}\) ÷ 5 = \(\frac{f}{1}\)

\(\frac{25}{4}\) ÷ 5 = \(\frac{f}{1}\)

\(\frac{25}{20}\) = \(\frac{f}{1}\)

By cross multiplication,

20 × f = 1 × 25

f = \(\frac{25}{20}\)

f = \(\frac{5}{4}\)

f = 1\(\frac{1}{4}\)

Therefore, Each batch of cookies used 1\(\frac{1}{4}\) cups of flour.

Using unit rates can help you compare two items.

Mike’s car can travel 425 miles on 10\(\frac{1}{2}\) gallons of gas. Jason’s car can travel 275 miles on 5\(\frac{4}{5}\) gallons of gas. Which car gets better gas mileage?

Let m represent Mike’s car and j represent Jason’s car.

Equivalent Ratio 1: = \(\frac{425}{10 \frac{1}{2}}\) = \(\frac{m}{l}\) m = 40\(\frac{10}{21}\) miles per gallon

Equivaient Ratio 2: \(\frac{275}{5 \frac{4}{5}}\) = \(\frac{j}{l}\) j = 47\(\frac{12}{29}\) miles per gallon

Jason’s car gets better gas mileage because it can go farther on one gallon of gas.

**Calculate unit rates to solve each problem.**

Question 1.

Cara can run 3 miles in 27\(\frac{1}{2}\) minutes. Melanie can run 6 miles in 53\(\frac{1}{3}\) minutes. Who can run faster?

Let c represent Cara’s speed and m represent Melanie’s speed.

Equivalent Ratio 1: _____________________

Equivalent Ratio 2: ______________________

_______________________can run faster.

Answer: Equivalent Ratio 1:\(\frac{3}{27 \frac{1}{2}}\) = \(\frac{d}{t}\) c

Equivalent Ratio 2: \(\frac{6}{53 \frac{1}{3}}\) = \(\frac{d}{t}\) m

Melanie can run faster.

Cara can run 3 miles in 27\(\frac{1}{2}\) minutes.

Melanie can run 6 miles in 53\(\frac{1}{3}\) minutes.

Let c represent Cara’s speed and m represent Melanie’s speed.

Equivalent Ratio 1: \(\frac{3}{27 \frac{1}{2}}\) = \(\frac{d}{t}\) c

\(\frac{3}{ \frac{55}{2}}\) = \(\frac{d}{t}\) c

= \(\frac{6}{55}\) = 0.109090

Equivalent Ratio 2: \(\frac{6}{53 \frac{1}{3}}\) = \(\frac{d}{t}\) m

\(\frac{6}{\frac{160}{3}}\) = \(\frac{d}{t}\) m

= \(\frac{18}{160}\) = 0.1125

Therefore, by comparing the above equivalent ratios, Melanie can run faster.

Question 2.

Bob goes to Shop and Save and buys 3\(\frac{1}{3}\) pounds of turkey for $10.50. Sonia goes to Quick Stop and buys 2\(\frac{1}{2}\) pounds of turkey for $6.25. Who got a better deal?

Let b represent Bob’s price and s represent Sonia’s price.

Equivalent Ratio 1: ____________________

Equivalent Ratio 2: _____________________

__________ got a better deal on turkey.

Answer: Equivalent Ratio 1: \(\frac{10.5}{3 \frac{1}{3}}\) = \(\frac{c}{q}\) b

Equivalent Ratio 2: \(\frac{6.25}{2 \frac{1}{2}}\) = \(\frac{c}{q}\) s

Sonia got a better deal on turkey.

Bob goes to Shop and Save and buys 3\(\frac{1}{3}\) pounds of turkey for $10.50.

Sonia goes to Quick Stop and buys 2\(\frac{1}{2}\) pounds of turkey for $6.25.

Let b represent Bob’s price and s represent Sonia’s price.

Equivalent Ratio 1: \(\frac{10.5}{3 \frac{1}{3}}\) = \(\frac{c}{q}\) b

\(\frac{10.5}{ \frac{10}{3}}\) = \(\frac{c}{q}\) b

= \(\frac{31.5}{10}\) = 3.15

Equivalent Ratio 2: \(\frac{6.25}{2 \frac{1}{2}}\) = \(\frac{c}{q}\) s

\(\frac{6.25}{ \frac{5}{2}}\) = \(\frac{c}{q}\) s

= \(\frac{12.5}{5}\) = 2.5

Therefore, by comparing the above equivalent ratios, Sonia got a better deal on turkey.

Question 3.

Thomas went for a long hike and burned 675 calories in 2\(\frac{1}{2}\) hours. Marvin decided to go for a bike ride and burned 1,035 calories in 3\(\frac{1}{4}\) hours. Who burned the most calories per hour?

Let t represent Thomas’s calories burned and m represent Marvin’s calories burned.

Equivalent Ratio 1: ______________________

Equivalent Ratio 2: _____________________

____________ burned the most calories per hour.

Answer: Equivalent Ratio 1: \(\frac{675}{2 \frac{1}{2}}\) = \(\frac{c}{t}\) t

Equivalent Ratio 2: \(\frac{1035}{3 \frac{1}{4}}\) = \(\frac{c}{t}\) m

Marvin burned the most calories per hour.

Thomas went for a long hike and burned 675 calories in 2\(\frac{1}{2}\) hours.

Marvin decided to go for a bike ride and burned 1,035 calories in 3\(\frac{1}{4}\) hours.

Let t represent Thomas’s calories burned and m represent Marvin’s calories burned.

Equivalent Ratio 1: \(\frac{675}{2 \frac{1}{2}}\) = \(\frac{c}{t}\) t

\(\frac{675}{ \frac{5}{2}}\) = \(\frac{c}{t}\) t

= \(\frac{1350}{5}\) = 270 calories per hour.

Equivalent Ratio 2: \(\frac{1035}{3 \frac{1}{4}}\) = \(\frac{c}{t}\) m

\(\frac{1035}{ \frac{13}{4}}\) = \(\frac{c}{t}\) m

= \(\frac{4140}{13}\) = 318.4615 calories per hour.

Therefore, by comparing the above equivalent ratios, Marvin burned the most calories per hour.