We included **H****MH Into Math Grade 7 Answer Key**** PDF** **Module 1 Lesson 3 Compute Unit Rates Involving Fractions **to make students experts in learning maths.

## HMH Into Math Grade 7 Module 1 Lesson 3 Answer Key Compute Unit Rates Involving Fractions

I Can compute unit rates associated with ratios of fractions.

**Spark Your Learning**

Rick and Tina hiked at different constant rates. Rick hiked \(\frac{1}{2}\) mile every 15 minutes, or \(\frac{1}{4}\) hour. It took Tina 10 minutes to hike \(\frac{1}{4}\) mile. Find the distance each hiked in 1 hour.

**Turn and Talk** How can you use the distance they each hiked in 1 hour to write a ratio for the distance they each would hike in 2 hours?

**Build Understanding**

1. Jessie loves to go hiking on rustic trails through trees and along rivers. One day in 20 minutes of hiking, she hiked 1 mile. If Jessie hiked at a constant rate, what would that rate be?

A. Write Jessie’s hiking rate in all the ways you can think of from the information given.

Answer:

One day in 20 minutes of hiking, she hiked 1 mile.

The one way is

The Jessie hiked at a constant rate is 20/1 minutes/mile.

Another way is

1/20 miles/minute.

= 1/mile/1/3 hours

= 1 x 3/1 ÷ 1/3 x 3/1 = 3miles/hour.

B. Complete the statements below to show how to write Jessie’s hiking rate as a unit rate in miles per hour.

You have to multiply \(\frac{1}{3}\) hour by ___ to make the second quantity in the unit rate ____ hour, so multiply the first quantity by _____ as well.

Answer:

1 mile/20 minutes = 1 mile/\(\frac{1}{3}\) hour = 1 mile/\(\frac{1}{3}\) hour × 3/3 = 3 miles/1 hour

You have to multiply \(\frac {1}{3} \) hour by 3/1 mile to make the second quantity in the unit rate 1/3 hour, so multiply the first quantity by 3/1 as well.

C. Amiya prefers hiking on more hilly trails. One time Amiya reached the mile marker pictured in 20 minutes hiking at a steady pace. Show how to find the unit rate in minutes per mile.

Answer:

Given that,

One time Amiya reached the mile marker pictured in 20 minutes hiking at a steady pace.

The unit rate in minutes per mile is 20 min/mile

**Turn and Talk** Compare the rates “minutes per mile” and “miles per minute.” Give an example of each rate.

Answer:

The ratio of minutes per mile = 20/1 = 20 miles per hour.

The ratio of miles per hour = 1/20 hours/mile.

**Step It Out**

2. A recipe says to use \(\frac{2}{3}\) cup of milk to make \(\frac{4}{5}\) serving of pudding. How many cups of milk are in 1 serving?

A. Recall that reciprocals are two numbers whose product is 1. Explain how reciprocals are used to find the unit rate.

____________________

____________________

Answer:

The reciprocals are 4/5 x 5/4 = 1

To find the unit rate the denominator is always 1.

So, will using the reciprocals the denominator becomes 1.

B. How can you use division to find this unit rate?

____________________

Answer:

For the unite rate divide the denominator with numerator then the denominator becomes 0.

**Turn and Talk** How can you find the number of servings of pudding for every cup of milk?

3. Jaylan makes limeade using \(\frac{3}{4}\) cup of water for every \(\frac{1}{5}\) cup of lime juice. Rene’s limeade recipe is different. He uses \(\frac{2}{3}\) cup of water for every \(\frac{1}{6}\) cup of lime juice. Whose limeade has a weaker flavor?

A. What do you need to know to solve this problem?

____________________

Answer:

Solve the unit rate of water to the lime juice in each limeade.

B. Compute the unit rate of water to lime juice in each limeade.

Answer:

Given that,

Jaylan makes limeade using \(\frac {3}{4} \) cup of water for every \(\frac {1}{5} \) cup of lime juice.

\(\frac {3}{4} \) = 3/4

\(\frac {1}{5} \) = 1/5

Rene’s limeade recipe is different. He uses \(\frac {2}{3} \) cup of water for every \(\frac {1}{6} \) cup of lime juice.

\(\frac {2}{3} \) = 2/3

\(\frac {1}{6} \) = 1/6

C. Whose limeade has a weaker flavor? Explain.

____________________

____________________

Answer:

The Jaylan limeade is 3.75 c water per 1 c lime juice.

The Rene Limeade is 4 c water per 1 c lime juice.

Therefore Jaylan Limeade is a weaker flavor than the Rene limeade.

4. The moon has a weaker gravitational pull than Earth, so objects weigh less on the moon. For example, Jaxon weighs 30\(\frac{5}{6}\) pounds on the moon and 185 pounds on Earth. Viola weighs 135 pounds on Earth and 22\(\frac{1}{2}\) pounds on the moon.

A. Show that the relationship between weight on the moon and weight on Earth is proportional.

The constant of proportionality for is _________

Answer:

The constant of proportionality for moon weight/earth weight = 0.135/0.846 = 0.159 lb.

B. Let x represent the weight on Earth. Let y represent the weight on the moon. Write an equation for the proportional relationship. Use it to find the weight of a 20-pound dog on the moon.

The equation is _____.

On the moon, the dog would weigh about ____ pounds.

Answer:

Given that,

x represents the earth

y represents the moon

The equation for the proportional relationship is = x/y.

The weight of the moon is the 20-pound dog.

Therefore the equation is x/20.

On the moon, the dog would weigh about x/20 pounds

**Turn and Talk** What would a dog that weighs 12 pounds on the moon weigh on Earth?

**Check Understanding**

Question 1.

A faucet leaks \(\frac{5}{8}\) quart of water in 15 minutes. How many quarts does the faucet leak per hour?

____________________

Answer:

Given that,

A faucet leaks \(\frac {5}{8} \) quart of water in 15 minutes.

\(\frac {5}{8} \) quart of water = 5/8 quart of water.

1 hour = 60 minutes

60/15 = 4

In 1 hour there are 4 quart = 5/8 x 4 = 2.5

In 4 quarts the faucet leak 2.5 of water per hour.

Question 2.

Toni ran \(\frac{4}{5}\) mile in \(\frac{1}{5}\) hour. Write an equation for the distance in miles y that she ran in x hours if she ran at a constant rate.

Answer:

Given that,

Toni ran \(\frac {4}{5} \) mile = 4/5 mile.

Toni ran in hours = \(\frac {1}{5} \) = 1/5 hours

The equation for the distance in miles y that she ran in x hours is

y/x = 4/5 ÷ 1/5

y/x = 0.8 ÷ 0.2

y/x = 0.4

The constant rate is 0.4 miles/hour.

Question 3.

Write \(\frac{3}{4}\) cup per \(\frac{1}{2}\) serving as a unit rate.

Answer:

Given that,

\(\frac {3}{4} \) cup = 3/4 cup.

\(\frac {1}{2} \) serving = 1/2 serving.

Divide the cup by serving.

3/4 ÷ 1/2

= 1.5

The unit rate is 1.5 cups/serving.

Question 4.

Write 2\(\frac{1}{4}\) miles in \(\frac{3}{4}\) hour as a unit rate.

Answer:

Given that,

2\(\frac {1}{4} \) miles = 2 x 1/4 miles.

\(\frac {3}{4} \) hour = 3/4 hours.

Know divide miles by hours

2 x 1/4 ÷ 3/4

= 0.5 ÷ 0.75

= 0.66 miles/hours.

The unit rate is 0.66 miles/hour.

**On Your Own**

Question 5.

**Health and Fitness** Jorge measured his heart rate after jogging. He counted 11 beats during a 6-second interval. What was the unit rate for Jorge’s heart rate in beats per minute? _________

Answer:

Given that,

Jorge measured his heart rate after jogging.

He counted 11 beats during a 6-second interval.

The unit rate for Jorge’s heart rate in beats per minute is

Divide beats by hours.

11/6 = beats/minute

The unit rate for Jorge’s heart rate in beats per minute is 1.83.

Question 6.

Chen bikes 2\(\frac{1}{2}\) miles in \(\frac{5}{12}\) hour, What is Chen’s unit rate in miles per hour?

____________________

Answer:

Given that,

2\(\frac {1}{2} \) miles = 2 x 1/2 miles.

\(\frac {5}{12} \) hour = 5/12 hours

Know divide miles by hours.

2 x 1/2 ÷ 5/12

= 1 ÷ 0.416

= 2.403 miles/hours.

The unit rate is 2.403 miles/hour.

Question 7.

Amal can run \(\frac{1}{8}\) mile in 1\(\frac{1}{2}\) minutes.

A. If he can maintain that pace, how long will it take him to run 1 mile?

_____________________

Answer:

Amal run \(\frac{1}{8}\) mile in 1\(\frac{1}{2}\) minutes.

1\(\frac{1}{2}\) = 1.5

1.5 × 8 = 12 minute

B. How long would it would take Amal to run 3 miles at that pace?

_____________________

Answer: It would also take him 12 minutes.

C. Naomi can run \(\frac{1}{4}\) mile in 2 minutes. Does Amal or Naomi run faster? How do you know?

_____________________

Answer:

Given,

Amal can run \(\frac{1}{8}\) mile in 1\(\frac{1}{2}\) minutes.

\(\frac{1}{8}\) ÷ 1\(\frac{1}{2}\)

Convert from mixed fraction to the improper fraction

\(\frac{1}{8}\) ÷ \(\frac{3}{2}\) = \(\frac{1}{12}\) mile/minute

Naomi can run \(\frac{1}{4}\) mile in 2 minutes.

\(\frac{1}{4}\) ÷ 2 = \(\frac{1}{8}\) mile/minute

Thus Naomi runs faster than Amal.

Question 8.

**Open Ended** When both quantities in a rate are fractions, what strategy do you use to write the rate as a unit rate?

________________________

________________________

Answer:

When both quantities in a rate are fractions, the strategy is.

First, write the ratios as the fraction and divide the numerator by the denominator.

Question 9.

**Reason** What is the ratio of dried fruit to sunflower seeds in the granola recipe? If you need to triple the recipe, will the ratio change? Explain.

________________________

________________________

Answer:

Given that,

Dried fruit = 1/2 cup.

Sunflower seeds = 1/8 cup.

The ratio of dried fruit to sunflower seeds in the granola recipe is 1/2 ÷ 1/8 = 4 cups

Triple the recipe = 4 x 4 x 4 = 64 cups.

Yes, when you triple the recipe the ratio will change.

Question 10.

The table shows the numbers of packages of peanut butter crackers y that can be made using various amounts of peanut butter x.

A. Show that the relationship is proportional.

_____________

Answer:

Yes, the relationship is proportional.

The equation is y = 4x.

B. Write an equation to represent the relationship, and find the amount of peanut butter used to make 25 cracker packages.

_____________

Answer:

The equation to represent the relationship y = 4x

Here, m = 4.

Find the amount of peanut butter used to make 25 cracker packages is

25 = 4x

x = 25/4

Therefore You need to use 25/4 teaspoons of peanut butter used to make the 25 cracker packages.

Question 11.

The relationship between adult dog weight x in pounds and the daily recommended amount of dog food y in cups is proportional.

Write an equation for the relationship. How much dog food is recommended for a 25-pound adult dog?

Answer:

Given that,

Food for the 50 pounds of the dog = 5/6 cups.

50 pounds = 50/2 = 25 pounds.

For 25 pounds of dog = 5/6 ÷ 2 = 5/12 cups.

The dog food recommended for a 25-pound adult dog is 5/12 cups.

**For Exercises 12—15, find each unit rate.**

Question 12.

\(\frac{5}{8}\) mile in \(\frac{1}{4}\) hour

Answer:

Given that,

\(\frac {5}{8} \) mile = 5/8 mile.

\(\frac {1}{4} \) hour = 1/4 hour.

Divide mile by hour.

5/8 ÷ 1/4

= 2.5 mile/hour

The unit rate is 2.5 miles/hour.

Question 13.

$68 for 8\(\frac{1}{2}\) hours

Answer:

Given that,

$68 for 8\(\frac {1}{2} \) hours

$68 for 8 x 1/2.

Divide dollars by hours.

68 ÷ 1/2

68 ÷ 0.5

= 136 dollars/hours.

The unit rate is 136 dollars/hour.

Question 14.

1\(\frac{1}{4}\) cup of flour per \(\frac{1}{8}\) cup of butter

Answer:

Given that.

1\(\frac {1}{4} \) cup of flour = 1 x 1/4

\(\frac {1}{8} \) cup of butter = 1/8

Know to divide a cup of flour with a cup of butter.

1 x 1/4 ÷ 1/8

= 0.25 ÷ 0.125

= 0.25 ÷ 0.125

= 2 cups of flour/cups of butter.

The unit rate is 2 cups of flour/cups of butter.

Question 15.

2\(\frac{1}{2}\) miles in \(\frac{3}{4}\) hour

____________________

Answer:

Given that,

2\(\frac {1}{2} \) miles = 2 x 1/2.

\(\frac {3}{4} \) hour = 3/4.

Divide miles by hours.

2 x 1/2 ÷ 3/4

= 1 ÷ 0.75

= 1.33 miles/hour.

The unit rate is 1.33 miles/hour.

**I’m in a Learning Mindset!**

What did I learn from peers when they shared their strategies with me for writing a unit rate?

**Lesson 1.3 More Practice/Homework**

**Compute Unit Rates Involving Fractions**

Question 1.

**STEM** Density is a unit rate measured in units of mass per unit of volume. The mass of a garnet is 5.7 grams. The volume is 1.5 cubic centimeters (cm^{3}). What is the density of the garnet?

Answer:

Given that,

The mass of the garnet is 5.7 grams.

The volume is 1.5 cubic centimetres.

The formula for the density is mass/volume.

The density of the grant is 5.7 grams/1.5 cubic centimeters.

The density of the grant = 3.8 grams/cubic centimeters.

Question 2.

**Math on the Spot** Jen and Kamlee are walking to school. After 20 minutes, Jen has walked \(\frac{4}{5}\) mile. After 25 minutes1 Kamlee has walked \(\frac{5}{6}\) mile. Find their speeds in miles per hour. Who is walking faster?

Answer:

Given that,

Jen and Kamlee are walking to school.

After 20 minutes, Jen has walked \(\frac{4}{5}\) mile.

\(\frac{4}{5}\) mile = 4/5 mile.

After 25 minutes1 Kamlee has walked \(\frac{5}{6}\) mile.

\(\frac{5}{6}\) = 5/6 mile.

The formula for the speed = distance/time

1 hour = 60 minutes.

The speed of Jan = 4/5 ÷ 20/60

= 12/5

= 2.4 miles per hour.

The speed of Kamlee = 5/6 ÷ 25/60

= 2 miles per hour.

Therefore Jan is 2.4 miles per hour and Kamlee is 2 hours per hour. So, Jan is walking faster.

Question 3.

**Reason** Maria and Franco are mixing sports drinks for a track meet. Maria uses \(\frac{2}{3}\) cup of powdered mix for every 2 gallons of water. Franco uses 1\(\frac{1}{4}\) cups of powdered mix for every 5 gallons of water. Whose sports drink is stronger? Explain how you found your answer.

________________________

________________________

________________________

Answer:

Given that,

Maria uses \(\frac{2}{3}\) cup of powdered mix for every 2 gallons of water.

\(\frac{2}{3}\) cup of powdered mix = 2/3.

Franco uses 1\(\frac{1}{4}\) cups of powdered mix for every 5 gallons of water.

1\(\frac{1}{4}\) cups of powdered mix = 1 x 1/4.

For maria the ratio of the powered mix to the water is

2/3 ÷ 2

= 1/3 cup/gallon

For Franco the ratio of the powered mix to the water is

1 x 1/4 ÷ 5

= 5/4 ÷ 5

= 1/4 cup/gallon

The 1/3 is bigger than the 1/4. So, Mria’s sports drink is stronger than the Franco sports drink.

Question 4.

**Model with Mathematics** Serena estimates that she can paint 60 square feet of wall space every half-hour. Write an equation for the relationship with time in hours as the independent variable. Can Serena paint 400 square feet of wall space in 3.5 hours? Why or why not?

________________________

Answer:

Given that,

Serena can paint = 60 square feet.

The time taken to paint = half-hour = 0.5 hours.

0.5 hours = 60 square feet’s.

For 1 hour = 60/0.5 = 120 sq. feet’s.

For I hours = 120i sq. feet’s.

Let us consider that the total area is y.

The relation between the I hours to the independent variable is

y = 120I

Can Serena paint 400 square feet of wall space in 3.5 hours?

Put y = 400 and I = 3.5 then

400 = 120(3.5)

400 is not equal to 420.

Therefore Serena cannot paint 400 square feet in 3.5 hours.

Question 5.

Cheri paid $6.50 for the bunch of grapes with the weight shown on the scale. What was the price per pound?

________________________

Answer:

Given that,

The cost for the bunch of graphs = $6.50

The weight of the grapes = 5 lb

Divide price by pound.

$6.50 ÷ 5

= 1.3

The unit rate is 1.3 price/lb

**Find the unit rate.**

Question 6.

\(\frac{1}{4}\) kilometer in \(\frac{1}{3}\) hour

Answer:

Given that,

\(\frac {1}{4} \) kilometre = 1/4 kilometre.

\(\frac {1}{3} \) hour = 1/3 hour

Divide the kilometre by hours

The unit rate is 1/4 ÷ 1/3 = 3/4 = 0.75 kilometre/hour.

Question 7.

\(\frac{7}{8}\) square foot in \(\frac{1}{4}\) hour

Answer:

Given that,

\(\frac {7}{8} \) square foot = 7/8 square foot.

\(\frac {1}{4} \) hour = 1/4 hour

Divide the square foot by the hour.

The unit rate is 7/8 ÷ 1/4 = 3.5 square feet/hour.

Question 8.

$6.50 for 3\(\frac{1}{4}\) pounds of grapes

Answer:

Given that,

Cost of the graphs = $6.50

The weight of the graphs = 3\(\frac {1}{4} \) pounds = 3 x 1/4 pounds.

Divide dollar by pounds.

The unit rate is $6.50 ÷ 3 x 1/4

= $6.50 ÷ 0.75

= 8.6 dollar/pounds

Question 9.

$49.50 for 5\(\frac{1}{2}\) hours

Answer:

Given that,

$49.50 for 5\(\frac {1}{2} \) hours.

$49.50 for 5 x 1/2 hours.

Divide dollar by hours

$49.50 ÷ 5 x 1/2

$49.50 ÷ 2.5

= 19.8

The unit rate is 19.8 dollars/hour.

Question 10.

247 heart beats in 6\(\frac{1}{2}\) minutes

Answer:

Given that,

247 heartbeats in 6\(\frac {1}{2} \) minutes.

247 heartbeats in 6 x 1/2.

Divide heartbeats by minutes.

247 ÷ 6 x 1/2

= 247 ÷ 3

= 82.33 heartbeats/minutes.

The unit rate is 82.33 heartbeats/minutes.

Question 11.

8\(\frac{1}{2}\) miles in \(\frac{1}{2}\) hour

Answer:

Given that,

8\(\frac {1}{2} \) miles = 8 x 1/2.

\(\frac {1}{2} \) hour = 1/2 hour.

Know divide miles by hours.

8 x 1/2 ÷ 1/2

= 4 ÷ 0.5

= 8

The unit rate is 8 miles/hour.

**Test Prep**

Question 12.

Select all the rates equivalent to the rate \(\frac{3}{4}\) cup per pound.

A. \(\frac{3}{8}\) cup per \(\frac{1}{2}\) pound

B. \(\frac{1}{4}\) cup per \(\frac{1}{2}\) pound

C. 3 cups for every 2 pounds

D. 1\(\frac{1}{2}\) cups for every 2 pounds

E. 0.1875 cup for every 0.25 pound

Answer:

Question 13.

Jordan cooked a 16-\(\frac{1}{5}\)-pound turkey in 5\(\frac{2}{5}\) hours. How many minutes per pound did it take to cook the turkey? Express your answer as a unit rate.

____________________

Answer:

Given that,

Jordan cooked 16-\(\frac {1}{5} \)-pound turkey in 5\(\frac {2}{5} \) hours.

Jordan cooked 16 x 1/5-pound turkey in 5 x 2/5 hours.

16 x 1/5 = 81/5 lb.

5 x 2/5 = 27/5 hours

1 hour = 60 minutes

27/5 x 60 = 324 minutes

Know Divide the total time by the total weight.

324/81/5 = 1.620/81 = 20 min/lb

The unit rate is 20 min/lb

Question 14.

Mr. March sells popcorn at his theater. He uses 3\(\frac{3}{4}\) cups of unpopped corn to make 15 bags of popped corn. Write an equation for the number of bags of popcorn b that can be made with c cups of unpopped corn.

Answer:

Given that,

Mr. March uses 3\(\frac{3}{4}\) cups of unpopped corn to make 15 bags of popped corn.

3\(\frac{3}{4}\) cups = 3 x 3/4.

The equation for the number of bags of popcorn b that can be made with c cups of unpopped corn is b/c =

Divide bags of popped corn into cups of unpopped corn.

b/c = 15 ÷ 3 x 3/4

b/c = 15 ÷ 2.25

b/c = 6.66 bags of popped corn/cups of unpopped corn.

Question 15.

Lucia uses 3 ounces of pasta to make \(\frac{3}{4}\) serving of pasta. How many ounces of pasta are there per serving? How many ounces of pasta should Lucia use to make 5 servings?

A. 3 ounces; 15 ounces

B. 4 ounces: 20 ounces

C. 6 ounces; 30 ounces

D. 9 ounces; 40 ounces

Answer:

Given that,

Lucia uses 3 ounces of pasta to make \(\frac {3}{4} \) serving of pasta.

3 ounces of pasta to make 3/4 serving of pasta.

Divide ounces by servings.

3 ÷ 3/4

= 3 ÷ 0.75

= 4 ounces/servings

The ounces of pasta should Lucia use to make 5 servings is 4 ounces x 5.

= 20 ounces.

4 ounces: 20 ounces.

Option B is the correct answer.

**Spiral Review**

Question 16.

John left school with $8.43. He found a quarter on his way home and then stopped to buy an apple for $0.89. How much money did he have when he got home?

Answer:

Given that

John left the school with $8.43.

On the way home, John buys an apple for $0.89.

Therefore $8.43 – $0.89 = $7.54

John has $7.54 when he got home.

Question 17.

Arian is making bracelets. For each bracelet, it takes \(\frac{1}{10}\) hour to pick out materials and \(\frac{1}{4}\) hour to braid it together. How many bracelets can Arian make in 5 hours?

Answer:

Given that,

For each bracelet to pick out the materials = \(\frac {1}{10} \) = 1/10 hours.

For each bracelet to band it together = \(\frac {1}{4} \) = 1/4 hours.

Therefore 1/10 + 1/4 = 7/20 hours.

For 1 bracelets = 7/10 hours.

Divide available time by given time.

5 ÷ 7/20 = 14.285

Arian can make 14 bracelets in 5 hours.

Question 18.

For a game, 3 people are chosen in the first round. Each of those people chooses 3 people in the second round, and so on. How many people are chosen in the sixth round?

____________________

Answer:

In the first round, 3 people are chosen.

In the second round, these 3 people choose 3 people. It means 3 x 3 = 9

In the third round again 3 people are chosen = 9 x 3 = 27

In the fourth round again 3 people are chosen = 27 x 3 = 81

In the fifth round again 3 people are chosen = 81 x 3 = 243

In the sixth round again 3 people are chosen = 243 x 3 = 729.

The 729 people are chosen in the sixth round.