We included HMH Into Math Grade 7 Answer Key PDF **Module 1 Lesson 2 Recognize Proportional Relationships in Tables **to make students experts in learning maths.

I Can identify proportional relationships in tables and equations, identify the constant of proportionality, and write the associated equation.

**Spark Your Learning**

For which tables can you predict the cost of 100 units of the item? Explain why you can make that prediction for some of the tables and not for others. For the tables that let you predict the cost of 100 units, find the cost.

**Turn and Talk** What characteristic of the table allows you to predict the cost of any number of items?

**Build Understanding**

1. Maxine walks dogs to earn extra money. The table of values shows the proportional relationship between the number of dogs Maxine walks and the amount of money she earns.

**Connect to Vocabulary**

Two quantities have a proportional relationship if the ratio of one quantity to the other is a constant.

In a proportional relationship, the constant unit rate is called the constant of proportionality and is usually represented by the letter k.

A. Show that the ratio of dogs walked to amount earned is a constant ratio.

_____________________________

B. Show that the ratio of amount earned to dogs walked is a constant ratio.

_____________________________

C. What is the unit rate of dogs walked per dollar earned? Does it make sense? Explain.

_____________________________

D. What is the unit rate of dollars earned per dog walked? Does it make sense? Explain.

_____________________________

E. What is the constant of proportionality k in this situation?

_____________________________

F. Describe what the unit rate k represents in this situation.

_____________________________

Answer:

A)

The ratio of dogs walked to amount earned = 2/7 = 0.28

4/14 = 0.28

6/21 = 0.28

Therefore the ratio of dogs walked to the amount earned is a constant ratio.

B)

The ratio of amount earned to dogs walked = 7/2 = 3.5

14/4 = 3.5

21/6 = 3.5

The ratio of the amount earned to dogs walked is a constant ratio.

C)

The unit rate of dogs walked to amount dollar earned = 2/7 = 0.28

4/14 = 0.28

6/21 = 0.28

The unit rate is 0.28 dogs walked/dollar.

D)

The unit rate of dollar earned to dogs walked = 7/2 = 3.5

14/4 = 3.5

21/6 = 3.5

The unit rate is 3.5 dollar/dogs walked.

E)

The constant of proportionality k = 3.5.

F)

The unit rate k = 3.5 dollar/dog walked.

**Turn and Talk** In a table of values that represents a proportional relationship, how can you find the constant of proportionality?

**Step It Out**

2. Two relationships are represented in the tables. Which table shows a proportional relationship, and which does not? Explain.

The ratios \(\frac{y}{x}\) (are/are not) equivalent. Therefore, the relationship shown in this table (is / is not) proportional.

The ratios \(\frac{y}{x}\) (are/are not) equivalent. Therefore, the relationship shown in this table (is / is not) proportional.

Answer:

The ratios \(\frac{y}{x} \) are equivalent. Therefore, the relationship shown in this table isproportional.

The ratios \(\frac{y}{x} \) are not equivalent. Therefore, the relationship shown in this table is notproportional.

3. The number of students whom Malcolm tutors is proportional to the amount earned. The calendar shows Malcolm’s earnings from tutoring in one week.

A. Complete the table.

Answer:

The equation is

y = 30x

If x = 1 then y = 30(1) = 30

If x = 3 then y = 30(3) = 90

If x = 4 then y = 30(4) = 120

If x = 5 then y = 30(5) = 150.

If x = 6 then y = 30(6) = 180

B. Find the constant of proportionality k.

Answer:

The constant of proportionality k = y/x = 30/1 = 30

The constant of proportionality is 30

C. The equation for a proportional relationship is y = kx, where k is the constant of proportionality. Use the value of k you found in Part B to write an equation for this proportional relationship.

y = x

Answer:

The equation is y = Kx

Therefore k = 30

Then y = 30x.

4. The equation y = 12x represents the number of inches in x feet.

A. The equation y = 12x (does / does not) represent a proportional relationship. If so, what is the constant of proportionality? How do you know?

_____________________________

_____________________________

Answer:

The equation y = 12x does not represent a proportional relationship. So, there is no constant of proportionality. We can know by using the equation.

The equation is y = 12x

If x= 0 then y = 12(0) = 0

If x = 1 then y = 12(1) = 12

If x = 2 then y = 12(2) = 24

If x = 3 then y = 12(3) = 36

If x = 4 then y = 12(4) = 48

If x= 5 then y = 12(5) = 60

B. Use the equation to complete the table of values for the relationship between inches and feet.

Answer:

The equation is y = 12x

If x= 0 then y = 12(0) = 0

If x = 1 then y = 12(1) = 12

If x = 2 then y = 12(2) = 24

If x = 3 then y = 12(3) = 36

If x = 4 then y = 12(4) = 48

If x= 5 then y = 12(5) = 60

**Turn and Talk** When would it be better to use an equation to represent a proportional relationship? When would it be better to use a table?

Answer: The equation is better to represent a proportional relationship.

**Check Understanding**

Question 1.

There are 4 quarters in $1.00.

A. Make a table of values to represent this relationship.

Answer:

Given that,

4 quarters in $1.00

The equation is y = 4x

If x = 1 then y = 4(1) =4

If x = 2 then y = 4(2) = 8

If x = 3 then y = 4(3) = 12

If x= 4 then y = 4(4) = 16.

If x = 5 then y = 4(5) = 20.

B. Is this a proportional relationship? If so, identify k and explain what it represents. If not, explain why not.

Answer:

Given that,

4 quarters in $1.00

The equation is y = 4x

If x = 1 then y = 4(1) =4

If x = 2 then y = 4(2) = 8

If x = 3 then y = 4(3) = 12

If x= 4 then y = 4(4) = 16.

If x = 5 then y = 4(5) = 20.

The ratio of Quarters and dollars is

k = y/x = 4/1 = 4

8/2 = 4

12/4 = 4

16/4 = 4

20/4 = 4

All ratios in the table are equal. So, it is a proportional relationship.

The value of the k = 4.

C. Write an equation for the situation. _____________________________

Answer:

Given that,

4 quarters in $1.00

The equation is y = 4x

If x = 1 then y = 4(1) =4

If x = 2 then y = 4(2) = 8

If x = 3 then y = 4(3) = 12

If x= 4 then y = 4(4) = 16.

If x = 5 then y = 4(5) = 20.

Question 2.

The equation y = 7x gives the cost y of x pounds of chicken at the grocery store. Complete the table for the given weights of chicken.

Answer:

Given that,

The equation is y = 7x

If x = 1 then y = 7(1) = 7

If x = 2 then y = 7(2) = 14

If x = 5 then y = 7(5) = 35

If x = 8 then y = 7(8) = 56

Question 3.

Is the relationship in the table proportional? If it is, write its equation.

_____________________________

Answer:

The ratio of x and y is k = y/x

5/1 = 5

10/2 = 5

15/3 = 5

20/4 = 5

All the ratios in the table are equal.

The constant of proportionality is 5.

Therefore the given table is Proportional.

**On Your Own**

Question 4.**Model with Mathematics** Reanna is making a scrapbook which holds 14 photos on each 2-page spread. Make a table of values to represent this relationship. Write an equation for the situation.

Answer:

Given that,

Reanna is making a scrapbook which holds 14 photos on each 2-page spread.

The is y = 14x

If x = 1 then y = 14(1) = 14

If x= 2 then y = 14(2) = 28

If x = 3 then y = 14(3) = 42

If x = 4 then y = 14(4) = 56

If x = 5 then y = 14(5) = 70

The ratio of x and y is k = y/x

14/1 = 14

28/2 = 14

42/3 = 14

56/4 = 14

70/4 = 14

The constant proportionality is 14.

Therefore the table is proportionality constant.

**Tell whether each table represents a proportional relationship. If it does, identify the constant of proportionality.**

Question 5.

_____________________________

Answer:

The ratio of x and y is k = y/x

63/3 = 21

147/7 = 21

189/9 = 21

The constant of proportionality is 21.

Therefore the table represents the constant of proportionality.

Question 6.

Answer:

The ratio of x and y is k = y/x

21/14 = 1.5

22.5/15 = 1.5

15/16 = 0.93

There is no constant proportionality.

Therefore the table does not represent the constant of proportionality.

Question 7.

Determine whether the table represents a proportional relationship. If it does, find the constant of proportionality and use it to write an equation to represent the table of values.

Answer:

The ratio of x and y is k = y/x

7/1 = 7

14/2 = 7

21/3 = 7

28/4 = 7

35/5 = 7

The constant of proportionality is 8

The equation is y = 7x.

All the ratios are equal in the above table. So, the table represents a proportional relationship.

Question 8.

The equation y = 8x gives the number of slices y in x pizzas. Make a table of values using the equation. Identify the constant of proportionality. Then complete each sentence.

There are ___ slices in 3 pizzas.

There are 16 slices in ___ pizzas.

Answer:

The equation is y = 8x

If x = 1 then y = 8(1) = 8

If x = 2 then y = 8(2) = 16

If x = 3 then y = 8(3) = 24

If x = 4 then y = 8(4) = 32

If x = 5 then y = 8(5) = 40

There are 24 slices in 3 pizzas.

There are 16 slices in 2 pizzas

The constant of proportionality is 8.

Question 9.**Reason** The table shows the relationship between the number of workers painting apartments in an apartment building and the number of days it takes to paint all 50 apartments. Determine whether the relationship is proportional. Explain your reasoning.

Answer:

The ratio of duration of job and workers is

k = y/x = 60/5 = 12

30/10 = 3

20/15 = 1.33

15/20 = 0.75

12/25 = 0.48.

The ratios of the table are not equal. So, the given table is not proportional.

**Model with Mathematics For Problems 10-12, use the description of a proportional relationship to make a table. Then identify the constant of proportionality, and write an equation for the situation.**

Question 10.

A 2-cup serving of chicken noodle soup has 1.5 ounces of noodles.

Answer:

Given that,

2 cup serving of chicken noodle soup has 1.5 ounces of noodles.

The equation is

y = 0.75x

If x = 2 then y = 0.75(2) = 1.5

If x = 3 then y = 0.75(3) = 2.25

If x = 4 then y = 0.75(4) = 3

The ratio of x and y is

k = y/x = 1.5/2 = 0.75

2.25/3 = 0.75

3/4 = 0.75

All the ratios in the table are equal. So, the table represents the proportional relationship.

The constant of proportionality is 0.75.

Question 11.

Rick is exercising at a constant pace.

Answer:

Given that,

Rick does 126 steps every 3 minutes.

The equation is y = 42x

If x = 3 then y = 42(3) = 126

If x = 4 then y = 42(4) = 168

If x = 5 then y = 42(5) = 210

The ratio of minutes and steps is

k =y/x = 126/3 = 42

168/4 = 42

210/5 = 42

Constant pace is 42.

Question 12.

Colin is preparing equal-sized care packages. He placed 34 items in 2 care packages he made.

Answer:

The equation is y = 17x

If x = 2 then y = 17(2) = 34.

If x = 3 then y = 17(3) = 51

If x = 4 then y = 17(4) = 68

If x = 5 then y = 17(5) = 85

Question 13.

The equation y = 100x gives the number of centimeters y in x meters. Make a table of values using the equation. Identify the constant of proportionality. Then complete each sentence.

k = ___; There are ___ centimeters in 3 meters. There are 200 centimeters in ___ meters.

Answer:

The equation is y =100x

If x = 1 then y = 100(1) = 100.

If x = 2 then y = 100(2) = 200

If x = 3 then y = 100(3) = 300

If x = 4 then y = 100(4) = 400

K= 100, there are 300 centimetres in 3 meters.

There are 200 centimetres in 2 meters.

**I’m in a Learning Mindset!**

What strategies do I use to decide if a relationship displayed in a table is proportional? How do I know when I am finished?

Answer:

You decide all the ratios of the table are equal and find the constant proportionality then the relationship displayed on the table is proportional.

**Lesson 1.2 More Practice/Homework**

**Recognize Proportional Relationships in Tables**

**Tell whether each table represents a proportional relationship. If it does, identify the constant of proportionality.**

Question 1.

_____________________________

Answer:

The ratio of x and y is

k = y/x = 18/2 = 9

45/5 = 9

63/7 = 9

All the ratios in the table is equal. So, the table represents the proportional relationship.

The constant of proportionality is 24.

Question 2.

_____________________________

Answer:

The ratio of x and y is

k = y/x

42/3 = 14

60/4 = 15

80/5 = 16

All the ratios in the table is not equal. So, the table does not represent the proportional relationship.

There is no constant of proportionality.

Question 3.**Math on the Spot** Determine whether the table represents a proportional relationship. If it does, find the constant of proportionality and use it to write an equation to represent the table of values.

_____________________________

Answer:

The ratio of Amount earned by number of lawns is

K = y/x = 24/1 = 24

48/2 = 24

72/3 = 24

96/4 = 24

All the ratios in the table is equal. So, the table represents the proportional relationship.

The constant of proportionality is 24.

The equation that represents the table is y = 24x.

Question 4.

The equation y = 6x gives the cost y of x of the tickets shown. Make a table of values. Identify the constant of proportionality. Then complete each sentence.

It costs ___ for 5 tickets.

It costs $24 for ____ tickets.

Answer:

Given that,

y = 6x

If x= 1 then y = 6(1) = 6

If x = 2 then y = 6(2) = 12

If x = 3 then y = 6(3) = 18

If x = 4 then y = 6(4) = 24

If x = 5 then y = 6(5) = 30.

It costs $30 for 5 tickets.

It costs $24 for 4 tickets.

**Model with Mathematics Use the description of a proportional relationship in each table. Identify the constant of proportionality, then write an equation to represent the situation in the table.**

Question 5.

Alison earned $24 by stocking shelves at the grocery store for 3 hours.

Answer:

Give that,

Alison earned $24 for 3 hours.

The ratio of total pay and time is

k = y/x = 8/1 = 8

16/2 = 8

24/3 = 8

48/6 = 8

All the ratios in the table is equal. So, the table represents the proportional relationship.

The constant of proportionality is 8.

The equation that represents the table is y = 8x.

Question 6.

Each cooler holds 18 water bottles.

Answer:

Given that,

Each cooler holds 18 water bottles.

The ratio of water bottles and coolers is

k = y/x = 18/1 = 18

36/2 = 18

72/4 = 18

126/7 = 18

All the ratios in the table is equal. So, the table represents the proportional relationship.

The constant of proportionality is 18.

The equation that represents the table is y = 18x.

**Test Prep**

Question 7.

Which table represents a proportional relationship?

Answer:

In the table 1

k = y/x = 23/2 = 11.5

33/3 = 11

43/4 = 10.75

In the table 2

k = y/x = 23/2 = 11.5

34.50/3 = 11.5

46/4 = 11.5

Table 2 has all the ratios that are equal. So, table 2 represents the proportional relationship.

Question 8.

Use the proportional relationship in the table.

A. Write an equation for the relationship.

_____________________________

Answer:

The equation for the relationship is

y = 16x

If x = 2 then y = 16(2) = 32

If x = 4 then y = 16(4) = 64

If x = 8 then y = 16(8) = 128.

B. There are 4 quarts in a gallon and 4 cups in a quart. How many cups are in one serving?

Answer:

Given that,

4 quarts in a gallon and 4 cups in a quart.

The number of cups in the serving is

4 x 4 =16 cups

there are 16 cups in one serving

C. There are 8 fluid ounces in a cup. How many fluid ounces are in one serving?

Answer:

Given that,

8 fluid ounces in a cup.

There are 8 fluid ounces in one serving.

Question 9.

What is the meaning of the constant of proportionality in this situation?

Answer:

The constant of the probability is the ratio between the two directly proportional quantities.

constant K = y/x = 48/2 = 24

72/3 = 24

120/5 = 24

The constant of proportionality is 24.

Question 10.

Describe a method for determining whether a table represents a proportional relationship.

_____________________________

_____________________________

_____________________________

Answer:

Yes, the table represents the proportional relationship. Because in the table all the ratios are equal.

Question 11.

Kevin uses \(\frac{2}{3}\) cup of flour to make 2 servings of biscuits. How many cups of

flour are there per serving? How many cups of flour should Kevin use to make 7 servings?

A. \(\frac{1}{3}\) cup; 2\(\frac{1}{3}\) cups

B. \(\frac{2}{3}\) cup; 2\(\frac{1}{3}\) cups

C. \(\frac{2}{3}\) cup; 4\(\frac{2}{3}\) cups

D. 1\(\frac{1}{3}\) cups; 4\(\frac{2}{3}\) cups

Answer:

Given that,

Kevin uses \(\frac {2}{3} \) cup of flour to make 2 servings of biscuits.

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

Divide cups by servings.

2/3 ÷ 2 = 1/3

1/3 cup of flour to make 1 serving of biscuit.

For 7 servings = 1/3 x 7 = 7/3.

Therefore the answer is \(\frac {1}{3} \) cup; 2\(\frac {1}{3} \) cups

Option A is the correct answer.

**Spiral Review**

Question 12.

Donya ran a 3k race at a constant speed in 21 minutes 30 seconds. At this speed, how long does it take her to run 1k?

_____________________________

Answer:

Given that,

Donya ran a 3k race at a constant speed in 21 minutes 30 seconds.

1 minute = 60 seconds

21 minutes 30 seconds = 21 + (30/60) = 21.5 minutes.

Donya ran a 1k race at a constant speed is 21.5/3 = 7.17 minutes.

7.17 minutes = 7 + (0.17 x 60) = 7 minutes 10 seconds.

Therefore Donya run 1k in 7 minutes 10 seconds.

Question 13.

It costs $20 for 4 play tickets and $35 for 7 play tickets. Is cost per ticket constant? Why or why not?

_____________________________

Answer:

Given that,

The cost of 4 play tickets is $20.

The cost of 7 play tickets is $35.

1 Play ticket cost = $20 ÷ 4 = $5.

7 Play ticket is $5 x 7 = $35.

The cost of the ticket for play is not changed so, it is a constant.

Therefore the cost per ticket is constant.