## Engage NY Eureka Math 7th Grade Module 1 Lesson 9 Answer Key

### Eureka Math Grade 7 Module 1 Lesson 9 Example Answer Key

Example 1.
Jackson’s Birdhouses
Jackson and his grandfather constructed a model for a birdhouse. Many of their neighbors offered to buy the birdhouses. Jackson decided that building birdhouses could help him earn money for his summer camp, but he is not sure how long it will take him to finish all of the requests for birdhouses. If Jackson can build 7 birdhouses in 5 hours, write an equation that will allow Jackson to calculate the time it will take him to build any given number of birdhouses, assuming he works at a constant rate.
a. Write an equation that you could use to find out how long it will take him to build any number of birdhouses
H = $$\frac{5}{7}$$B

Define the variables. B represents the number of birdhouses, and H represents the number of hours (time constructing birdhouses).

→ Does it matter which of these variables is independent or dependent?
→ No. The number of birdhouses made could depend on how much time Jackson can work, or the amount of time he works could depend on how many birdhouses he needs to make.
→ If it is important to determine the number of birdhouses that can be built in one hour, what is the constant of proportionality?
→ $$\frac{\text { number of birdhouses }}{\text { number of hours }}$$ is $$\frac{7}{5}$$, or 1.4.
→ What does that mean in the context of this situation?
→ It means that Jackson can build 1.4 birdhouses in one hour or one entire birdhouse and part of a second birdhouse in one hour.
→ If it is important to determine the number of hours it takes to build one birdhouse, what is the constant of proportionality?
→ $$\frac{\text { number of hours }}{\text { number of birdhouses }}$$ is $$\frac{5}{7}$$, or 0.71, which means that it takes him $$\frac{5}{7}$$ of an hour to build one birdhouse or ($$\frac{5}{7}$$)(60) = 43 minutes to build one birdhouse.
→ This part of the problem asks you to write an equation that will let Jackson determine how long it will take him to build any number of birdhouses, so we want to know the value of H. This forces H to be the dependent variable and B to be the independent variable. Our constant of proportionality will be $$\frac{\text { dependent }}{\text { independent }}$$, which is $$\frac{H}{B}$$ or $$\frac{y}{x}$$. $$\frac{y}{x}$$ is $$\frac{5}{7}$$, so we will use the equation H = $$\frac{5}{7}$$ B.
Use the equation above to determine the following:

b. How many birdhouses can Jackson build in 40 hours?
If H = $$\frac{5}{7}$$ B and H = 40, then substitute 40 in the equation for H and solve for B since the question asks for the number or birdhouses.
40 = ($$\frac{5}{7}$$)B
($$\frac{7}{5}$$)40 = ($$\frac{7}{5}$$)($$\frac{5}{7}$$)B
56 = B
Jackson can build 56 birdhouses in 40 hours.

c. How long will it take Jackson to build 35 birdhouses? Use the equation from part (a) to solve the problem.
If H = $$\frac{5}{7}$$B and B = 35, then substitute 35 into the equation for B; H = ($$\frac{5}{7}$$)(35); H = 25. It will take Jackson 25 hours to build 35 birdhouses.

d. How long will it take to build 71 birdhouses? Use the equation from part (a) to solve the problem.
If H = $$\frac{5}{7}$$B and B = 71, then substitute 71 for B into the equation; H = ($$\frac{5}{7}$$)(71); H = 50.7 (rounded to the nearest tenth). It will take Jackson 50 hours and 42 minutes (60(0.7)) to build 71 birdhouses.

Remind students that while one may work for a fractional part of an hour, a customer will not want to buy a partially built birdhouse. Tell students that some numbers can be left as non-integer answers (e.g., parts of an hour that can be written as minutes), but others must be rounded to whole numbers (e.g., the number of birdhouses completed or sold). All of this depends on the context. We must consider the real-life context before we determine if and how we round.

Example 2.
Al’s Produce Stand
Al’s Produce Stand sells 6 ears of corn for $1.50. Barbara’s Produce Stand sells 13 ears of corn for$3.12. Write two equations, one for each produce stand, that model the relationship between the number of ears of corn sold and the cost. Then, use each equation to help complete the tables below.
Al’s Produce Stand: y = 0.25x; where x represents the number of ears of corn, and y represents the cost
Barbara’s Produce Stand: y=0.24x; where x represents the number of ears of corn, and y represents the cost

→ If you use E to represent the number of ears of corn and C to represent the cost for the variables instead of x and y, how would you write the equations?
→ C = 0.25E and C = 0.24E

### Eureka Math Grade 7 Module 1 Lesson 9 Problem Set Answer Key

Question 1.
A person who weighs 100 pounds on Earth weighs 16.6 lb. on the moon.
a. Which variable is the independent variable? Explain why.
Weight on Earth is the independent variable because most people do not fly to the moon to weigh themselves first. The weight on the moon depends on a person’s weight on Earth.

b. What is an equation that relates weight on Earth to weight on the moon?
M = ($$\frac{16.6}{100}$$)E
M = 0.166E

c. How much would a 185-pound astronaut weigh on the moon? Use an equation to explain how you know.
30.71 lb.

d. How much would a man who weighs 50 pounds on the moon weigh on Earth?
301 lb.

Question 2.
Use this table to answer the following questions.
Number of Gallons of Gas Number of Miles Driven

 Number of Gallons of Gas Number of Miles Driven 0 0 2 62 4 124 10 310

a. Which variable is the dependent variable, and why?
The number of miles driven is the dependent variable because the number of miles you can drive depends on the number of gallons of gas you have in your tank.

b. Is the number of miles driven proportionally related to the number of gallons of gas? If so, what is the equation that relates the number of miles driven to the number of gallons of gas?
Yes, the number of miles driven is proportionally related to the number of gallons of gas because every measure of gallons of gas can be multiplied by 31 to get every corresponding measure of miles driven.
M = 31G

c. In any ratio relating the number of gallons of gas and the number of miles driven, will one of the values always be larger? If so, which one?
Yes, the number of miles will be larger except for the point (0,0). The point (0,0) means 0 miles driven uses 0 gallons of gas.

d. If the number of gallons of gas is known, can you find the number of miles driven? Explain how this value would be calculated.
Yes, multiply the constant of proportionality (31 mpg) by the number of gallons of gas.

e. If the number of miles driven is known, can you find the number of gallons of gas used? Explain how this value would be calculated.
Yes, divide the number of miles driven by the constant of proportionality (31 mpg).

f. How many miles could be driven with 18 gallons of gas?
558 miles

g. How many gallons are used when the car has been driven 18 miles?
$$\frac{18}{31}$$ gallons

h. How many miles have been driven when half a gallon of gas is used?
$$\frac{31}{2}$$ = 15.5 miles

i. How many gallons have been used when the car has been driven for a half mile?
$$\frac{1}{62}$$ gallons

Question 3.
Suppose that the cost of renting a snowmobile is $37.50 for 5 hours. a. If c represents the cost and h represents the hours, which variable is the dependent variable? Explain why. Answer: c is the dependent variable because the cost of using the snowmobile depends on the number of hours you use it. c = 7.5h b. What would be the cost of renting 2 snowmobiles for 5 hours? Answer:$75

Question 4.
In Katya’s car, the number of miles driven is proportional to the number of gallons of gas used. Find the missing value in the table.x

a. Write an equation that will relate the number of miles driven to the number of gallons of gas.
M = 28G, where M is the number of miles, and G is the number of gallons of gas.

b. What is the constant of proportionality?
28

c. How many miles could Katya go if she filled her 22-gallon tank?
616 miles

d. If Katya takes a trip of 600 miles, how many gallons of gas would be needed to make the trip?
21$$\frac{3}{7}$$ gallons

e. If Katya drives 224 miles during one week of commuting to school and work, how many gallons of gas would she use?
Oscar wrote the equation k=1.6m, and he said that the unit rate $$\frac{1.6}{1}$$ represents kilometers per mile.