## Engage NY Eureka Math 8th Grade Module 4 Lesson 22 Answer Key

### Eureka Math Grade 8 Module 4 Lesson 22 Exercise Answer Key

Exercises

Exercise 1.
Peter paints a wall at a constant rate of 2 square feet per minute. Assume he paints an area y, in square feet, after x minutes.
a. Express this situation as a linear equation in two variables.
$$\frac{y}{x}$$ = $$\frac{2}{1}$$
y = 2x

b. Sketch the graph of the linear equation.

c. Using the graph or the equation, determine the total area he paints after 8 minutes, 1 $$\frac{1}{2}$$ hours, and 2 hours. Note that the units are in minutes and hours.
In 8 minutes, he paints 16 square feet.
y = 2(90)
= 180

In 1 $$\frac{1}{2}$$ hours, he paints 180 square feet.
y = 2(120)
= 240

In 2 hours, he paints 240 square feet.

Exercise 2.
The figure below represents Nathan’s constant rate of walking.

a. Nicole just finished a 5-mile walkathon. It took her 1.4 hours. Assume she walks at a constant rate. Let y represent the distance Nicole walks in x hours. Describe Nicole’s walking at a constant rate as a linear equation in two variables.
$$\frac{y}{x}$$ = $$\frac{5}{1.4}$$
y = $$\frac{25}{7}$$ x

b. Who walks at a greater speed? Explain.
Nathan walks at a greater speed. The slope of the graph for Nathan is 4, and the slope or rate for Nicole is $$\frac{25}{7}$$. When you compare the slopes, you see that 4 > $$\frac{25}{7}$$.

Exercise 3.
a. Susan can type 4 pages of text in 10 minutes. Assuming she types at a constant rate, write the linear equation that represents the situation.
Let y represent the total number of pages Susan can type in x minutes. We can write $$\frac{y}{x}$$ = $$\frac{4}{10}$$ and y = $$\frac{2}{5}$$ x.

b. The table of values below represents the number of pages that Anne can type, y, in a few selected x minutes. Assume she types at a constant rate.

Anne types faster. Using the table, we can determine that the slope that represents Anne’s constant rate of typing is $$\frac{2}{3}$$. The slope or rate for Nicole is $$\frac{2}{5}$$. When you compare the slopes, you see that $$\frac{2}{3}$$ > $$\frac{2}{5}$$.

Exercise 4.
a. Phil can build 3 birdhouses in 5 days. Assuming he builds birdhouses at a constant rate, write the linear equation that represents the situation.
Let y represent the total number of birdhouses Phil can build in x days. We can write $$\frac{y}{x}$$ = $$\frac{3}{5}$$ and y = $$\frac{3}{5}$$ x.

b. The figure represents Karl’s constant rate of building the same kind of birdhouses.

Who builds birdhouses faster? Explain.
Karl can build birdhouses faster. The slope of the graph for Karl is $$\frac{3}{4}$$, and the slope or rate of change for Phil is $$\frac{3}{5}$$. When you compare the slopes, $$\frac{3}{4}$$ > $$\frac{3}{5}$$.

Exercise 5.
Explain your general strategy for comparing proportional relationships.
When comparing proportional relationships, we look specifically at the rate of change for each situation. The relationship with the greater rate of change will end up producing more, painting a greater area, or walking faster when compared to the same amount of time with the other proportional relationship.

### Eureka Math Grade 8 Module 4 Lesson 22 Problem Set Answer Key

Question 1.
a. Train A can travel a distance of 500 miles in 8 hours. Assuming the train travels at a constant rate, write the linear equation that represents the situation.
Let y represent the total number of miles Train A travels in x minutes. We can write $$\frac{y}{x}$$ = $$\frac{500}{8}$$ and y = $$\frac{125}{2}$$ x.

b. The figure represents the constant rate of travel for Train B.

Which train is faster? Explain.
Train B is faster than Train A. The slope or rate for Train A is $$\frac{125}{2}$$, and the slope of the line for Train B is $$\frac{200}{3}$$. When you compare the slopes, you see that $$\frac{200}{3}$$ > $$\frac{125}{2}$$.

Question 2.
a. Natalie can paint 40 square feet in 9 minutes. Assuming she paints at a constant rate, write the linear equation that represents the situation.
Let y represent the total square feet Natalie can paint in x minutes. We can write $$\frac{y}{x}$$ = $$\frac{40}{9}$$, and y = $$\frac{40}{9}$$ x.

b. The table of values below represents the area painted by Steven for a few selected time intervals. Assume Steven is painting at a constant rate.

Who paints faster? Explain.
Natalie paints faster. Using the table of values, I can find the slope that represents Steven’s constant rate of painting: $$\frac{10}{3}$$. The slope or rate for Natalie is $$\frac{40}{9}$$. When you compare the slopes, you see that $$\frac{40}{9}$$ > $$\frac{10}{3}$$.

Question 3.
a. Bianca can run 5 miles in 41 minutes. Assuming she runs at a constant rate, write the linear equation that represents the situation.
Let y represent the total number of miles Bianca can run in x minutes. We can write $$\frac{y}{x}$$ = $$\frac{5}{41}$$, and y = $$\frac{5}{41}$$ x.

b. The figure below represents Cynthia’s constant rate of running.

Who runs faster? Explain.
Cynthia runs faster. The slope of the graph for Cynthia is $$\frac{1}{7}$$, and the slope or rate for Nicole is $$\frac{5}{41}$$. When you compare the slopes, you see that $$\frac{1}{7}$$ > $$\frac{5}{41}$$.

Question 4.
a. Geoff can mow an entire lawn of 450 square feet in 30 minutes. Assuming he mows at a constant rate, write the linear equation that represents the situation.
Let y represent the total number of square feet Geoff can mow in x minutes. We can write $$\frac{y}{x}$$ = $$\frac{450}{30}$$, and y = 15x.

b. The figure represents Mark’s constant rate of mowing a lawn.

Who mows faster? Explain.
Geoff mows faster. The slope of the graph for Mark is $$\frac{14}{2}$$ = 7, and the slope or rate for Geoff is $$\frac{450}{30}$$ = 15. When you compare the slopes, you see that 15 > 7.

Question 5.
a. Juan can walk to school, a distance of 0.75 mile, in 8 minutes. Assuming he walks at a constant rate, write the linear equation that represents the situation.
Let y represent the total distance in miles that Juan can walk in x minutes. We can write $$\frac{y}{x}$$ = $$\frac{0.75}{8}$$, and y = $$\frac{3}{32}$$ x.

b. The figure below represents Lena’s constant rate of walking.

Who walks faster? Explain.
Lena walks faster. The slope of the graph for Lena is $$\frac{1}{9}$$, and the slope of the equation for Juan is $$\frac{0.75}{8}$$, or $$\frac{3}{32}$$. When you compare the slopes, you see that $$\frac{1}{9}$$ > $$\frac{3}{32}$$.

### Eureka Math Grade 8 Module 4 Lesson 22 Exit Ticket Answer Key

Question 1.
Water flows out of Pipe A at a constant rate. Pipe A can fill 3 buckets of the same size in 14 minutes. Write a linear equation that represents the situation.
Let y represent the total number of buckets that Pipe A can fill in x minutes. We can write $$\frac{y}{x}$$ = $$\frac{3}{14}$$ and y = $$\frac{3}{14}$$ x.
Pipe A fills the same-sized buckets faster than Pipe B. The slope of the graph for Pipe B is $$\frac{1}{5}$$, and the slope or rate for Pipe A is $$\frac{3}{14}$$. When you compare the slopes, you see that $$\frac{3}{14}$$ > $$\frac{1}{5}$$.