Resources

Resources

Write and solve equations for proportional relationships

On a set of steps, the first step from the base of the steps is labeled

1

, the second is labeled

2

, and so on. Each step has the same height. The table below shows the step number and step's height from the base of the steps.

\newline

\begin{tabular}{cc}

\newline

Step number & Height from the base (meters) \\

\newline

50

10

\newline

90

18

\newline

110

22

\newline

\newline

Write an equation to describe the relationship between

n

, the step number, and

h

, the height from the base.

\newline

\square

The following table shows a proportional relationship between

. Write an equation to describe the relationship between

Audrey and Gabriel, the boy she was babysitting, were playing basketball together. Her score was

30

points, and his score was

10

points. Audrey wanted to make the game more fair, so she called a time-out and modified the rules a bit. Audrey explained that, for the rest of the game, she would get

3

points per basket, and Gabriel would get

7

points per basket. Then they played a bit longer. After the time-out, they both made the same number of baskets and ended up with a tied score. How many baskets did each person make after the time out?

\newline

Write a system of equations, graph them, and type the solution.

\newline

\newline

Read the description of a proportional relationship.

\newline

The basketball gym at Darrell's school was built in the

1940

s and needs a lot of repairs. So, the school is asking families for donations to build a new gym. Darrell convinced some local businesses to match a fraction of the families' donations. There is a proportional relationship between the amount of money a family donates (in dollars),

x

, and the matching contribution (in dollars) from local businesses,

y

\newline

When a family donates

\$10

, the businesses contribute

\$5

. Write the equation for the relationship between

x

y

\newline

y = \_\_

Read the description of a proportional relationship.

\newline

Sean wants to save up some money to buy a new smartphone, so he babysits on the weekends. There is a proportional relationship between the time Sean spends babysitting (in hours),

x

, and the amount of money he earns babysitting (in dollars),

y

\newline

1

hour of babysitting, Sean earns

\$6

. Write the equation for the relationship between

x

y

\newline

y =

Read the description of a proportional relationship.

\newline

Kimi uses her phone primarily to take a lot of selfies. The storage space on her phone is limited, so Kimi needs to keep track of how many selfies she takes. There is a proportional relationship between the number of selfies Kimi takes,

x

, and the amount of storage (in megabytes) she uses taking selfies,

y

\newline

When Kimi takes

5

selfies, they take up

10

megabytes of storage. Write the equation for the relationship between

x

y

\newline

y = \_

Read the description of a proportional relationship.

\newline

Clara owns and operates Stain-B-Gone Carpet Cleaners. She uses a special non-toxic cleaning solution to clean carpets. There is a proportional relationship between the amount of cleaning solution (in quarts) Clara uses,

x

, and the area of carpet (in square yards) she cleans with the solution,

y

\newline

3

quarts of cleaning solution to clean

12

square yards of carpet. Write the equation for the relationship between

x

y

\newline

y = \_

Read the description of a proportional relationship.

\newline

In The Tall Tales of Timothy Toad, a witch brews a magical invisibility potion. There is a proportional relationship between the amount of potion the witch drinks (in milliliters),

x

, and the amount of time she is invisible (in minutes),

y

\newline

1

milliliter of potion, the witch is invisible for

3

minutes. Write the equation for the relationship between

x

y

\newline

y = \_\_

Read the description of a proportional relationship.

\newline

There's a huge sale at the grocery store, so Carla stocks up on her favorite breakfast cereal, Frosted Crunchy Bunches. As she puts the cereal boxes on a shelf in her pantry, she wonders if she has enough shelf space. There is a proportional relationship between the number of cereal boxes Carla stores on her pantry shelf,

x

, and the amount of shelf space (in inches) the boxes take up,

y

\newline

3

12

inches of shelf space. Write the equation for the relationship between

x

y

\newline

y = \_\_\_

Read the description of a proportional relationship.

\newline

Jill is thrilled to be cast as Juliet in her school's production of Romeo and Juliet, but she has lot of lines to memorize! There is a proportional relationship between the number of days Jill has been memorizing her lines,

x

, and the total number of lines she has memorized,

y

\newline

2

days of practice, Jill has memorized

10

lines. Write the equation for the relationship between

x

y

\newline

y = \_

Pick the story that can be modeled by the equation

3 + x = 9

\newline

\newline

(A) For his camp's Neon Night, Dale bought

9

boxes of glow sticks. He split the boxes evenly among

x

cabins. Each cabin got

3

boxes of glow sticks.

\newline

(B) For his camp's Neon Night, Dale bought

9

boxes of glow sticks. He bought

3

boxes of pink glow sticks and

x

boxes of green glow sticks.

A recipe book shows measurement conversions for tablespoons to cups. It shows that

8

tablespoons converts to

0.5

24

tablespoons converts to

1.5

cups. Write an equation that shows the proportional relationship between cups and tablespoons where

c

represents cups and

n

represents tablespoons.

Read the description of a proportional relationship.

\newline

Dana is finding the perimeter of different-sized squares. There is a proportional relationship between the side length of the square in inches,

x

, and the perimeter of the square in inches,

y

\newline

2

-inch side lengths has a perimeter of

8

inches. Write the equation for the relationship between

x

y

\newline

y =

Read the description of a proportional relationship.

\newline

The Picture and Sound electronics store has hired Allie to work in the warehouse. She uses a pulley with a hand crank to lift heavy boxes. There is a proportional relationship between the number of times Allie turns the crank to lift a box,

x

, and the height the box has been lifted (in feet),

y

\newline

She turns the crank

4

times to lift a box

1

foot. Write the equation for the relationship between

x

y

\newline

y = \_

Read the description of a proportional relationship.

\newline

Every Sunday, Jack and his friends get together to play their favorite board game, The Gems of Rawumba. Whoever collects the most gems wins! Players can earn gems by building mines. There is a proportional relationship between the number of mines Jack builds,

x

, and the number of gems he earns,

y

\newline

5

mines, he earns

10

gems. Write the equation for the relationship between

x

y

\newline

y = \_

Read the description of a proportional relationship.

\newline

In a flash of sheer brilliance, Bridget invents a time machine! The machine uses a small nuclear reactor to generate the electricity it needs to travel back in time. There is a proportional relationship between how many years Bridget wants to travel back in time,

x

, and how much electricity (in megawatts) her time machine needs,

y

\newline

7

years, Bridget's time machine needs

14

megawatts of electricity. Write the equation for the relationship between

x

y

\newline

y = \_

Read the description of a proportional relationship.

\newline

Neil measures the lengths of some objects in both yards and feet. There is a proportional relationship between the number of yards,

x

, and the number of feet,

y

\newline

An object that is

2

6

feet long. Write the equation for the relationship between

x

y

\newline

y =

Read the description of a proportional relationship.

\newline

When a truck full of boxes arrives at the Weston Post Office, Bill unloads the truck by placing the boxes on a conveyor belt. Bill's coworkers are impressed with how fast Bill does his job. There is a proportional relationship between the time (in minutes) Bill spends unloading the truck,

x

, and the number of boxes he unloads from the truck,

y

\newline

1

minute, Bill unloads

6

boxes from the truck. Write the equation for the relationship between

x

y

\newline

y = \_\_

Read the description of a proportional relationship.

\newline

Maggie's space shuttle crew accidentally left her behind on a mysterious planet. She soon realizes that her body is aging much faster on this planet than it did on Earth. There is a proportional relationship between the number of weeks Maggie spends on the planet,

x

, and the number of years she ages,

y

\newline

3

weeks on the planet, Maggie's body has aged

12

years. Write the equation for the relationship between

x

y

\newline

y = \_

Read the description of a proportional relationship.

\newline

To help the environment and make some extra money, Dan takes his empty cans to the Cash Not Trash recycling shop. The shop pays cash based on the total weight of the cans Dan brings in for recycling. There is a proportional relationship between the weight (in pounds) of the cans Dan brings into the shop,

x

, and the amount (in dollars) the shop pays Dan,

y

\newline

Last month, Dan brought

4

pounds of cans into the shop and was paid

\$1

. Write the equation for the relationship between

x

y

\newline

y = \_\_

Read the description of a proportional relationship.

\newline

Tanner has outgrown the bike he rode when he was small. So, for his twelfth birthday, his parents surprise him with a five-speed Blaze Rider

2000

! Tanner has never had a bike with multiple speeds before, and he is eager to learn how it works. He starts by testing the middle speed. There is a proportional relationship between the number of times Tanner pedals,

x

, and the number of times the wheels rotate,

y

\newline

4

times, the wheels rotate

8

times. Write the equation for the relationship between

x

y

\newline

y = \_\_

Read the description of a proportional relationship.

\newline

Scott loves to play the piano but doesn't like practicing the exercises his piano teacher assigns. Scott knows the exercises will help him play better though, so he tries to motivate himself using his favorite treat \mdash chocolate chip cookies. There is a proportional relationship between the amount of time (in hours) that Scott practices the piano,

x

, and how many cookies he gives himself,

y

\newline

When Scott practices for

1

hour, he gives himself

4

cookies. Write the equation for the relationship between

x

y

\newline

y = \_

Read the description of a proportional relationship.

\newline

Over the summer, Belleville Science Academy renovates its building. The academy's principal hires Aubrey to lay new tile in the main hallway. There is a proportional relationship between the length (in feet) of hallway Aubrey covers with tiles,

x

, and the number of tiles she needs,

y

\newline

To cover the first

2

feet of the hallway, Aubrey needs

8

tiles. Write the equation for the relationship between

x

y

\newline

y = \_\_

Read the description of a proportional relationship.

\newline

Joy is hooked on a new book series, The Galaxy. Each book in the series is the same length and chronicles a different year in the Waka Waka Galaxy. Joy has cleared off the top shelf of her bookcase to leave room for each of the books as they come out. There is a proportional relationship between the number of books on the shelf,

x

, and how much shelf space the books take up (in inches),

y

\newline

3

books in the series take up

6

inches of space along the shelf. Write the equation for the relationship between

x

y

\newline

y = \_\_

Ehsan is having a big going-out-ofbusiness sale and he wants to advertise his sale. He starts by making a large number of advertising fliers for his sale and gives

5

of his friends many fliers. He asks each of them to give fliers to

2

of their friends the next day, and to tell each of their friends to give fliers to

2

of their friends the day after, and so on. Let

F

be the number of people who will be given fliers on day

t

. Which of the following best explains the relationship between

t

F

\newline

1

\newline

(A) The relationship is linear because the values of

F

always increase as the values of

t

\newline

(B) The relationship is exponential because

F

increases by a factor of

5

t

1

\newline

(C) The relationship is linear because

F

increases by a factor of

5

t

1

\newline

(D) The relationship is exponential because

F

increases by a factor of

2

t

1

Dyami planted a palm tree in the back yard of his house several years ago. Initially it was

200

centimeters high and its height increased by

30

centimeters each year. Let

H

be the height of the tree in centimeters

t

years after it was planted. Which of the following best explains the relationship between

t

H

\newline

1

\newline

(A) The relationship is exponential because

H

increases by a factor of

30

t

1

\newline

(B) The relationship is linear because

H

30

t

1

\newline

(C) The relationship is exponential because

H

always increases as

t

\newline

(D) The relationship is linear because

H

200

t

1

The physical education instructor asked each student to do a total of

36

1

minute. The instructor wanted students to do

8

times as many push-ups as pull-ups. Write a system of equations of linear equations that represents this situation. How many pull-ups and push-ups were required in

1

Juan and Rob are selling cookie dough for a school fundraiser. Juan has

t

cookie dough orders, and Rob has

40

cookie dough orders. They have a total of

75

cookie dough orders all together.

\newline

Write an equation to describe this situation.

300

-page book. She read at a rate of

15

pages per day in

d

\newline

Write an equation to describe this situation.

There's a roughly linear relationship between the number of times a species of cricket will chirp in one minute and the temperature outside. For a certain type of cricket, this relationship can be expressed using the formula

T=0.35 c+40

T

represents the temperature in degrees Fahrenheit and

c

represents the number of times the cricket chirps in one minute. What is the meaning of the

c

T=95 ?

\newline

The number of times the cricket could be expected to chirp in one minute if it's

95^{\circ} \mathrm{F}

\newline

The change in temperature in degrees Fahrenheit for each additional cricket chirp in one minute.

\newline

The expected temperature in degrees Fahrenheit if the cricket has chirped o times per minute.

\newline

The expected temperature in degrees Fahrenheit if the cricket has chirped

95

times per minute.

There's a roughly linear relationship between the number of times a species of cricket will chirp in one minute and the temperature outside. For a certain type of cricket, this relationship can be expressed using the formula

T=0.25 c+37

T

represents the temperature in degrees Fahrenheit and

c

represents the number of times the cricket chirps in one minute. What is the meaning of the

c

T=77

\newline

The expected temperature in degrees Fahrenheit if the cricket has chirped o times per minute.

\newline

The change in temperature in degrees Fahrenheit for each additional cricket chirp in one minute.

\newline

The expected temperature in degrees Fahrenheit if the cricket has chirped

77

times per minute.

\newline

The number of times the cricket could be expected to chirp in one minute if it's

77^{\circ} \mathrm{F}

Melissa and Rachel like to make funny hats. Melissa has made

20

zebra-printed hats, and Rachel has made

h

striped hats. Together they have made a total of

42

\newline

Write an equation to describe this situation.

Melissa and Rachel like to make funny hats. Melissa has made

20

zebra-printed hats, and Rachel has made

h

striped hats. Together they have made a total of

42

\newline

Write an equation to describe this situation.

Melissa and Rachel like to make funny hats. Melissa has made

20

zebraprinted hats, and Rachel has made

h

striped hats. Together they have made a total of

42

\newline

Write an equation to describe this situation.

Brandon and Jerry like to play video games. Brandon has

v

video games, and Jerry has

29

video games. Together they have a total of

73

\newline

Write an equation to describe this situation.

Chas and Tyler have been collecting video games. Chas has

25

video games, and Tyler has

m

video games. Together they have a total of

65

\newline

Write an equation to describe this situation.

Richard and Jordan went to see a movie. Richard spent

m

dollars at the movie theater, and Jordan spent

\$ 12

at the movie theater. Together they spent a total of

\$ 26

\newline

Write an equation to describe this situation.

Ha-yoon can dive to a depth of

7

meters. Ji-woo cannot dive as deep as

\mathrm{Ha}

\newline

Write an inequality that describes

j

, the depths Ji-woo can dive to in meters.

A baker needs more than

120

minutes to let their dough rise.

\newline

Write an inequality that describes

m

, the number of minutes needed for the dough to rise.

On her way home from the laboratory, Duru realized that she left a test tube containing

50

000

bacteria in the lab. Each minute that passes,

\frac{1}{3}

of the total number of bacteria duplicate. If the number of bacteria reaches

100

000

, the test tube will explode! Naturally, she turned around and rushed back to the lab.

\newline

t

minutes to return to the lab, and she found the test tube intact.

\newline

Write an inequality in terms of

t

that models the situation.

3

trucks to deliver a total of

T

liters of cement. Each truck can transport

x

liters of cement per trip, and makes a total of

y

\newline

Write an equation that relates

T

x

y

A certain factory produces and sells

C

computers per month. It costs

x

dollars to produce a single computer, and the factory sells each computer for

y

dollars. Their monthly net profit is

10

000

\newline

Write an equation that relates

C

x

y

Fen completed a triathlon race whose total distance is

D

kilometers. First, she swam a distance of

0

5

kilometer. Then, she rode a bicycle for

x

hours at an average speed of

20

kilometers per hour. Finally, she ran for

0

5

hour at an average speed of

v

kilometers per hour.

\newline

Write an equation that relates

D

x

v

A certain factory produces and sells

8000

cars per month, making a monthly net profit of

P

dollars. They sell each car for

n

dollars and it costs them

c

dollars to produce a single car.

\newline

Write an equation that relates

P

n

c

x

delivery persons carry a total of

B

bags of groceries up the stairs. They can each carry

y

bags at a time, and they each make

2

trips up the stairs.

\newline

Write an equation that relates

x

B

y

A certain factory produces and sells

x

kilograms of ice cream per month. It costs

c

dollars to produce one kilogram of ice cream, which is then sold for

\mathbf{0 . 1 6}

dollars. The factory's monthly net profit is

P

\newline

Write an equation that relates

x, c

P

Wilmer went up the hill for

x

minutes at a speed of

y

kilometers per minute. Then he went down the same path at a speed of

z

kilometers per minute, and it took

\operatorname{him} w

minutes to do it.

\newline

Write an equation that relates

x, y

z

w

Each Christmas elf makes

x

toys in an hour.

n

t

hours to produce

1000

\newline

Write an equation that relates

x, n

t

25

-meter-long fence that she plans to use to enclose a rectangular garden of width

w

. The fencing will be placed around all four sides of the garden so that its area is

25

\newline

Write an equation in terms of

w

that models the situation.

20

-meter-long fence that she plans to use to enclose a rectangular garden of width

w

. The fencing will be placed around all four sides of the garden so that its area is

18

75

\newline

Write an equation in terms of

w

that models the situation.