Resources

Resources

Evaluate two-variable equations: word problems

A stage crew is assembling a three level semi circular platform on a stage for a performance. The platform has the dimensions shown in the diagram and a total volume of

448 \times \pi

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A. what is the volume, in terms of

x

, of each of three levels of the platform?

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B. Use what you know about the total volume to write an equation involving

x

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C. Solve the equation from part B.

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D. Use your solution from part C to calculate the dimensions (radius and height) of each of the levels of the platform.

28

. Theater A stage crew is assembling a three-level semi-circular platform on a stage for a performance. The platform has the dimensions shown in the diagram and a total volume of

448 \pi

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a. What is the volume, in terms of

x

, of each of the three levels of the platform? Hint:

V=\pi r^{2} h

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b. Use what you know about the total volume to write an equation involving

x

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c. Solve the equation from part (b).

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d. Use your solution from part (c) to calculate the dimensions (radius and height) of each of the levels of the platform.

The plot in the

dI

-plane shows the light intensity,

I

, in watts per square meter, at a depth of

d

meters below the surface of a lake. Which of the following equations best models the relationship between

I

d

1

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I = 950 \times 1.65^d

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I = -950 \times 1.65^d

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I = 1610 \times 1.65^{-d}

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I = -1610 \times 1.65^{-d}

Li Na split up the area of a rectangle to help her solve a division equation. area model split into

4

parts, each with a width of

4

units. From top to bottom the parts have a height of

500

units and an area of

2,000

square units; a height of

100

units and an area of

400

square units; a height of

80

units and an area of

320

square units; a height of

3

units and an area of

12

2,000

400

320

12

4

500

100

80

3

units What equation did Li Na solve? Choose

1

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1

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2,732

683

683

2,732

683

683

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2,732

3

683

2,732

3

683

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2,732

4

683

2,732

4

683

Anumeha is mowing lawns for a summer job. For every mowing job, she charges an initial fee of

\$10

plus a constant fee for each hour of work. Her fee for a

5

-hour job, for instance, is

\$35

y

represent Anumeha's fee (in dollars) for a single job that took

x

hours for her to complete.

The cost of each house in New Mexico is summarized in the histogram. Houses that are worth more than

1

million dollars are not shown due to space limitations. Above

1

million dollars, the number of houses continues to decrease steadily as the cost increases. What is the range of house costs?

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1

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12

Properties of operations on rational and irrational numbers

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7

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y

z

are both irrational. Which statement about

y \cdot z

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y \cdot z

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y \cdot z

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y \cdot z

can be rational or irrational, depending on the values of

y

z

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Seema is managing her monthly budget and wants to analyze her spending patterns. With a total monthly income of ₹

60,000

, she has allocated her budget as follows:

30\%

15\%

10\%

for utilities, and

25\%

for savings. The remaining amount is designated for miscellaneous expenses. Seema seeks to understand how her spending aligns with these allocations and to assess whether she is staying within her financial goals.

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(a) How much does Seema spend on rent in rupees?

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(b) What is the amount spent on groceries in rupees?

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(c) Calculate the amount allocated to utilities.

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(d) How much does Seema save each month?

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(e) Determine the amount spent on miscellaneous expenses.

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Raj's catering company is serving an event for

35

guests. Each guest will choose either a salad or a bowl of soup.

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1

. Write an equation that represents how many total salads Raj needs to prepare

(s)

b

guests choose a bowl of soup.

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2

. How many salads does Raj need to prepare if

21

guests choose a bowl of soup? salads

Raj's catering company is serving an event for

35

guests. Each guest will choose either a salad or a bowl of soup.

\newline

1

. Write an equation that represents how many total salads Raj needs to prepare

(s)

b

guests choose a bowl of soup.

\newline

2

. How many salads does Raj need to prepare if

21

guests choose a bowl of soup? salads

A group of friends ran in a

200

-meter race. The graph in the

t

d

plane shows Ryan's distance,

R

, Beatriz's distance,

B

, and Grace's distance,

G

, all in meters,

t

seconds since the start of the race. Who stopped running temporarily before the race was over?

A veterinarian prescribes flea-prevention medicine for cats based on the weight of the cat. The dosage is

0.03

milliliters of medicine per pound the cat weighs. How many milliliters of medicine would the veterinarian prescribe for a

15

Fang and Meng have a large rose garden. When the roses first bloomed, they initially picked

4

fresh roses. Each day thereafter, as more roses bloomed, they quadrupled the number of roses they picked. Let

R

be the number of roses picked on day

t

. Which of the following best explains the relationship between

t

R

? The relationship is exponential because the values of

R

always increase as the values of

t

increase.The relationship is exponential because

R

increases by a factor of

4

t

1

Pure water freezes at

0^\text{o}\text{C}

. Yoshimi thought that water would freeze at lower temperatures if salt was added to it. To test this, Yoshimi added different amounts of salt to cups that contained

1

liter of water. The table shows the data that Yoshimi collected.

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What is the dependent variable?

Prizes are to be awarded to the best pupils in each class of an elementary school. The number of students in each grade is shown in the table, and the school principal wants the number of prizes awarded in each grade to be proportional to the number of students. If there are twenty prizes, how many should go to fifth-grade students?

Problem At a pond there were ducks swimming. The tape diagram shows the ratio of ducklings to adult ducks. A tape diagram with

2

tapes of unequal lengths. The first tape has

5

equal parts. A curved bracket above the first tape is labeled Ducklings. The second tape has

1

part of the same size as in the first tape. A curved bracket below the second tape is labeled Adult ducks. Complete the table. Original ratio

\text{Number in the pond}

\quad

\quad

Laverne, a hair dresser, is going to work at most

7

hours in the salon tomorrow. It takes her

0.7

hour to do a haircut and

2

hours to color someone's hair. Select the inequality in standard form that describes this situation. Use the given numbers and the following variables.

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x

= the number of haircuts Laverne can fit in

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y

= the number of color appointments Laverne can fit in

Urpi collects rainwater in a barrel for her garden. One week, it did not rain, so Urpi used

234

liters of the rainwater. The next week it rained a little, so Urpi got

280

more liters in her barrel, then used

137

of them. Urpi wonders how much the rainwater volume in her barrel has changed. Which of the following equations matches the situation above?

Jada sells ground paprika. Her weekly profit (in dollars) as a function of the amount of paprika she sold that week (in kilograms) is graphed. The first and fourth quadrants of a coordinate plane. The horizontal axis is from

0

100

with a scale of

5

and is titled Paprika in kilograms. The vertical axis is from

-300

500

with a scale of

100

and is titled Profit in dollars. The graph of the line is

y = 8x - 280

. How much does Jada make for selling

5

kilograms of paprika? Choose

1

The following data points represent last year's revenue (in thousands of dollars) for each Herman's Hoagies location.

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2

questions about the data points.

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1

. Sort the data from least to greatest.

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2

. Find the median revenue.

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\square

thousand dollars

The following data points represent last year's revenue (in thousands of dollars) for each Herman's Hoagies location.

\newline

2

questions about the data points.

\newline

1

. Sort the data from least to greatest.

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2

. Find the median revenue.

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\square

thousand dollars

The following data points represent last year's revenue (in thousands of dollars) for each Herman's Hoagies location.

\newline

2

questions about the data points.

\newline

1

. Sort the data from least to greatest.

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2

. Find the median revenue.

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\square

thousand dollars

Yak Travel Agency arranges trips for climbing Mount Everest. For each trip, they charge an initial fee in addition to

ext{‍}

for each vertical meter climbed. For instance, the price for climbing all the way to the summit, which is

ext{‍}

meters above the base of the mountain, is

ext{‍}

ext{‍}

represent the fee (in dollars) of a trip where they climbed

ext{‍}

vertical meters. Complete the equation for the relationship between the fee and vertical distance.

Yak Travel Agency arranges trips for climbing Mount Everest. For each trip, they charge an initial fee in addition to

ext{‍}

for each vertical meter climbed. For instance, the price for climbing all the way to the summit, which is

ext{‍}

meters above the base of the mountain, is

ext{‍}

ext{‍}

represent the fee (in dollars) of a trip where they climbed

ext{‍}

vertical meters. Complete the equation for the relationship between the fee and vertical distance.

Jacque needs to buy some pizzas for a party at her office. She's ordering from a restaurant that charges a

\$7.50

delivery fee and

\$14

per pizza. She wants to buy as many pizzas as she can, and she also needs to keep the delivery fee plus the cost of the pizzas under

\$60

. Each pizza is cut into

8

slices, and she wonders how many total slices she can afford. Let

p

represent the number of pizzas that Jacque buys.

Mrs. Tyler is researching what it would cost to order flower arrangements for a fancy party. She wants one large centerpiece for the head table, and smaller arrangements for the smaller tables. Dayton Florist charges

\$12

for each smaller arrangement, plus

\$40

for the large centerpiece. Grace's Flowers, in contrast, charges

\$50

for the large centerpiece and

\$10

per arrangement for the rest. If Mrs. Tyler orders a certain number of small arrangements, the cost will be the same at either flower shop. What would the total cost be?

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

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The volleyball team and the wrestling team at Stafford High School are having a joint car wash today, and they are splitting the revenues. The volleyball team gets

\$2

per car. In addition, they have already brought in

\$30

from past fundraisers. The wrestling team has raised

\$35

in the past, and they are making

\$1

per car today. After washing a certain number of cars together, each team will have raised the same amount in total. What will that total be?

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

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Martin works in the shipping department of a toy factory that makes radio-controlled helicopters. Small helicopters weigh

2

pounds each, and are shipped in a container that weighs

20

pounds. Large ones, on the other hand, weigh

4

pounds apiece, and are shipped in a container that weighs

10

pounds. If these boxes can hold a certain number of helicopters each, all of the packed containers will have the same shipping weight. What would the total weight be?

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

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Sarah has whole milk, which contains

3.25\%

butterfat by volume, and one type of low-fat milk, which contains

1\%

butterfat by volume. Which of the following systems could be used to determine the amount of whole milk,

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w

, in ounces, and the amount of low-fat milk,

l

, in ounces, that Sarah should mix to obtain

32

ounces of a low-fat milk with

2\%

butterfat by volume?

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1

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0.01 l+0.0325 w=32

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\space

\space

\space

\space

\space

\space

\space

l+w=0.02

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0.01 l+0.0325 w=32

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\space

\space

\space

\space

\space

\space

\space

l+w=0.64

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0.01 l+0.0325 w=0.2

\newline

\space

\space

\space

\space

\space

\space

\space

l+w=32

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0.01 l+0.0325 w=0.64

\newline

\space

\space

\space

\space

\space

\space

\space

l+w=32

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This equation shows how the size of Beth's sticker collection is related to the number of sticker packs she buys.

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s = 20p

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p

represents the number of sticker packs she buys, and the variable

s

represents the total number of stickers she has. After buying

1

sticker pack, how many stickers will Beth have in total?

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This equation shows how the size of Steve's sticker collection is related to the number of sticker packs he buys.

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s = 20p

\newline

p

represents the number of sticker packs he buys, and the variable

s

represents the total number of stickers he has. After buying

1

sticker pack, how many stickers will Steve have in total?

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This equation shows how the distance Pablo has cycled depends on the number of trips he has taken to work.

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d = t + 9

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t

represents the number of trips he has made, and the variable

d

represents the total distance cycled in kilometers. After

6

trips to work, how many kilometers will Pablo have cycled in total?

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_____ kilometers

This equation shows how the distance Janice has cycled depends on the number of trips she has taken to work.

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d = t + 14

\newline

t

represents the number of trips she has made, and the variable

d

represents the total distance cycled in kilometers. After

5

trips to work, how many kilometers will Janice have cycled in total?

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_____ kilometers

This equation shows how the time required to screen-print a batch of shirts is related to the number of shirts in the batch.

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t = 3s

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s

represents the number of shirts in the batch, and the variable

t

represents the time required to screen-print the shirts. How long does it take to screen-print a batch of

5

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This equation shows how the length of Darnell's essay depends on the number of hours he spends writing this week.

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p = h

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h

represents the number of hours he spends writing this week, and the variable

p

represents the total number of pages that have been written. After spending

4

hours writing this week, how many pages will Darnell have written in total?

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This equation shows how the number of flowers Malik can have in his garden is related to the number of seed packets he purchases.

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f = s + 7

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s

represents the number of seed packets he purchases, and the variable

f

represents the total number of flowers in the garden. With

10

seed packets, how many total flowers can Malik have in his garden?

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This equation shows how the amount Billy earns from dog walking depends on the number of hours he works.

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d = 15h

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h

represents the number of hours spent walking dogs, and the variable

d

represents the amount of money earned. After a total of

1

hour of walking dogs, how much money will Billy have earned?

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\$

\_\_\_\_

Pick the story that can be modeled by the equation

x + 2 = 8

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(A)At Hayland Farms, Leo picked a total of

8

small pumpkins. He picked

x

orange pumpkins and

2

white pumpkins.

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(B)At Hayland Farms, Leo picked

2

white pumpkins. Each pumpkin cost

x

dollars, and Leo paid

\$8

Pick the story that can be modeled by the equation

\frac{x}{5} = 15

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(A)Leah and her friends went out to dinner last night. Including tip, the bill came out to

x

dollars. The group split the bill evenly among the

5

of them. They each paid

\$15

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(B)Leah and her friends went out to dinner last night. Leah's meal cost

x

dollars, but she used a

\$5

coupon. Leah only spent

\$15

Kendall made a Dance Mix playlist for her birthday party. She added

s

songs from each of her

5

favorite bands. In all, Kendall added

50

songs to her Dance Mix playlist.

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Which diagram models the story?

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Which equation models the story?

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\newline

5s = 50

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\frac{s}{5} = 50

Camilla is moving next week, and she is comparing rental vehicles that will fit all her furniture. She is deciding between a cargo van that is

7

feet long and a box truck that is

10

feet long. Camilla wants as much space as possible, so she decides to rent the box truck since it is

b

feet longer than the cargo van.

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Which diagram models the story?

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Which equation models the story?

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b - 10 = 7

\newline

7 + b = 10

Mr. Newton wears science-themed ties to school. In all, he has

34

ties with scientific designs. Of all the ties in his collection,

b

are bow ties and

21

are straight ties.

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Which diagram models the story?

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Which equation models the story?

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(A)

b + 21 = 34

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(B)

b - 21 = 34

Alice signed up to bake chocolate chip and oatmeal raisin cookies for a bake sale. After she made

c

chocolate chip cookies, Alice decided to make

12

fewer oatmeal raisin cookies, since they are less popular. She only made

24

oatmeal raisin cookies.

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Which diagram models the story?

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Which equation models the story?

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c + 12 = 24

\newline

c - 12 = 24

Victoria teaches a dog-training course at her local community center. After every trick, she rewards each dog with a treat. During Saturday's class, she gave

t

treats to each of the

6

dogs. In all, Victoria gave out

30

treats during the class.

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Which diagram models the story?

\newline

\newline

Which equation models the story?

\newline

\newline

(A)

\frac{t}{6} = 30

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(B)

6t = 30

Azalea Botanical Gardens is adding new tropical greenhouses. The gardeners will split

t

tropical plants evenly among the

3

new greenhouses. They will put

20

tropical plants in each greenhouse.

\newline

Which diagram models the story?

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Which equation models the story?

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\newline

3t = 20

\newline

\frac{t}{3} = 20

Craig makes bottles of bubble bath and scented soap bars. He sells them at his local farmers' market. On Saturday, he sold

b

bottles of bubble bath and

25

soap bars. Craig sold

38

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Which diagram models the story?

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Which equation models the story?

\newline

\newline

\newline

b + 25 = 38

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b - 25 = 38

Edmond bought a new cage for his pet rabbit since he's gotten so big. The area of the old cage's floor was

6

square feet. The area of the new cage's floor is

n

square feet larger, for a total of

8

\newline

Which diagram models the story?

\newline

Which equation models the story?

\newline

\newline

n - 6 = 8

\newline

6 + n = 8

The Wrap & Roll food truck is the most popular lunch spot in the city. They have

v

vegetarian wraps and

8

wraps containing meat on today's menu. There are

12

wraps on the menu in all.

\newline

Which diagram models the story?

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Which equation models the story?

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v + 8 = 12

\newline

v - 8 = 12

Mary's favorite hobby is tending to her vegetable garden. The garden bed has a total area of

48

square feet. She has planted cherry tomatoes in

t

square feet of the garden and cucumbers in the other

36

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Which diagram models the story?

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Which equation models the story?

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t + 36 = 48

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t - 36 = 48

Ann is making her grandfather a scrapbook for his birthday. The scrapbooking kit she bought has

15

fewer pieces of colored paper than pieces of white paper. The scrapbooking kit has

w

pieces of white paper and

35

pieces of colored paper.

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Which diagram models the story?

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Which equation models the story?

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w + 15 = 35

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w - 15 = 35

...