Resources

Resources

Solve a system of equations using any method: word problems

Ivy is downloading a computer program from the Internet. After

8

minutes, the computer program is

35 \%

downloaded. If the computer program continues to download at the current rate, about how much longer will it take for Ivy's computer to finish downloading the program?

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1

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12

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\mathbf{1 5}

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18

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23

A grocery store normally sells lemonade for

\$ 3.50

per bottle. The grocery store is currently having a sale on lemonade which advertises

6

bottles for only

\$ 13.50

. How much cheaper is the lemonade on sale per bottle compared to the normal price per bottle?

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1

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\$ 0.75

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\$ 1.25

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\$ 1.75

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\$ 2.25

An umbrella sprinkler is positioned on a ceiling at a point whose

x

0

. Negative values of

x

indicate distances, in meters, to the left of the position of the sprinkler, and positive values indicate distances to the right.

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The path of water from the sprinkler can be modeled by the quadratic function

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w(x)=-\frac{1}{4}\left(x^{2}-12\right)

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w(x)

is the height of the water, in meters, at position

x

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Which of the following equivalent expressions displays the height of the ceiling as a constant or coefficient?

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1

A forest ranger travels up a river by boat to collect soil samples. The boat travels at a speed of

v

miles per hour for a distance of

2

miles. After collecting the samples, the ranger travels the same distance back, but the boat travels

5

miles per hour slower than it did on the way up the river. If the ranger spent

3

hours travelling up and back, which of the following equations could be used to determine the speed of the boat on the way up the river?

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1

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2 v^{2}-13 v+17=0

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3 v^{2}-15 v-4=0

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3 v^{2}-11 v+10=0

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3 v^{2}-19 v+10=0

In game theory, the average branching factor refers to the average number of legal moves available to a player at each turn. In chess, the average branching factor is about

35

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If Jules wants to write a chess simulation program that represents every possible sequence of moves in a game that lasts

n

moves, the following function approximates the number of distinct sequences

s(n)

he would simulate.

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s(n)=35^{n}

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How many times greater does the number of sequences become each time the game lasts

2

The Capulet and Montague families love writing.

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Last year, each Capulet wrote

4

essays, each Montague wrote

6

essays, and both families wrote

100

essays in total.

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This year, each Capulet wrote

8

essays, each Montague wrote

12

essays, and both families wrote

200

essays in total.

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How many Capulets and Montagues are there?

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1

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(A) There is not enough information to determine the exact number of Capulets and Montagues.

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(B) The given information describes an impossible situation.

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16

6

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6

16

When Famke charges her phone, it gains

2

percentage points each minute. When Famke uses her phone, it loses

5

percentage points each minute.

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Once, Famke used her phone for some time, and then charged it for twice the amount of time she used it. In the end, her phone lost

10

percentage points.

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How long did Famke use her phone and how long did she charge it?

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1

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(A) There is not enough information to determine the exact time Famke used and charged her phone.

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(B) The given information describes an impossible situation.

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(C) Famke used her phone for

10

minutes, and charged her phone for

20

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(D) Famke used her phone for

20

minutes, and charged her phone for

\mathbf{1 0}

Liam's bookstore sold

40

20

newspapers for a total of

\$ 130

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A day later, the bookstore sold

8

4

newspapers at the same prices, for a total of

\$ 28

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How much does a notebook and a newspaper cost at Liam's bookstore?

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1

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(A) There is not enough information to determine the exact cost of a notebook and a newspaper.

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(B) The given information describes an impossible situation.

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(C) A notebook costs

\$ 2.50

and a newspaper costs

\$ 1.50

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(D) A notebook costs

\$ 1.50

and a newspaper costs

\$ 2.50

6

tubes of red paint,

2

tubes of yellow paint,

13

tubes of white paint, and

9

tubes of blue paint for a mural.

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What is the ratio of tubes of blue paint to all of the tubes of paint used?

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1

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30

9

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30

21

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9

30

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21

30

Erin loves to play sports. In all, she has earned

3

tennis trophies,

4

basketball trophies,

7

soccer trophies, and

1

volleyball trophy.

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What is the ratio of Erin's tennis trophies to all of her trophies?

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1

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3: 4

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3: 7

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3: 12

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3: 15

F=P+P r t

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For an investment that bears simple interest, one can predict future value

F

t

years if the principal or initial amount invested

P

and annual interest rate

r

are known, by using the given equation. Which of the following correctly expresses the principal amount in terms of the annual rate, number of years invested, and future value?

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1

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P=\frac{F}{1+r t}

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P=\frac{F}{1-r t}

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P=\frac{F+P}{r t}

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P=\frac{F-P}{r t}

\frac{P_{1} V_{1}}{T_{1}}=\frac{P_{2} V_{2}}{T_{2}}

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Commonly known as the Combined Gas Law, the given equation relates the pressure,

P

V

, and temperature,

T

, of an ideal gas between its initial (

1

2

) states. Which of the following shows the final volume in terms of the initial and final pressure, the initial and final temperatures, and the initial volume?

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1

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V_{1}=V_{2}

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V_{2}=\frac{P_{1} T_{2} V_{1}}{\left(T_{1} / P_{2}\right)}

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V_{2}=\frac{P_{1} T_{2} V_{1}}{P_{2} T_{1}}

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V_{2}=\frac{P_{1} V_{1}}{T_{1}}-\frac{T_{2}}{P_{2}}

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For safety, each climber has

8

hooks when climbing the mountain. A large group of climbers climbs up a mountain and puts all their supplies in a heap in a cabin. One of the climbers wishes to know the number of climbers in the group, so he counts the number of hooks in the supply heap. If the climber counts

h

hooks, which of the following functions can be used to find the number of climbers,

c(h)

, in the group?

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1

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c(h)=\frac{h}{64}

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c(h)=\frac{h}{8}

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c(h)=8 h

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c(h)=64 h

Video games create a sense of motion by displaying still images, called frames, in rapid succession. A popular video game displays

60

frames per second. Which of the following equations shows the total number of frames,

f

, that the video game will display in

s

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1

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f=s+60

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f=\frac{s}{60}

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f=\frac{60}{s}

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f=60 s

Miles's Internet service provider lets him use up to

1

229

gigabytes of data in a month and charges him an additional fee if he goes over the data limit. If Miles already used

984

gigabytes of data this month, which of the following represents

d

, the amount of data, in gigabytes, he can use in the rest of the month without incurring an additional fee?

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1

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d \leq 245

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d \geq 245

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d \leq 2,213

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d \geq 2,213

3.5 c-1.5 d \geq 50

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Esa's Pastries sells cupcakes for

\$ 3.50

each and donuts for

\$ 1.50

each. The given inequality represents the difference, in dollars, between cupcake sales and donut sales on a typical day based on

c

, the number of cupcakes sold and

d

, the number of donuts sold. If Esa sold

200

donuts on a typical day, what is the minimum number of cupcakes she sold on that day?

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1

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25

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50

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100

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350

Jules is making a peanut butter sandwich with two slices of bread and chunky peanut butter. If the mass of two slices of bread is

60

grams and the mass of the entire sandwich is

92

grams, which of the following equations can be used to find

x

, the amount of peanut butter Jules used in grams?

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1

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30+x=92

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60+x=92

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120+x=92

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60 x=92

Jasmine sleeps for an average of

6

hours each day. If Jasmine sleeps for

\frac{1}{3}

the amount of time that her cat sleeps each day, which of the following equations can be used to find

x

, the number of hours Jasmine's cat sleeps each day?

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1

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6 x=\frac{1}{3}

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3 x=6

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x-\frac{1}{3}=6

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\frac{x}{3}=6

Aria has two pet snakes, Mian and Udon. The combined length of the two snakes is

34

inches, and Udon is

12

inches shorter than Mian. Which of the following systems of equations could be used to find

m

, the length of Mian in inches, and

u

, the length of Udon in inches?

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1

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m+u=12

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m-u=34

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m+u=12

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u-m=34

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m+u=34

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m-u=12

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m+u=34

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u-m=12

\begin{array}{l} x+y=11 \\ x+3=y \end{array}

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Bex and Yin are running errands together. The given system of equations relates

x

, the number of errands Bex has to run, and

y

, the number of errands Yin has to run. Based on the system, which of the following statements is true?

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1

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11

more errands to run than Yin does.

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11

more errands to run than Bex does.

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3

more errands to run than Yin does.

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3

more errands to run than Bex does.

Which of the following statements about the graph of

y=12(0.75)^{x}

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1

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x

y

increases at an increasing rate.

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x

y

increases at a decreasing rate.

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x

y

decreases at an increasing rate.

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x

y

decreases at a decreasing rate.

Anna would like to surround an

8

\times 10

in photograph with a border of uniform width. Let

x

be the width of the border, in inches. Which of the following functions could she use to determine the area,

A

, in square inches

\left(\mathrm{in}^{2}\right)

, of the picture and border combined?

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1

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A=(8-x)(10-x)

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A=(8-2 x)(10-2 x)

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A=(8+x)(10+x)

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A=(8+2 x)(10+2 x)

Wang Lei would like to build a

144 \mathrm{ft}^{2}

rectangular garden. He plans to enclose this area with exactly

50 \mathrm{ft}

of fencing. Which of the following equations could be used to find the width,

x

, of Wang Lei's garden?

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1

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x(72-x)=50

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x(25-x)=144

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x(144-2 x)=50

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x(50-2 x)=144

p(x)=(x-0.95)(500-100 x)

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The equation models

p

, the daily profit, in dollars, a food truck makes by selling their signature tacos at a price of

x

dollars each. What is the daily profit if the tacos are sold at

\$ 3

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1

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\$ 410

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\$ 600

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\$ 820

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\$ 2,460

h(t)=-16.1 t^{2}+100 t

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The equation models

h

, the height of a firework shell in feet,

t

seconds after launch. What is the height, in feet, of the firework shell

2

seconds after launch?

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1

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135

6

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167

8

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232

2

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264

4

P(x)=2,500(0.98)^{x}

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The function models

P

, the population of the village of Kalm

x

1997

. Based on the model, what is the population of Kalm

18

1997

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1

\newline

2

450

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2

127

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1

846

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1

738

Josh is hiking Glacier National Park. He has now hiked a total of

17 \mathrm{~km}

2 \mathrm{~km}

\frac{1}{2}

of the way done with his hike. If

h

is the total length of Josh's hike, which of the following equations best describes the situation?

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1

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\frac{1}{2} h-2=17

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\frac{1}{2} h+2=17

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2 h-2=17

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2 h+2=17

The rapid growth rate of Moso bamboo makes it an environmentally important plant, as it stores massive amounts of carbon dioxide

\left(\mathrm{CO}_{2}\right)

. Each year, one hectare of Moso bamboo in China absorbs

5

1

tons of carbon dioxide, which can compensate for the annual

\mathrm{CO}_{2}

3

people. Which of the following equations best models the number of hectares,

h

, of Moso bamboo that must be planted to compensate for the annual

\mathrm{CO}_{2}

100

people in China?

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1

\newline

100=3 h

\newline

100=5.1 h

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100=\frac{5.1}{3} h

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100=\frac{3}{5.1} h

Felipe is saving money for a class trip. He already has saved

\$ 250

that he will put toward the trip. To save more money for the trip, Felipe gets a job where each month he can add

\$ 350

to his savings for the trip. Let

m

be the number of months that Felipe has worked at his new job. If Felipe needs to save

\$ 2700

to go on the trip, which equation best models the situation?

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1

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250 m-350=2700

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250 m+350=2700

\newline

350 m-250=2700

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350 m+250=2700

A reservoir can be filled to

\frac{1}{4}

1

day by Pump A, and to

\frac{1}{3}

1

day by Pump B. Both pumps have a constant rate of flow. Which of the following inequalities best describes the number of days,

d

, it will take for both pumps working at the same time to fill the reservoir to over

\frac{7}{8}

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1

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\frac{12}{7} d>\frac{7}{8}

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\frac{4}{d}+\frac{3}{d} \leq \frac{7}{8}

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\frac{7}{12} d>\frac{7}{8}

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\frac{d}{4}+\frac{d}{3} \leq \frac{7}{8}

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ar equations word problems | Lesson

At a particular college, a film club and a chess club were founded at the beginning of the same month. The film club began with

16

members and gained

1

new member per month and the chess club began with

4

members and gained

4

new members per month. At this rate, at the beginning of which month did the two clubs have the same membership?

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1

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2^{\text {nd }}

\newline

3^{\text {rd }}

\newline

4^{\text {th }}

\newline

5^{\text {th }}

John read the first

114

pages of a novel, which was

3

pages less than

\frac{1}{3}

of the novel. If

p

is the total number of pages in the novel, which of the following equations best describes the situation?

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1

\newline

\frac{1}{3} p-3=114

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\frac{1}{3} p+3=114

\newline

3 p-3=114

\newline

3 p+3=114

\begin{array}{l} -x+\frac{1}{2} y=-3 \\ 0=c x-2 y+10 \end{array}

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In the system of equations,

c

is a constant. For what value of

c

does the system of linear equations have no solutions?

A utility service company has a fleet of cars and trucks. The gas tank of each car has a volume of

16

gallons, and the gas tank of each truck has a volume of

30

gallons. If it takes

2

000

gallons of gas to fill the empty gas tanks of the entire fleet, which equation shows the possible number of cars,

c

, and number of trucks,

t

, in the fleet?

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1

\newline

16 t+30 c=2,000

\newline

30 t+16 c=2,000

\newline

t+30 c=2,000

\newline

t+16 c=2,000

A manufacturer of pickles has a fixed monthly cost of

\$ 10,000

. Additionally, each jar of pickles that the manufacturer produces costs

\$ 0.75

. If these are the only monthly costs incurred by the manufacturer, which of the following functions best models the manufacturer's total monthly cost,

C

, when it produces

p

jars of pickles?

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1

\newline

C(p)=10,000+0.75 p

\newline

C(p)=0.75+10,000 p

\newline

C(p)=10,000 \cdot 0.75 \cdot p

\newline

C(p)=0.75(10,000+p

A professional baseball team sells regular tickets and premium tickets to all of its home games. At a recent game, the average price for a regular ticket was

\$ 52.46

, and the average price for a premium ticket was

\$ 175.85

. Which of the following equations shows a relationship where the number of regular tickets,

r

, and the number of premium tickets,

p

, that the team sells generates

\$ 5,000,000

in ticket sales?

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1

\newline

\frac{p}{52.46}+\frac{r}{175.85}=5,0 c

\newline

\frac{r}{52.46}+\frac{p}{175.85}=5,0 \mathrm{C}

\newline

52.46 p+175.85 r=5,01

\newline

52.46 r+175.85 p=5,01

When Lúcia plays golf at the Rolling Hills golf course, she loses about

12

balls on average. When she plays golf at the Meandering Meadows golf course, she loses about

8

balls on average. If Lúcia lost approximately

100

balls after playing at Rolling Hills

r

times and at Meandering Meadows

m

times, which of the following equations best represents the relationship between

r

m

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1

\newline

8 r+12 m=100

\newline

12 r+8 m=100

\newline

12 r-8 m=100

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8 r-12 m=100

A clothier who only makes shirts and pants can make a shirt in

4

hours and a pair of pants in

6

hours. Which of the following equations shows the number of shirts,

s

, and the number of pants,

p

, that the clothier can make in a

40

hour work week?

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1

\newline

6 s+4 p=40

\newline

4 s+6 p=40

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\frac{s}{4}+\frac{p}{6}=40

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\frac{s}{6}+\frac{p}{4}=40

A farmer's rectangular plot with a length of

l

meters and a width of

w

meters is shown in the figure. The perimeter, or distance around the plot, is

700

meters. Which of the following equations shows the perimeter of the plot in terms of length and width?

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1

\newline

2 l+2 w=700

\newline

2 l+2 w=350

\newline

l+w=700

\newline

l w=700

A cashier counts

\$ 15.75

in quarters and dimes at the end of a workday. Which of the following equations shows the relationship between the number of quarters,

q

d

, that the cashier counted?

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1

\newline

\frac{0.25}{q}+\frac{0.10}{d}=15.75

\newline

\frac{q}{0.25}+\frac{d}{0.10}=15.75

\newline

0.35(q+d)=15.75

\newline

0.25 q+0.10 d=15.75

D=27.4 m+12,100

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D

, in millions of miles, of the Voyager

1

spacecraft from the Sun

m

months after January

1

2015

is approximated by the equation. What is the best interpretation of

27

4

as shown in the given equation?

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1

\newline

1

27

4

months before January

1

2015

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(B) The Voyager

1

27

4

million miles per hour.

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(C) The Voyager

1

was a distance of

27

4

million miles from the Sun on January

1

2015

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(D) The distance between Voyager

1

and the Sun increases by

27

4

million miles every month.

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linear relationships | Lesson

T=0.25 q+87

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Manon's statistics professor puts a bonus question on every test, which adds

0

25

points to a student's overall grade at the end of the term. The equation gives Manon's current overall grade,

T

, after taking into account

q

extra credit questions answered correctly. What does the

87

mean in this equation?

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1

\newline

(A) Manon must get an

87

or above to pass.

\newline

(B) Manon's average is an

87

before adding the extra credit.

\newline

(C) After adding in her extra credit, Manon's average is an

87

\newline

(D) Manon's professor gave

87

opportunities for extra credit this term.

\newline

linear relationships | Lesson

When Li Juan's auto yard is filled to capacity with only cars, it has

60

cars. When it is filled to capacity with only vans, it has

50

vans. Which linear equation models the number of cars,

c

v

, that could be in Li Juan's auto yard when it is filled to capacity?

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1

\newline

\frac{c}{60}+\frac{v}{50}=1

\newline

60 c+50 v=1

\newline

\frac{c}{50}+\frac{v}{60}=1

\newline

50 c+60 v=1

Noah is having his driveway resurfaced, which costs

\$ 25

per square meter. The driveway will also be given new edging around its perimeter, which costs

\$ 6.50

a

be the area, in square meters, of the driveway; and let

c

be the total cost of resurfacing the driveway and installing the new edging. If the perimeter of the driveway is

36

meters, which of the following equations correctly relates

a

c

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1

\newline

a=234-25 c

\newline

c=234-25 a

\newline

a=234+25 c

\newline

c=234+25 a

A forest that is populated by crows and doves currently contains a total of

1

500

crow and dove nests. Crows use approximately

8

grams of twigs when building a nest, and doves use approximately

6

grams of twigs when building a nest. Which of the following equations best approximates the relationship between

C

, the total amount, in grams, of twigs used for crow nests and

D

, the total amount, in grams, of twigs used for dove nests?

\newline

1

\newline

\frac{C}{8}+\frac{D}{6}=1,500

\newline

\frac{C}{6}+\frac{D}{8}=1,500

\newline

6 C+8 D=1,500

\newline

8 C+6 D=1,500

Elon is purchasing materials for a gardening project. Mulch costs

\$ 33

per cubic yard. Edge material costs

\$ 60

5

yards. Elon needs

60

yards worth of edge material. Let

v

be the volume, in cubic yards, of mulch that Elon buys; and let

c

be the total cost, in dollars, of his purchase. Which of the following equations correctly relates

v

c

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1

\newline

33 v+720=c

\newline

33 v+1,980=c

\newline

33 v+3,600=c

\newline

33 v+18,000=c

Dew point temperature, in degrees Celsius

\left({ }^{\circ} \mathrm{C}\right)

, is defined as the temperature to which the air would have to cool in order to reach saturation. For a temperature of

35^{\circ} \mathrm{C}

, an estimate of the dew point can be obtained by first subtracting

20^{\circ} \mathrm{C}

from the temperature, and then adding

1^{\circ}

for every increase of

5

points in the relative humidity,

R

. If the dew point is

30^{\circ} \mathrm{C}

, which of the following equations best models the situation?

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1

\newline

30=15-\frac{1}{5} R

\newline

30=15+5 R

\newline

30=15-5 R

\newline

30=15+\frac{1}{5} R

A landscaping company sells bags that hold up to

2

3

\left(\mathrm{ft}^{3}\right)

of mulch. The company guarantees that there is at least

2 \mathrm{ft}^{3}

of mulch inside each bag. Which of the following functions gives the nonnegative difference between the maximum and minimum mulch volumes, in cubic feet, that could be contained in

b

bags purchased from the landscaping company?

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1

\newline

f(b)=0.3 b

\newline

f(b)=2 b

\newline

f(b)=2.3 b

\newline

f(b)=4.3 b

Jacob's classroom contains tables and chairs, each with

4

legs. Jacob puts a pad on the end of every leg so that the tables and chairs will not make noise when they are moved around. If Jacob needs exactly

80

pads for his classroom, which of the following equations correctly relates the number of chairs,

c

, to the number of tables,

t

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1

\newline

c=80-4 t

\newline

c=80-t

\newline

c=20-4 t

\newline

c=20-t

A charging station can only charge cell phones and laptops. Cell phones use approximately

3

joules per second

\left(\frac{\mathrm{J}}{\mathrm{s}}\right)

when charging, and laptops use approximately

40 \frac{\mathrm{J}}{\mathrm{s}}

when charging. Which of the following equations represents the relationship between the number of cell phones,

c

, and the number of laptops,

l

, that could be charging when the station's output is exactly

160 \frac{\mathrm{J}}{\mathrm{s}}

\newline

1

\newline

c=-40 l+160

\newline

c=-\frac{40}{3} l+\frac{160}{3}

\newline

c=-3 l+160

\newline

c=-\frac{3}{40} l+\frac{160}{40}

...