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Math Problems
Grade 8
Write linear functions: word problems
A researcher studying the environmental benefits of green roofs receives a grant of
$
250
,
000
\$250,000
$250
,
000
. The researcher estimates that the study will cost on average
$
15
,
000
\$15,000
$15
,
000
per month.
\newline
Which of the following equations shows the amount of grant funds,
g
g
g
, that the researcher will have left after
m
m
m
months of conducting the study?
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C =
1
1
1
.
5
5
5
b +
67
67
67
.
5
5
5
\newline
The total cost,
C
C
C
, in dollars, to produce
b
b
b
books is given by the equation. What is the meaning of
1.5
1.5
1.5
in the equation?
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Jack's mother gave him
50
50
50
chocolates to give to his friends at his birthday party. He gave
3
3
3
chocolates to each of his friends and still had
2
2
2
chocolates left. Write an equation to determine the number of friends
(
x
)
(x)
(
x
)
at Jack's party. Find the number of friends at Jack's party.
\newline
friends
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Suraj took a slice of pizza from the freezer and put it in the oven. The graph shows the pizza's temperature,
y
y
y
, in degrees Celsius, as a function of time,
x
x
x
, in minutes. How many minutes did it take for the pizza to increase its temperature by
4
4
4
degrees Celsius?
\newline
□
\square
□
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Larry is on his school's cross-country team. He runs
8
8
8
miles during each practice. Write an equation that shows the relationship between the number of practices Larry attends,
x
x
x
, and the total number of miles he runs,
y
y
y
.
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A caterpillar eats
1400
%
1400 \%
1400%
of its birth mass in one day. The caterpillar's birth mass is
m
m
m
grams.
\newline
Which of the following expressions could represent the amount, in grams, the caterpillar eats in one day?
\newline
Choose
2
\mathbf{2}
2
answers:
\newline
A
1400
m
1400 m
1400
m
\newline
B
1400
100
m
\frac{1400}{100} m
100
1400
m
\newline
c
140
m
140 \mathrm{~m}
140
m
\newline
D
1.4
m
1.4 m
1.4
m
\newline
E
14
m
14 m
14
m
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A caterpillar eats
1400
%
1400 \%
1400%
of its birth mass in one day. The caterpillar's birth mass is
m
m
m
grams.
\newline
Which of the following expressions could represent the amount, in grams, the caterpillar eats in one day?
\newline
Choose
2
\mathbf{2}
2
answers:
\newline
A
1400
m
1400 m
1400
m
\newline
B
1400
100
m
\frac{1400}{100} m
100
1400
m
\newline
c
140
m
140 \mathrm{~m}
140
m
\newline
D
1.4
m
1.4 m
1.4
m
\newline
E
14
m
14 m
14
m
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A passenger train and a freight train start toward each other at the same time from two points
300
300
300
miles apart. If the rate of the passenger train exceeds the rate of the freight train by
15
15
15
mph and they meet after
4
4
4
hours, what is the rate of each train?
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A book publishing company hired a group of writers to improve their already existing content. The total cost,
C
C
C
, in dollars, incurred by the company for
h
h
h
hours of work is given by the equation below.
\newline
c
=
1000
+
25
h
c=1000+25h
c
=
1000
+
25
h
\newline
What is the best interpretation of
250
250
250
in the equation?
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Brendan walked his dog
b
b
b
blocks in the morning and
6
6
6
blocks in the afternoon. Write an expression that shows the number of blocks Brendan walked his dog in all.
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C
=
40
(
5
−
s
)
C = 40( 5 - s )
C
=
40
(
5
−
s
)
After Hiro's family photoshoot, the photographer sells the printed photos by the sheet. Hiro has a
$
200
\$200
$200
credit from a prior session, and the equation gives the total credit left,
C
C
C
, in dollars, if he buys
s
s
s
sheets. How many sheets can Hiro purchase with his credit?
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A bank efficiency ratio is the ratio of a bank's expenses to its revenue, expressed as a percentage. If a bank has
$
222
\$222
$222
million dollars in expenses and an efficiency ratio of
75
%
75\%
75%
, what is its revenue in millions of dollars?
\newline
◻
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100
−
10
m
100-10m
100
−
10
m
\newline
Dante's goal this year is to read
100
100
100
books. The expression above models the number of books Dante has left to read $m months after the beginning of the year. Based on the expression, how many books does Dante have left to read after
4
4
4
months?
\newline
◻
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Alex learned
80
80
80
new vocabulary words for his English exam. The following function gives the number of words he remembers after
t
t
t
days:
\newline
W
(
t
)
=
80
(
1
−
0.1
t
)
3
W(t)=80(1-0.1 t)^{3}
W
(
t
)
=
80
(
1
−
0.1
t
)
3
\newline
What is the instantaneous rate of change of the number of words Alex remembers after
5
5
5
days?
\newline
Choose
1
1
1
answer:
\newline
(A)
10
\mathbf{1 0}
10
days
\newline
(B)
10
\mathbf{1 0}
10
words per day
\newline
(C)
−
6
-6
−
6
days
\newline
(D)
−
6
-6
−
6
words per day
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A water tank is drained. The following function gives the volume, in liters, of the water remaining in the tank
t
t
t
minutes after the drain is opened:
\newline
V
(
t
)
=
3000
(
1
−
0.05
t
)
2
V(t)=3000(1-0.05 t)^{2}
V
(
t
)
=
3000
(
1
−
0.05
t
)
2
\newline
What is the instantaneous rate of change of the volume after
10
10
10
minutes?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
150
-150
−
150
liters per minute
\newline
(B)
−
150
-150
−
150
minutes per liter
\newline
(C)
750
750
750
liters per minute
\newline
(D)
750
750
750
minutes per liter
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Kajal holds a ruler in some wavy water. The depth of the water
t
t
t
seconds after she starts measuring it, in
c
m
\mathrm{cm}
cm
, is given by
\newline
D
(
t
)
=
50
−
23
sin
(
π
(
t
+
0.23
)
)
.
D(t)=50-23 \sin (\pi(t+0.23)) .
D
(
t
)
=
50
−
23
sin
(
π
(
t
+
0.23
))
.
\newline
After she starts measuring, when is the first time the depth of the waves is at its average value? Give an exact answer.
\newline
When
t
=
□
t= \square
t
=
□
seconds
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An express train takes
1
1
1
hour less than a passenger train to travel
132
k
m
132 \mathrm{~km}
132
km
between Mysore and Bangalore. If the average speed of the express train is
11
k
m
/
h
r
11 \mathrm{~km} / \mathrm{hr}
11
km
/
hr
more than that of the passenger train, form the quadratic equation to find the average speed of express train.
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The sum of the ages of a man and his wife is
81
81
81
years. The ratio of their ages is
5
:
4
5:4
5
:
4
. Find the age of the younger person.
\newline
A.
30
30
30
years
\newline
B.
36
36
36
years
\newline
C.
45
45
45
years
\newline
D.
51
51
51
years
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The sum of the ages of a man and his wife is \(81 years. The ratio of their ages is
5
:
4
5: 4
5
:
4
. Find the age of the younger person.
\newline
A.
30
30
30
years
\newline
B.
36
36
36
years
\newline
C.
45
45
45
years
\newline
D.
51
51
51
years
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Point A is located at
−
19
-19
−
19
. Point
B
\mathrm{B}
B
is
6
6
6
less than Point
A
\mathrm{A}
A
. Where is
B
\mathrm{B}
B
located?
\newline
B
=
□
B= \square
B
=
□
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Point
X
\mathrm{X}
X
is located at
−
14
-14
−
14
. Point
Y
\mathrm{Y}
Y
is
6
6
6
less than Point
X
\mathrm{X}
X
. Where is
Y
\mathrm{Y}
Y
located?
\newline
Y
=
□
Y= \square
Y
=
□
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Tyshawn was out at a restaurant for dinner when the bill came. He wanted to leave a tip of
28
%
28 \%
28%
. What number should he multiply the cost of the meal by to find the total plus tip in one step?
\newline
Answer:
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Larry is on his school's cross-country team. He runs
8
8
8
miles during each practice. Write an equation that shows the relationship between the number of practices Larry attends,
x
x
x
, and the total number of miles he runs,
y
y
y
.
\newline
y = ____
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In an ATM if the currency notes of denominations of Rs.
500
500
500
, Rs.
100
100
100
and Rs.
50
50
50
respectively the notes are in the ratio of
3
:
3
:
4
3:3:4
3
:
3
:
4
. The total cash in the ATM is Rs.
400
,
000
400,000
400
,
000
. How many notes of each denominations that ATM contains?
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Seb bikes at a constant rate of
20
20
20
kilometers per hour.
1
1
1
. Write an equation that represents the distance, in kilometers, Seb travels
(
b
)
(b)
(
b
)
based on how many hours he bikes
(
t
)
(t)
(
t
)
at this rate.
2
2
2
. How far will Seb travel if he bikes at this rate for
3
3
3
hours?
_
\_
_
kilometers
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P
(
t
)
=
20
(
0.95
)
t
P(t)=20(0.95)^t
P
(
t
)
=
20
(
0.95
)
t
\newline
The function models
P
P
P
, the population of Leetown in thousands,
t
t
t
years after
2007
2007
2007
. What was the population of Leetown in
2007
2007
2007
?
\newline
Choose
1
1
1
answer:
\newline
(A)
\text{(A)}
(A)
5
5
5
thousand
\newline
(B)
\text{(B)}
(B)
19
19
19
thousand
\newline
(C)
\text{(C)}
(C)
20
20
20
thousand
\newline
(D)
\text{(D)}
(D)
95
95
95
thousand
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Montraie goes to a restaurant and the subtotal on the bill was
x
x
x
dollars. A tax of
7
%
7 \%
7%
is applied to the bill. Montraie decides to leave a tip of
21
%
21 \%
21%
on the entire bill (including the tax). Write an expression in terms of
x
x
x
that represents the total amount that Montraie paid.
\newline
Answer:
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To get the
10
%
10 \%
10%
discount, a shopper must spend at least
$
500
\$ 500
$500
.
\newline
Use
d
d
d
to represent the spending (in dollars) of a shopper who gets the discount.
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P
=
19
2
+
4
25
d
P=\frac{19}{2}+\frac{4}{25} d
P
=
2
19
+
25
4
d
\newline
The amount of paint,
P
P
P
, in pints, that an art studio can buy with
d
d
d
dollars is given by the equation. What is the best interpretation of
4
25
\frac{4}{25}
25
4
as shown in the equation?
\newline
Choose
1
1
1
answer:
\newline
(A) The paint costs
4
25
\frac{4}{25}
25
4
of a dollar per pint.
\newline
(B) One dollar can buy
4
25
\frac{4}{25}
25
4
of a pint of paint.
\newline
(C) The art studio started with
4
25
\frac{4}{25}
25
4
of a pint of paint.
\newline
(D) The art studio can purchase at most
4
25
\frac{4}{25}
25
4
of a pint of paint.
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There are
4
4
4
squares that are shaded on grid. This represents
20
%
20\%
20%
of the total squares that are jn the grid. How many squares are there on the grid
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Camacho is buying a monster truck. The price of the truck is
x
x
x
dollars, and he also has to pay a
13
%
13\%
13%
monster truck tax.
\newline
Which of the following expressions could represent how much Camacho pays in total for the truck?
\newline
Choose
2
2
2
answers:
\newline
(A)
(
1
+
0.13
)
x
(1+0.13)x
(
1
+
0.13
)
x
\newline
(B)
13
100
x
\frac{13}{100}x
100
13
x
\newline
(C)
13
x
13x
13
x
\newline
(D)
1.13
x
1.13x
1.13
x
\newline
(E)
13
x
+
x
13x+x
13
x
+
x
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A particle travels along the
x
x
x
-axis such that its velocity is given by
v
(
t
)
=
t
2
sin
(
2
t
)
v(t)=t^{2} \sin (2 t)
v
(
t
)
=
t
2
sin
(
2
t
)
. Find all times when the speed of the particle is equal to
3
3
3
on the interval
0
≤
t
≤
4
0 \leq t \leq 4
0
≤
t
≤
4
. You may use a calculator and round your answer to the nearest thousandth.
\newline
Answer:
t
=
t=
t
=
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Emil is monitoring the temperature during a cold font. At
7
7
7
:
00
00
00
a.m. the temperature is
−
1
1
∘
C
-11^{\circ} \mathrm{C}
−
1
1
∘
C
. At
11
11
11
:
00
00
00
a.m., Emil notices that the temperature has risen
5
∘
C
5^{\circ} \mathrm{C}
5
∘
C
.
\newline
What temperature does Emil observe at
11
11
11
:
00
00
00
a.m.?
\newline
∘
C
{ }^{\circ} \mathrm{C}
∘
C
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Ethan rollerbladed a total of
623
k
m
623 \mathrm{~km}
623
km
over
d
d
d
days. He rollerbladed the same distance each day.
\newline
How many kilometers did Ethan rollerblade each day?
\newline
Write your answer as an expression.
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The expression
20
l
+
25
g
−
10
20 l+25 g-10
20
l
+
25
g
−
10
gives the number of dollars that Josie makes from mowing
l
l
l
lawns and raking
g
g
g
gardens.
\newline
How many dollars does Josie make from raking
4
4
4
gardens and mowing
3
3
3
lawns?
\newline
dollars
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Alexandra paid
$
7
\$ 7
$7
to park her car for
3
3
3
hours at the parking garage. The garage charges a constant hourly parking rate.
\newline
Write an equation that shows the relationship between
p
p
p
, the number of hours parked, and
c
c
c
, the cost in dollars.
\newline
Do NOT use a mixed number.
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A tropical punch recipe calls for
300
m
l
300 \mathrm{ml}
300
ml
of sugar for every
2
2
2
flavor packages.
\newline
Write an equation that shows the relationship between
s
s
s
, the amount of sugar in milliliters, and
f
f
f
, the number of flavor packages for this recipe.
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Justin runs at a constant rate, traveling
17
k
m
17 \mathrm{~km}
17
km
in
2
2
2
hours.
\newline
Write an equation that shows the relationship between
d
d
d
, the distance he runs in kilometers, and
h
h
h
, the time he spends running in hours.
\newline
Do NOT use a mixed number.
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Carlos harvests cassavas at a constant rate. He needs
35
35
35
minutes to harvest a total of
15
15
15
cassavas.
\newline
Write an equation to describe the relationship between
t
t
t
, the time, and
c
c
c
, the total number of cassavas.
\newline
Do NOT use a mixed number.
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The Green Goober, a wildly unpopular superhero, mixes
3
3
3
liters of yellow paint with
5
5
5
liters of blue paint to make
8
8
8
liters of special green paint for his costume.
\newline
Write an equation that relates
y
y
y
, the amount of yellow paint in liters, and
b
b
b
, the amount of blue paint in liters, needed to make the Green Goober's special green paint. Do NOT use a mixed number.
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Hannah reads at a constant rate of
3
3
3
pages every
8
8
8
minutes.
\newline
Write an equation that shows the relationship between
p
p
p
, the number of pages she reads, and
m
m
m
, the number of minutes she spends reading.
\newline
Do NOT use a mixed number.
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Larry read a
400
400
400
-page book. He read at a rate of
10
10
10
pages per day for
d
d
d
days.
\newline
Write an equation that could be used to find out how many days it took Larry to read the book.
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Mr. Herman's class is selling candy for a school fundraiser. The class has a goal of raising
$
500
\$ 500
$500
by selling
c
c
c
boxes of candy. For every box they sell, they make
$
2.75
\$ 2.75
$2.75
.
\newline
Write an equation that the students could solve to figure out how many boxes of candy they need to sell.
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We want to solve the following equation.
\newline
x
+
2
=
5
x
+
5
x+2=\sqrt{5 x+5}
x
+
2
=
5
x
+
5
\newline
One of the solutions is
x
≈
−
0.6
x \approx-0.6
x
≈
−
0.6
.
\newline
Find the other solution.
\newline
Hint: Use a graphing calculator.
\newline
Round your answer to the nearest tenth.
\newline
x
≈
x \approx
x
≈
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We want to solve the following equation.
\newline
x
3
−
4
x
=
2
x
x^{3}-4 x=2^{x}
x
3
−
4
x
=
2
x
\newline
Three of the solutions are
x
≈
−
2.0
,
x
≈
−
0.2
x \approx-2.0, x \approx-0.2
x
≈
−
2.0
,
x
≈
−
0.2
, and
x
≈
9.8
x \approx 9.8
x
≈
9.8
.
\newline
Find the other solution.
\newline
Hint: Use a graphing calculator.
\newline
Round your answer to the nearest tenth.
\newline
x
≈
x \approx
x
≈
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We want to solve the following equation.
\newline
3
log
2
(
x
)
=
x
3
−
4
x
2
+
4
x
3 \log _{2}(x)=x^{3}-4 x^{2}+4 x
3
lo
g
2
(
x
)
=
x
3
−
4
x
2
+
4
x
\newline
One of the solutions is
x
≈
3.3
x \approx 3.3
x
≈
3.3
.
\newline
Find the other solution.
\newline
Hint: Use a graphing calculator.
\newline
Round your answer to the nearest tenth.
\newline
x
≈
x \approx
x
≈
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We want to solve the following equation.
\newline
x
3
+
3
x
2
−
x
=
∣
x
−
1
∣
x^{3}+3 x^{2}-x=|x-1|
x
3
+
3
x
2
−
x
=
∣
x
−
1∣
\newline
Two of the solutions are
x
≈
−
0.7
x \approx-0.7
x
≈
−
0.7
and
x
≈
0.5
x \approx 0.5
x
≈
0.5
.
\newline
Find the other solution.
\newline
Hint: Use a graphing calculator.
\newline
Round your answer to the nearest tenth.
\newline
x
≈
x \approx
x
≈
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Charlotte is solving the following equation for
x
x
x
.
\newline
(
x
+
3
)
2
−
10
=
2
(x+3)^{2}-10=2
(
x
+
3
)
2
−
10
=
2
\newline
Her first few steps are given below.
\newline
(
x
+
3
)
2
=
12
x
+
3
=
±
12
\begin{aligned} (x+3)^{2} & =12 \\ x+3 & = \pm \sqrt{12} \end{aligned}
(
x
+
3
)
2
x
+
3
=
12
=
±
12
\newline
Is it necessary for Charlotte to check her answers for extraneous solutions?
\newline
Choose
1
1
1
answer:
\newline
(A) Yes
\newline
(B)
N
o
\mathrm{No}
No
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Tyson uploaded
500
500
500
videos to the Internet. Sarita uploaded fewer videos than Tyson.
\newline
Write an inequality that compares the number of videos Sarita uploaded,
v
v
v
, to the number of videos Tyson uploaded.
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A small pizza has an area of
730
730
730
square centimeters.
\newline
Write an inequality that describes
p
p
p
, the area of a pizza that is larger than a small pizza.
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