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Math Problems
Grade 7
Evaluate two-variable equations: word problems
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
6
6
square feet. The area of the new cage's floor is
n
n
n
square feet larger, for a total of
8
8
8
square feet.
\newline
Which diagram models the story?
\newline
Which equation models the story?
\newline
Choices:
\newline
(A)
n
−
6
=
8
n - 6 = 8
n
−
6
=
8
\newline
(B)
6
+
n
=
8
6 + n = 8
6
+
n
=
8
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Mr. Newton wears science-themed ties to school. In all, he has
34
34
34
ties with scientific designs. Of all the ties in his collection,
b
b
b
are bow ties and
21
21
21
are straight ties.
\newline
Which diagram models the story?
\newline
Which equation models the story?
\newline
Choices:
\newline
(
A
)
(A)
(
A
)
b
+
21
=
34
b + 21 = 34
b
+
21
=
34
\newline
(
B
)
(B)
(
B
)
b
−
21
=
34
b - 21 = 34
b
−
21
=
34
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Over the weekend, Julian visited the Wondrous Wings Butterfly Museum with his family. In the Discovery Zone, he was able to observe
c
c
c
caterpillars and
8
8
8
butterflies. That's a total of
15
15
15
insects he got to see up close in all.
\newline
Which diagram models the story?
\newline
Which equation models the story?
\newline
Choices:
\newline
(A)
c
+
8
=
15
c + 8 = 15
c
+
8
=
15
\newline
(B)
c
−
8
=
15
c - 8 = 15
c
−
8
=
15
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At Amps Arcade, Anne is about to play her favorite game, Road Dash. She comes in first place half of the time. If she comes in first place for every race in a tournament, she will get her name added to the winners' board. Anne will play a
4
4
4
-race tournament today. How likely is it that her name will be added to the winners' board? Anne simulates the situation by rolling a six-sided die
4
4
4
times. Each time an even number appears, it represents Anne coming in first place. This table shows the results of
100
100
100
trials: Number of even numbers rolled
0
0
0
1
1
1
2
2
2
3
3
3
4
4
4
Number of trials
5
5
5
27
27
27
4
4
4
0
0
0
4
4
4
1
1
1
4
4
4
2
2
2
Based on Anne's results, what is the probability that her name will be added to the winner's board?
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After Batman brought the Joker to justice, the crime rate in Arkham City decreased by
12
%
12\%
12%
. Previously, the crime rate was
c
c
c
incidents per
1000
1000
1000
people. Which of the following expressions could represent the crime rate in Arkham City after Batman brought the Joker to justice?
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Kajal holds a ruler in some wavy water. The depth of the water
\newline
t
t
t
seconds after she starts measuring it, in
\newline
cm
\text{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
\newline
t
=
t=
t
=
\newline
□
\square
□
seconds
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Francesca tried to evaluate an expression. Here is her work:
\newline
3
(
5
)
2
−
18
÷
3
3(5)^2 - 18 \div 3
3
(
5
)
2
−
18
÷
3
\newline
3
(
25
)
−
18
÷
3
{
Step 1
}
3(25) - 18 \div 3 \quad \{\text{Step 1}\}
3
(
25
)
−
18
÷
3
{
Step 1
}
\newline
75
−
18
÷
3
{
Step 2
}
75 - 18 \div 3 \quad \{\text{Step 2}\}
75
−
18
÷
3
{
Step 2
}
\newline
57
÷
3
{
Step 3
}
57 \div 3 \quad \{\text{Step 3}\}
57
÷
3
{
Step 3
}
\newline
19
{
Step 4
}
19 \quad \{\text{Step 4}\}
19
{
Step 4
}
\newline
Is Francesca's work correct?
\newline
(A) No, Francesca made a mistake going from the given expression to Step
1
1
1
. Francesca should have multiplied
3
3
3
and
5
5
5
before evaluating the exponent. The order of operations says to perform operations inside parentheses before you evaluate the exponent.
\newline
(B) No, Francesca made a mistake going from Step
2
2
2
to Step
3
3
3
. Francesca should have divided
18
18
18
by
3
3
3
before subtracting. The order of operations says to divide before you subtract.
\newline
(C) Yes, Francesca's work is correct.
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Martha wants to buy a charm bracelet. Princeton Fine Jewelry charges
$
30
\$30
$30
per charm, plus
$
50
\$50
$50
for the bracelet. Wong Jewelers, in contrast, charges
$
20
\$20
$20
per charm and
$
100
\$100
$100
for the bracelet. If Martha wants to add a certain number of charms to her bracelet, the cost will be the same at either Jewelry shop. What would the total cost of the bracelet be? Write a system of equations.
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The Green Goober, a wildly unpopular superhero, mixes
y
y
y
liters of yellow paint with
b
b
b
liters of blue paint to make
y
+
b
y + b
y
+
b
liters of special green paint for his costume. 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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A boat is heading towards a lighthouse, where Dalvin is watching from a vertical distance of
138
138
138
feet above the water. Dalvin measures an angle of depression to the boat at point
A
A
A
to be
13
13
13
degrees. At some later time, Dalvin takes another measurement and finds the angle of depression to the boat (now at point
B
B
B
) to be
45
45
45
degrees. Find the distance from point
A
A
A
to point
B
B
B
.
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A boat is heading towards a lighthouse, where Dalvin is watching from a vertical distance of
138
138
138
feet above the water. Dalvin measures an angle of depression to the boat at point
A
A
A
to be
13
13
13
degrees. At some later time, Dalvin takes another measurement and finds the angle of depression to the boat (now at point
B
B
B
) to be
45
45
45
degrees. Find the distance from point
A
A
A
to point
B
B
B
.
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After Batman brought the Joker to justice, the crime rate in Arkham City decreased by
x
%
x\%
x
%
. Previously, the crime rate was
y
y
y
incidents per
z
z
z
people. Which of the following expressions could represent the crime rate in Arkham City after Batman brought the Joker to justice?
\newline
Choose
2
2
2
answers:
\newline
a)
y
(
100
−
x
)
z
\frac{y(100-x)}{z}
z
y
(
100
−
x
)
\newline
b)
y
z
(
100
−
x
)
%
\frac{y}{z}(100-x)\%
z
y
(
100
−
x
)
%
\newline
c)
y
−
x
z
\frac{y-x}{z}
z
y
−
x
\newline
d)
y
z
−
x
\frac{y}{z-x}
z
−
x
y
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This equation shows how the total length Linda has knit depends on the number of nights she has spent knitting.
\newline
t
=
n
+
5
t = n + 5
t
=
n
+
5
\newline
The variable
n
n
n
represents the number of nights Linda has spent knitting, and the variable
t
t
t
represents the total length she has knit in centimeters. After
3
3
3
nights of knitting, how many centimeters of scarf will Linda have knit in total?
\newline
_____ centimeters
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This equation shows how the amount of money Brody makes from selling lemonade is related to the number of cups sold.
\newline
d
=
2
c
d = 2c
d
=
2
c
\newline
The variable
c
c
c
represents the number of cups sold, and the variable
d
d
d
represents earnings. After selling a total of
4
4
4
cups of lemonade, how much money will Brody have made in all?
\newline
$
\$
$
_
_
_
_
\_\_\_\_
____
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This equation shows how the cost of Mark's birthday party depends on the number of guests.
\newline
c
=
g
+
14
c = g + 14
c
=
g
+
14
\newline
The variable
g
g
g
represents the number of guests, and the variable
c
c
c
represents the cost in dollars. If there are
4
4
4
guests, how much will Mark's birthday party cost?
\newline
$
\$
$
_
_
_
_
\_\_\_\_
____
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This equation shows how the cost of Joy's birthday party depends on the number of guests.
\newline
c
=
g
+
8
c = g + 8
c
=
g
+
8
\newline
The variable
g
g
g
represents the number of guests, and the variable
c
c
c
represents the cost in dollars. If there are
10
10
10
guests, how much will Joy's birthday party cost?
\newline
$
\$
$
_
_
_
_
\_\_\_\_
____
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This equation shows how the number of pictures Ryan has taken so far this year is related to the number of days he spends on vacation.
\newline
p
=
10
d
p = 10d
p
=
10
d
\newline
The variable
d
d
d
represents the number of days he spends on vacation, and the variable
p
p
p
represents the total number of pictures taken this year. After
2
2
2
days of vacation, how many pictures will Ryan have taken?
\newline
_____ pictures
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Xavier and Shannon both tutor on the weekends. Xavier made
63
63
63
$
\$
$
after
3
3
3
hours of tutoring. Shannon made
95
95
95
$
\$
$
after
5
5
5
hours of tutoring. Who charges more per hour?
\newline
Choices:
\newline
(A)Xavier
\newline
(B)Shannon
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Rafael purchased his car for
$
12
,
000
\$12,000
$12
,
000
. The value of Rafael's car depreciates over time, decreasing by
10
%
10\%
10%
every year. Which of the following graphs could model this relationship if
t
t
t
is elapsed time in years, and
V
V
V
is the value of Rafael's car in thousands of dollars?
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Mason ate
c
c
c
more crackers than Brooke. Brooke ate only
8
8
8
crackers. Write an expression that shows how many crackers Mason ate.
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A store sells all compact discs for
$
14.99
\$14.99
$14.99
. The following equation represents the revenue,
y
y
y
, (in dollars) generated by selling
x
x
x
CDs.
\newline
y
=
14.99
x
y=14.99x
y
=
14.99
x
(
x
≥
0
)
(x \geq 0)
(
x
≥
0
)
\newline
y
=
□
y=\square
y
=
□
when
x
=
8
x=8
x
=
8
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Stem and Leaf is one of the quick way of finding _____And _____ of a distribution
\newline
A. Median and Mode
\newline
B. Median and skewness
\newline
C. Mean and median
\newline
D. Mean and Mode
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Given the cone dimensions below:
\newline
cone's slant height
=
20
m
=20 \mathrm{~m}
=
20
m
\newline
cone's radius
=
12
m
=12 \mathrm{~m}
=
12
m
\newline
Find the height of the cone and the cone's volume.
\newline
cone height
=
=
=
\qquad
\newline
cone volume
=
=
=
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It's the end of the summer, and Sandeep is draining his family's swimming pool. The function
y
=
−
600
x
+
12
,
000
y = -600x + 12,000
y
=
−
600
x
+
12
,
000
represents the total number of gallons in the pool,
y
y
y
, as a function of the number of hours the pool has been draining,
x
x
x
.
\newline
What is true about the function?
\newline
Choices:
\newline
(A)It is nonlinear because it decreases at a constant rate.
\newline
(B)It is nonlinear because it is always decreasing.
\newline
(C)It is linear because it decreases at a constant rate.
\newline
(D)It is linear because it is always decreasing.
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A party store sells tablecloths for
$
1
\$1
$1
each,
5
5
5
for
$
5
\$5
$5
, or
10
10
10
for
$
8
\$8
$8
. Is the relationship between the number of tablecloths purchased and total cost a linear relationship?
\newline
Choices:
\newline
(A)yes
\newline
(B)no
\newline
Now, justify your answer.
\newline
Choices:
\newline
(A)Tablecloths cost a constant amount,
$
1
\$1
$1
each.
\newline
(B)The three money amounts don't increase by a constant amount.
\newline
(C)The three quantities of tablecloths don't increase by a constant amount.
\newline
(D)The cost per tablecloth can vary, depending on how many you buy.
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w
=
108
w
w=\sqrt{108w}
w
=
108
w
\newline
What is the sum of all solutions to the given equation?
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T
T
T
=
0
0
0
.
25
q
25q
25
q
+
87
+ 87
+
87
\newline
Manon's statistics professor puts a bonus question on every test, which adds
0.25
0.25
0.25
points to a student's overall grade at the end of the term. The equation gives Manon's current overall grade,
T
T
T
, after taking into account
q
q
q
extra credit questions answered correctly. What does the
87
87
87
mean in this equation?
\newline
Choose
1
1
1
answer:
\newline
(A) Manon must get an
87
87
87
or above to pass.
\newline
(B) Manon's average is an
87
87
87
before adding the extra credit.
\newline
(C) After adding in her extra credit, Manon's average is an
87
87
87
.
\newline
(D) Manon's professor gave
87
87
87
opportunities for extra credit this term.
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During a mayoral election, candidate A spent
$
960
,
000
\$960,000
$960
,
000
on her campaign, and candidate B spent
$
720
,
000
\$720,000
$720
,
000
on her campaign. What is the ratio of candidate A's campaign spending to candidate B's campaign spending?
\newline
Choose
1
1
1
answer:
\newline
3
:
4
3:4
3
:
4
\newline
4
:
3
4:3
4
:
3
\newline
7
:
9
7:9
7
:
9
\newline
9
:
7
9:7
9
:
7
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Fernando built a robot to filter air and water efficiently. He expects the robot to filter more than
512
512
512
liters of air and water while using less than
9
9
9
Joules of energy.
\newline
A
A
A
represents the number of minutes the robot filters air, and
\newline
W
W
W
represents the number of minutes the robot filters water.
\newline
21
A
+
14
W
>
512
21A + 14W > 512
21
A
+
14
W
>
512
represents the number of minutes the robot filters air and water to filter more than
512
512
512
liters of air and water.
\newline
0.36
A
+
0.6
W
<
9
0.36A + 0.6W < 9
0.36
A
+
0.6
W
<
9
represents the number of minutes the robot filters air and water while using less than
9
9
9
Joules of energy.
\newline
Does the robot meet both of Fernando's expectations by filtering air for
15
15
15
minutes and filtering water for
5
5
5
minutes?
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Dina is conducting a study on the amount of electricity used per person in the country of Qatar. She finds that from the year
2000
2000
2000
to
2008
2008
2008
, the amount of electricity used per person grew by an average of
6
%
6\%
6%
from one year to the next. Let
e
(
t
)
e(t)
e
(
t
)
be the amount of electricity used per person in kilowatt-hours and
t
t
t
the number of years after
2000
2000
2000
. If the amount of electricity used per person in
2000
2000
2000
was approximately
12
,
300
12,300
12
,
300
kilowatt-hours, which of the following mathematical models will give Dana how much electricity was used per person between
2000
2000
2000
and
2008
2008
2008
?
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During the youth baseball season, Carter grills and sells hamburgers and hot dogs at the Hillview baseball field. On Saturday, he sold
30
30
30
hamburgers and
25
25
25
hot dogs and earned a total of
$
195
\$195
$195
. On Sunday, he sold
15
15
15
hamburgers and
20
20
20
hot dogs and earned a total of
$
120
\$120
$120
.
\newline
This system of equations can be used to represent the situation.
\newline
\newline
Complete the sentence.
\newline
How much does Carter charge for a hot dog?
\newline
$
\$
$
____
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Hanson is at a ball game with his family. He goes to the concession stand to buy soft pretzels and water bottles for everyone. A water bottle costs
2
$
2\$
2$
, and a pretzel costs
3.50
$
3.50\$
3.50$
. Hanson buys
16
16
16
items and spends
47
$
47\$
47$
in all.
\newline
This system of equations can be used to represent the situation:
\newline
x
+
y
=
16
x + y = 16
x
+
y
=
16
\newline
2
x
+
3.5
y
=
47
2x + 3.5y = 47
2
x
+
3.5
y
=
47
\newline
Which statement is correct?
\newline
(A) In the system of equations,
x
x
x
represents the amount of money Hanson spends on pretzels, and
y
y
y
represents the amount of money he spends on water bottles.
\newline
(B) In the system of equations,
x
x
x
represents the number of water bottles Hanson buys, and
y
y
y
represents the number of pretzels he buys.
\newline
(C) In the system of equations,
x
x
x
represents the number of pretzels Hanson buys, and
y
y
y
represents the number of water bottles he buys.
\newline
(D) In the system of equations,
x
x
x
represents the amount of money Hanson spends on water bottles, and
y
y
y
represents the amount of money he spends on pretzels.
\newline
How much money does Hanson spend on pretzels?
\newline
3.50
$
3.50\$
3.50$
2
2
2
3.50
$
3.50\$
3.50$
3
3
3
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Susan set up a lemonade stand to raise money for a children's hospital. She's selling cups of lemonade for
2.50
2.50
2.50
$
\$
$
each and brownies for
1.50
1.50
1.50
$
\$
$
each. She sells
280
280
280
items and raises
540
540
540
$
\$
$
. This system of equations can be used to represent the situation:
\newline
x
+
y
=
280
x + y = 280
x
+
y
=
280
\newline
2.5
x
+
1.5
y
=
540
2.5x + 1.5y = 540
2.5
x
+
1.5
y
=
540
\newline
Which statement is correct?
\newline
(A) In the system of equations,
x
x
x
represents the amount of money Susan raises from selling brownies, and
$
\$
$
0
0
0
represents the amount of money she raises from selling lemonade.
\newline
(B) In the system of equations,
x
x
x
represents the number of brownies Susan sells, and
$
\$
$
0
0
0
represents the number of cups of lemonade she sells.
\newline
(C) In the system of equations,
x
x
x
represents the number of cups of lemonade Susan sells, and
$
\$
$
0
0
0
represents the number of brownies she sells.
\newline
(D) In the system of equations,
x
x
x
represents the amount of money Susan raises from selling lemonade, and
$
\$
$
0
0
0
represents the amount of money she raises from selling brownies.
\newline
How much money does Susan raise from selling lemonade?
\newline
$
\$
$
____
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Mr. Lee wants to hire a balloon artist for the carnival day at school, and he is considering two companies that each charge a fixed hourly rate. A balloon artist from The Amusing Muse charges
$
549
\$549
$549
for
3
3
3
hours. This equation describes how much a balloon artist from Laugh Time Parties charges. In the equation,
y
y
y
represents the cost in dollars for
x
x
x
hours.
\newline
\newline
y
=
190
x
y = 190x
y
=
190
x
\newline
Which company charges more per hour?
\newline
Choices:
\newline
(A)The Amusing Muse
\newline
(B)Laugh Time Parties
\newline
How much would that company charge for
2
2
2
hours?
\newline
$
\$
$
_____
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Translate the sentence into an equation.
\newline
The sum of
8
8
8
times a number and
9
9
9
is
2
2
2
.
\newline
Use the variable
b
b
b
for the unknown number.
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Cyce city charges an initial price of
$
\$
$
15
15
15
plus
$
\$
$
0
0
0
.
75
75
75
per hour for bike rentals, which is represented by the expression
$
\$
$
15
15
15
−
0
-0
−
0
.
75
75
75
h
h
h
. Complete each statement to describe the terms. The variable
h
h
h
represents ? The constant
15
15
15
represents ? The variable term
0.75
h
0.75h
0.75
h
represents ?
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The monthly cost (in dollars) of water use is a linear function of the amount of water used (in hundreds of cubic feet, HCF). The cost for using
14
14
14
HCF of water is
$
39.89
\$39.89
$39.89
, and the cost for using
51
51
51
HCF is
$
100.94
\$100.94
$100.94
. What is the cost for using
21
21
21
HCF of water?
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The monthly cost (in dollars) of a long-distance phone plan is a linear function of the total calling time (in minutes). The monthly cost for
40
40
40
minutes of calls is
$
15.60
\$15.60
$15.60
and the monthly cost for
62
62
62
minutes is
$
18.46
\$18.46
$18.46
. What is the monthly cost for
56
56
56
minutes of calls?
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The ratio boys : girls in a park is
1
:
4
1:4
1
:
4
.
\newline
65
65
65
girls leave and the ratio becomes
4
:
3
4:3
4
:
3
.
\newline
Work out how many boys there are in the park.
\newline
□
\square
□
boys
\newline
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1.4
×
1
0
5
1.4 \times 10^{5}
1.4
×
1
0
5
bacteria are measured to be in a dirt sample that weighs
1
1
1
gram. Use scientific notation to express the number of bacteria that would be in a sample weighing
4
4
4
grams.
\newline
Answer:
□
×
1
0
□
\square \times 10^ \square
□
×
1
0
□
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The expression
1.05
S
+
1.02
B
1.05S + 1.02B
1.05
S
+
1.02
B
predicts the end-of-year value of a financial portfolio, where
S
S
S
is the value of stocks in dollars and
B
B
B
is the value of bonds in dollars in the portfolio at the beginning of the year. What is the predicted end-of-year value, in dollars, of a portfolio that begins the year with
S
S
S
dollars in stocks and
B
B
B
dollars in bonds?
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Olivia has a
20
20
20
-meter-long fence that she plans to use to enclose a rectangular garden of width
w
w
w
. The fencing will be placed around all four sides of the garden so that its area is
18.75
18.75
18.75
square meters.
\newline
Write an equation in terms of
w
w
w
that models the situation.
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Lennox owns a big apple orchard. She ships her apples to various markets using a fleet of trucks. Every week, each truck goes on
3
3
3
trips, and for each trip Lennox gets
$
300
\$300
$300
. On a single trip, a truck delivers
50
50
50
packs, and each pack contains
12
12
12
kilograms of apples. Overall, Lennox sells
$
4500
\$4500
$4500
worth of apples in a week.
\newline
How much does Lennox get for a single kilogram of apples?
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The planet Uranus is about
3
×
1
0
9
3 \times 10^9
3
×
1
0
9
kilometers from the sun. The dwarf planet Pluto is about
6
×
1
0
9
6 \times 10^9
6
×
1
0
9
kilometers from the sun. Use these numbers to complete the sentence below. Write your answer in standard form, without using exponents. Uranus is about ____ times as far from the sun as Pluto.
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This equation shows how the number of pictures Clarence has taken so far this year is related to the number of days he spends on vacation.
\newline
p
=
d
+
1
p = d + 1
p
=
d
+
1
\newline
The variable
d
d
d
represents the number of days he spends on vacation, and the variable
p
p
p
represents the total number of pictures taken this year. After
6
6
6
days of vacation, how many pictures will Clarence have taken?
\newline
_____ pictures
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This equation shows how the money Oxford Middle School spends on workbooks is related to the total number of workbooks they buy.
\newline
d
=
18
t
d = 18t
d
=
18
t
\newline
The variable
t
t
t
represents the number of workbooks bought by the school, and the variable
d
d
d
represents the total cost of those workbooks. How much would it cost Oxford Middle School to buy a total of
1
1
1
workbook?
\newline
$
\$
$
_
_
_
_
\_\_\_\_
____
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This equation shows how the number of cakes Naomi can bake is related to the number of additional sticks of butter she buys.
\newline
c
=
3
s
c = 3s
c
=
3
s
\newline
The variable
s
s
s
represents the number of additional sticks of butter Naomi buys, and the variable
c
c
c
represents the total number of cakes she can bake. With
3
3
3
additional sticks of butter, how many total cakes can Naomi bake?
\newline
_____ cakes
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This equation shows how the cost of Deion's birthday party depends on the number of guests.
\newline
c
=
4
g
c = 4g
c
=
4
g
\newline
The variable
g
g
g
represents the number of guests, and the variable
c
c
c
represents the cost in dollars. If there are
5
5
5
guests, how much will Deion's birthday party cost?
\newline
$
\$
$
_____
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This equation shows how the length of Alana's essay depends on the number of hours she spends writing this week.
\newline
p
=
h
+
9
p = h + 9
p
=
h
+
9
\newline
The variable
h
h
h
represents the number of hours she spends writing this week, and the variable
p
p
p
represents the total number of pages that have been written. After spending
7
7
7
hours writing this week, how many pages will Alana have written in total?
\newline
_____ pages
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A hot air balloon rose at a constant rate. After
3
3
3
minutes, the balloon was
44
44
44
meters above the ground. Then,
2
2
2
minutes later, the hot air balloon had risen to
70
70
70
meters above the ground.
\newline
How much did the hot air balloon rise each minute?
\newline
____ meters
\newline
How far above the ground was the hot air balloon at the start of the ride?
\newline
____ meters
\newline
Complete the equation that describes the relationship between the altitude of the hot air balloon in meters,
A
A
A
, and the elapsed time in minutes,
t
t
t
.
\newline
Write your answer using whole numbers or decimals rounded to the nearest tenth.
\newline
A = ____ t + ____
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