For the Classic ACT exam:
The ACT Mathematics test is a timed exam...60 questions in 60 minutes
This implies that you have to solve each question in one minute.
Each of the first 20 questions (less challenging) will typically take less than a minute a solve.
Each of the next 20 questions (medium challenging) may take about a minute to solve.
Each of the last 20 questions (more challenging) may take more than a minute to solve.
The goal is to maximize your time.
You use the time saved on the questions you solve in less than a minute to solve questions that will take more
than a minute.
So, you should try to solve each question correctly and timely.
So, it is not just solving a question correctly, but solving it correctly on time.
Please ensure you attempt all ACT questions.
There is no negative penalty for a wrong answer.
Also: please note that unless specified otherwise, geometric figures are drawn to scale. So, you can figure out
the correct answer by eliminating the incorrect options.
Other suggestions are listed in the solutions/explanations as applicable.
These are the solutions to the ACT past questions on the topics: Expressions and Equations.
When applicable, the TI-84 Plus CE calculator (also applicable to TI-84 Plus calculator) solutions are provided
for some questions.
The link to the video solutions will be provided for you. Please
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(2.) Wally buys dog collars online for $4 each.
The shipping and handling fee is $5 total, regardless of the number of dog collars ordered.
Which of the following equations represents the relationship between x, the number of dog collars
ordered, and y, Wally's total bill in dollars?
$
A.\;\; y = \dfrac{x}{5} \\[5ex]
B.\;\; y = 4x \\[3ex]
C.\;\; y = 9x \\[3ex]
D.\;\; y = 4x + 5 \\[3ex]
E.\;\; y = 5x + 4 \\[3ex]
$
Cost for dog collars @ $4 per dog collar = $4 \cdot x = 4x$
Shipping and Handling Fee = $5
∴ $y = 4x + 5$
(3.) What number, when added to $\dfrac{1}{3}$, gives a sum that is equal to the sum of $\dfrac{1}{4}$ and
$\dfrac{1}{6}$?
Teacher: The answer is Option F. Student: How do you know?
Just by looking at it? Teacher: Yes...all other options do not have any term in y² Student: But I would like to know how to expand it. Teacher: Sure. We have at least two ways to do it.
But I recommend the process of eliminating incorrect options on the ACT due to the time limit of 1 minute
per question.
This question takes about 2 seconds to do.
The remaining 58 seconds should be used for questions that may take more than a minute to solve.
(9.) One watch gains $2\dfrac{1}{2}$ minutes per hour while another watch loses $1\dfrac{1}{4}$ minutes per
hour.
If both watches are set correctly at noon, after how many hours will the times shown by the watches be
exactly
1 hour apart?
(11.) Yesterday, there was no snow on the ground at 8:00 a.m. when snow began to fall.
Snow fell from 8:00 a.m. to 8:00 p.m. at a constant rate of $\dfrac{3}{4}$ inch per hour.
Kate built a snowman made of 2 spherical snowballs, with the smaller placed on top of the larger.
When she built the snowman, the diameter of the larger snowball was 3 times the diameter of the smaller
snowball.
Today, the day after she built the snowman, there is a 50% chance of rain.
If it rains today, then there is a 60% chance that the snowman will melt.
If it does not rain today, then there is a 10% chance the snowman will melt.
Kate could not build the snowman until there was 6 inches of snow on the ground.
At how many hours after 8:00 a.m. yesterday was there exactly 6 inches of snow on the ground?
(12.) At a local pet store, 50 shoppers were polled to see if they owned cats or dogs.
Among the polled shoppers, 31 owned at least 1 dog, 20 owned at least 1 cat, 7 owned at least 1 dog and at
least 1 cat, and 6 owned neither
a dog nor a cat.
Every polled shopper was given at least 1 gift for taking part in the survey.
Those who owned at least 1 dog were given 1 dog toy valued at $4, those who owned at least 1
cat
were given 1 cat toy valued at $3, and those who
owned neither were given a gift card with a dollar value of g.
A total dollar value of t was given to the shoppers.
Which of the following expressions gives the value of g in terms of t?
(13.) Chris can paint $\dfrac{1}{6}$ of a fence per hour.
Sandy can paint $\dfrac{1}{8}$ of the same fence per hour.
What fraction of the fence will be painted when Chris and Sandy work at these rates for 2 hours?
$
\text{Let the measure of the fence } = p \\[3ex]
Chris:\;\; \dfrac{1}{6} \;\;of\;\;p\;\;\text{per hour for 2 hours} = \dfrac{1}{6} \cdot p \cdot 2 =
\dfrac{p}{3} \\[5ex]
Sandy\;\; \dfrac{1}{8} \;\;of\;\;p\;\;\text{per hour for 2 hours} = \dfrac{1}{8} \cdot p \cdot 2 =
\dfrac{p}{4} \\[5ex]
\text{Chris and Sandy: } \implies \\[3ex]
= \dfrac{p}{3} + \dfrac{p}{4} \\[5ex]
= \dfrac{4p + 3p}{12} \\[5ex]
= \dfrac{7p}{12} \\[5ex]
$
∴ when working together:
The fence is: p
The fraction of the fence is $\dfrac{7}{12}$
(14.) If $x^2 - 5x - 6 = 0$, what is the sum of the 2 possible values of x?
(18.) To estimate the number of bass in a small lake, a biology class caught and tagged 32 bass from the lake.
The next day the class caught 60 bass from the lake and found that 9 of those bass were tagged.
Assuming the 60 bass caught are representative of the entire population of bass in the lake, which of the
following is closest to the number of bass in the entire population?
1st Sample:
Number of tagged bass in the 1st sample = 32
Let the entire population of the bass in the lake = n
2nd Sample:
Number of tagged bass in the 2nd sample = 9
Sample size of 2nd sample = 60
$
\underline{Capture-Recapture\;\;Method} \\[3ex]
\dfrac{32}{n} = \dfrac{9}{60} \\[5ex]
9n = 60 \cdot 32 \\[3ex]
n = \dfrac{60 \cdot 32}{9} \\[5ex]
n = 213.3\bar{3} \\[3ex]
$
The correct answer is Option C.
(19.) The rational equation $\dfrac{y}{y - 4} + \dfrac{2}{y + 1} = \dfrac{y - 5}{y^2 - 3y - 4}$ has the same
solution set as which of the following equations?
(21.) For what positive value of k will the expression $9x^2 + kx + 25$ factor into the form $(ax +
b)^2$ for some real number a and some real number b?
$
2x + 3y = 7 \\[3ex]
\text{Subtract 2x from both sides} \\[3ex]
3y = 7 - 2x \\[3ex]
\text{Divide both sides by 3} \\[3ex]
y = \dfrac{7 - 2x}{3} \\[5ex]
y = \dfrac{-2x + 7}{3}
$
(25.) A conference presenter earned $48.50 for attending a conference and $15.35 per
hour for the hours she spent preparing for her presentation.
Let y be the amount of money, in dollars, earned by the presenter when she spent x hours
preparing for her presentation.
Which of the following equations gives the relationship between x and y?
$
F.\;\; y = 15.35x \\[3ex]
G.\;\; y = 33.15x \\[3ex]
H.\;\; y = 63.85x \\[3ex]
J.\;\; y = 15.35x + 48.50 \\[3ex]
K.\;\; y = 48.50x + 15.35 \\[3ex]
$
$
\text{Attending the conference} = \$48.50 \\[3ex]
x \text{ hours of preparation of presentation } @ \$15.35 \text{ per hour} = \$15.35x \\[3ex]
Earnings = 15.35x + 48.50
$
(26.) Christopher works in a clothing store.
He earns $7.50 per hour, plus 6% of his sales.
Which of the following expressions gives Christopher's earnings, in dollars, when he works x hours and
y dollars in sales?
(28.) In a 145-member choir of only altos and sopranos, there are 37 more altos than sopranos.
What is the ratio of altos to sopranos?
A. 54:91 B. 54:145 C. 91:54 D. 91:145 E. 108:37
$
\text{Let the number of sopranos} = p \\[3ex]
\text{This implies that the number of altos} = p + 37 \\[3ex]
\therefore p + (p + 37) = 145 \\[3ex]
p + p + 37 = 145 \\[3ex]
2p = 145 - 37 \\[3ex]
2p = 108 \\[3ex]
p = \dfrac{108}{2} \\[5ex]
p = 54 \\[5ex]
sopranos:\;\;p = 54 \\[3ex]
altos:\;\; p + 37 = 54 + 37 = 91 \\[3ex]
\text{Ratio of altos to sopranos} = 91:54
$
Use the following information to answer questions 29 – 31.
RunOnline sells shoes and accessories to members and nonmembers of their club.
Members are charged a onetime fee of $40 in order to pay the member price.
The nonmember and member prices of certain items are given below.
Item
Nonmember price
Member price
Pair of shoes
Hat
Water bottle
Workout bag
$100 $12 $15 $27
$90 $10 $14 $20
All given prices include tax and shipping.
Last year, RunOnline had a total of 1,000 customers.
Of those customers, 25% were members.
(29.) Ed and Sarah both bought p pairs of shoes from RunOnline.
Ed is a member and Sarah is a nonmember.
Ed’s total cost, including the onetime fee, was equal to Sarah’s total cost.
What is p?
(30.) From RunOnline, a nonmember purchased a total of 5 items: 2 hats, 2 water bottles, and 1 workout bag.
Which of the following dollar amounts is closest to the mean cost per item purchased by this nonmember?
$
\underline{\text{Nonmember}}\\[3ex]
\text{2 hats } @ \text{\$12 per hat} = 2(12) = \$24 \\[3ex]
\text{2 water bottles } @ \text{\$15 per water bottle} = 2(15) = \$30 \\[3ex]
\text{1 workut bag } @ \text{\$27 per bag} = 1(27) = \$27 \\[3ex]
\text{Total cost} = 24 + 30 + 27 = \$81 \\[5ex]
\text{Number of items} = 5 \\[3ex]
\text{Mean cost per item} \\[3ex]
= \dfrac{\text{Total cost}}{\text{Number of items}} \\[5ex]
= \dfrac{\$81}{5} \\[5ex]
= \$16.2 \\[3ex]
\approx \$16...\text{to the nearest dollar}
$
(31.) RunOnline is currently selling 100 water bottles to members each year.
It was predicted that for every $0.50 decrease in the member price, RunOnline will sell 10 more
water bottles to members each year.
RunOnline decides to lower the member price to $10 per water bottle.
Based on the prediction and excluding the onetime fee, what will be the total revenue from water bottles sold
to members the year following the price reduction?
For Members
Current number of water bottles sold each year = 100
Current price for each water bottle sold = $14
Prediction:
For $0.50 decrease from $14,
RunOnline will sell: 100 + 10 = 110 water bottles the following year
For 2(0.5) = $1 decrease from $14,
RunOnline will sell: 100 + 2(10) = 120 water bottles the following year
For 3(0.5) = $1.5 decrease from $14,
RunOnline will sell: 100 + 3(10) = 130 water bottles the following year
Therefore,
For p(0.5) = $0.5p decrease from $14,
RunOnline will sell: 100 + p(10) = (100 + 10p) water bottles the following year
RunOnline decides to lower the member price to $10 per water bottle.
This implies that: $0.5p decrease from $14 = $10
This implies that: $14 - 0.5p = 10$
In other words, how many times was $0.5 reduced from $14 to give $10
We need to find this number of times so we can know the expected projection in sales
$
\underline{\text{Reduction in price by how much?}}
14 - 0.5p = 10 \\[3ex]
14 - 10 = 0.5p \\[3ex]
0.5p = 4 \\[3ex]
p = \dfrac{4}{0.5} \\[5ex]
p = 8 \\[3ex]
$
The price was decreased by 50 cents eight times
So, we project the increase in sales to be: 10 times eight = 80
This will make the total sales to be 100 + 80 = 180
$
\underline{Sales} \\[3ex]
100 + 10p \\[3ex]
100 + 10(8) \\[3ex]
100 + 80 \\[3ex]
180 \\[3ex]
$
So, by decreasing the price from $14 to $10, RunOnline expects to increase sales
from 100 water bottles to 180 bottle waters.
Now, let us calcuate the total revenue
$
-2 \le x \le 4 \\[3ex]
-1 \le y \le 5 \\[5ex]
\underline{\text{Check the endpoints}} \\[3ex]
|x - y| \\[3ex]
1st:\;\; |-2 - (-1)| \\[3ex]
|-2 + 1| \\[3ex]
|-1| \\[3ex]
1 \\[5ex]
2nd:\;\; |-2 - 5| \\[3ex]
|-7| \\[3ex]
7 \\[5ex]
3rd:\;\; |4 - (-1)| \\[3ex]
|4 + 1| \\[3ex]
|5| \\[3ex]
5 \\[5ex]
4th:\;\; |4 - 5| \\[3ex]
|-1| \\[3ex]
1 \\[3ex]
$
The maximum value of $|x - y|$ is 7
(33.) In the 3 equations below, A, B, and C are positive real numbers.
Each equation will be graphed in the standard (x, y) coordinate plane.
Which of the following equations will result in lines that have a negative slope?
$
\text{I.}\;\;\; Ax + By = C \\[3ex]
\text{II.}\;\;\; Ax - By = C \\[3ex]
\text{III.}\;\;\; -Ax - By = C \\[3ex]
$
F. I only G. II only H. III only J. I and II only K. I and III only
$
y = mx + b...\text{Slope - Intercept Form} \\[3ex]
\text{Slope = coefficient of }x = m...\text{in this case} \\[3ex]
$
Note: In Slope–Intercept Form: Slope is always the coefficient of x A, B, and C are positive real numbers.
$
(I.) \\[3ex]
Ax + By = C \\[3ex]
By = -Ax + C \\[3ex]
y = \dfrac{-Ax + C}{B} \\[5ex]
y = -\dfrac{A}{B}x + \dfrac{C}{B} \\[5ex]
Slope = -\dfrac{A}{B} \\[5ex]
\text{Negative Slope} \\[5ex]
(II.) \\[3ex]
Ax - By = C \\[3ex]
Ax - C = By \\[3ex]
By = Ax - C \\[3ex]
y = \dfrac{Ax - C}{B} \\[5ex]
y = \dfrac{A}{B}x - \dfrac{C}{B} \\[5ex]
Slope = \dfrac{A}{B} \\[5ex]
\text{Positive Slope} \\[5ex]
(III.) \\[3ex]
-Ax - By = C \\[3ex]
-(Ax + By) = C \\[3ex]
-1 \cdot (Ax + By) = -1 \cdot C \\[3ex]
Ax + By = -C \\[3ex]
By = -Ax - C \\[3ex]
y = \dfrac{-Ax - C}{B} \\[5ex]
y = -\dfrac{A}{B}x - \dfrac{C}{B} \\[5ex]
Slope = -\dfrac{A}{B} \\[5ex]
\text{Negative Slope} \\[3ex]
$
I. and III. will result in a negative slope for positive real numbers of A, B, and C
(34.) The trinomial $x^2 + 11x + 24$ can be factored into 2 binomilas with positive integer coefficients.
Which of the following binomials is 1 of the factors?
$
F.\;\; x + 1 \\[3ex]
G.\;\; x + 2 \\[3ex]
H.\;\; x + 3 \\[3ex]
J.\;\; x + 4 \\[3ex]
K.\;\; x + 5 \\[3ex]
$
$
x^2 + 11x + 24 \\[4ex]
\text{Factors of 24 whose sum is 11 are 3 and 8} \\[3ex]
(x + 3)(x + 8) \\[5ex]
(x + 3) \text{ is one of the factors}
$
(35.) A drum contains 40 liters of a 6% potassium bromide solution.
This solution is mixed with 80 liters of pure water to produce a new potassium bromide solution.
What percent of the new solution is potassium bromide?
This is a case of Mixtures: an application of linear equations
C — V — A
Concentration (C) * Volume (V) = Amount (A)
Pure water is 100% water but 0% potassium bromide
6% = 6 ÷ 100 = 0.06
Let us represent this information using a table.
Solution
C
V
A = C × V
1st
0.06
40
2.4
2nd
0
80
0
Mixture
what?
40 + 80 = 120
2.4
The volume of the mixture increases, however, the concentration of the potassium bromide decreases in the
mixture.
To calculate it:
$
\text{Compare to: } y = mx + b ...\text{Slope – Intercept Form} \\[3ex]
y = 4x + 2 \\[3ex]
m = slope = 4
$
(37.) Aman left his bicycle at his friend's house last night.
Today, he decided to walk from home to his friend's house, visit, and then ride the bicycle back home along
the same route he walked.
Aman walked at 3 mph, visited for 1.5 hours, and then rode the bicycle at 8 mph.
Aman arrived home 3 hours after he started walking.
Which of the following values is closest to the number of miles Aman walked?
Aman
Walked: speed = 3 miles per hour
Visited: time = 1.5 hours
Rode Bicycle: speed = 8 miles per hour
Arrived home 3 hours after he started walking
This implies that:
From the time he started walking to the time he arrived home: time = 3 hours
But he visited for 1.5 hours
This implies that he walkedandrode bicycle for 1.5 hours (3 − 1.5 = 1.5)
So,: walking and riding bicycle took 1.5 hours
How do we break it down?
His speed was faster when riding a bicycle than when walking
So, he spent more time (of that 1.5 hours) walking that riding a bicycle
Let's assume that he walked for 1 hour
How many miles did he walk? ...this refers to the distance
s.......t.......d
speed * time = distance
distance = speed * time
$
distance = \dfrac{3\;miles}{hour} * 1\;hour = 3\;miles \\[5ex]
$
Answer close to 3 miles would be the ideal answer.
Let us analyze the options to determine the answer.
Remove Option E. because even if he walked for 1.5 hours, at 3 miles per hour, the distance = 4.5 miles
But Option E. is 8.3 miles
This is not possible because 8.3 > 4.5
Remove Option D. because it is not possible for him to use all the time (for both walking and riding
bicycle) for walking alone.
Remove Option A. because he used more time to walk than ride the bicycle.
Remove Option C. because it is not "really" feasible that he rode the bicycle for about 0.14 hours in this
scenario
The real option would be that he walked for 1.1 hours and rode the bicycle 0.4 hours
This implies that he walked: 3 * 1.1 = 3.3 miles
(38.) In a 73-member choir of only altos and sopranos, there are 19 more altos than sopranos.
What is the ratio of altos to sopranos?
F. 27:46 G. 27:73 H. 46:27 J. 46:73 K. 54:19
Let the:
Number of sopranos = s
Number of altos = a
A 73-member choir of only altos and sopranos ⇒ a + s = 73 ...equation (1.)
There are 19 more altos than sopranos ⇒ a = 19 + s ...equation (2.)
Substitute a = 19 + s into equation (1.) ⇒
$
a + s = 73 \\[3ex]
19 + s + s = 73 \\[3ex]
2s = 73 - 19 \\[3ex]
2s = 54 \\[3ex]
s = \dfrac{54}{2} \\[5ex]
s = 27 \\[5ex]
a + s = 73 \\[3ex]
a = 73 - s \\[3ex]
a = 73 - 27 \\[3ex]
a = 46 \\[5ex]
\text{Ratio of altos to sopranos} = 46:27
$
(39.) Camp counselors will combine $5\dfrac{1}{2}$ pounds of peanuts with $1\dfrac{1}{4}$ pounds of cashews to
make trail mix for their campers.
Each camper will get a small bag filled with $\dfrac{1}{8}$ pound of the trail mix.
What is the maximum number of bags of trail mix that the camp counselors can fill?
Combine $5\dfrac{1}{2}$ pounds of peanuts with $1\dfrac{1}{4}$ pounds of cashews
= $5\dfrac{1}{2}\;lb + 1\dfrac{1}{4}\;lb$
Let the maximum number of bags of trail mix that the camp counselors can fill = p
Each camper will get a small bag filled with $\dfrac{1}{8}$ pound of the trail mix = $\dfrac{1}{8}\;lb \cdot
p$
(46.) The price of 4 meals for Erin and her 3 friends was $60 before the addition of tax and tip.
Erin and 2 friends each had meals of equal price.
The price of the 3rd friend's meal was twice the price of Erin's meal.
What was the price, in dollars, of Erin's meal?
(50.) Sam works at Glendale Hospital and earns $12 per hour for the first 40 hours and
$18 per hour for every additional hour he works each week.
Last week, Sam earned $570.
To the nearest whole number, how many hours did he work?
F. 32 G. 35 H. 38 J. 45 K. 48
Let the number of hours Sam worked = h
Additional hours over 40 hours = h − 40
