# Praxis® Core Math Practice Test

Use our free Praxis Core Math practice test to review the most important skills and concepts that you must know for this section of your certification exam. The Praxis Core Math test includes a total of 56 questions that must be answered within 85 minutes. An on-screen calculator will be available. The test covers 4 major topics: Numbers & Quantities, Algebra & Functions, Geometry, Statistics & Probability. You may use a calculator.

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 Question 1

### Solve for $x$: $3(x + 1) = 5(x − 2) + 7$

 A $−2$ B $2$ C $\dfrac{1}{2}$ D $3$ E $−3$
Question 1 Explanation:
The correct answer is (D). Begin by distributing the $3$ and the $5$ through their respective parentheses, then combine like terms on each side of the equal sign:

$3(x + 1) = 5(x − 2) + 7$
$3x + 3 = 5x − 10 + 7$
$3x + 3 = 5x − 3$

Add $3$ to both sides to maintain the equality:

$3x + 6 = 5x$

Subtract $3x$ from both sides and then divide the resulting equation by $2$ to solve for $x$ as follows:

$6 = 2x$
$3 = x$
 Question 2

### Which of the following numbers is the greatest? $\dfrac{2}{3} ~~ 0.6 ~~ \dfrac{13}{22} ~~ 0.08$

 A $\dfrac{2}{3}$ B $0.6$ C $\dfrac{13}{22}$ D $0.08$
Question 2 Explanation:
The correct answer is (A). To determine which value is the greatest, convert the fractions to decimals and compare each value. The fraction $\frac{2}{3}$ is equivalent to $2 ÷ 3$ or $0.66$; the fraction $\frac{13}{22}$ evaluates to $0.59$.
 Question 3

### Liam is driving to Utah. He travels at $70$ kilometers per hour for $2$ hours, and $63$ kilometers per hour for $5$ hours. Over the $7$ hour time period what was Liam's average speed?

 A $64$ km/h B $65$ km/h C $66$ km/h D $67$ km/h E $68$ km/h
Question 3 Explanation:
The correct answer is (B). To find the average, use the following formula:

$\text{Average Speed} = \text{Total Distance} ÷ \text{Total Time}$

Total Distance $= 2 \text{hours} × 70 \text{km/h} + 5 \text{hours} × 63 \text{km/h}$
$= 140 \text{km} + 315 \text{km}$
$= 455 \text{km}$

Total Time $= 7$ hours

Average Speed $= 455 \text{km} ÷ 7 \text{hours}$
$= 65 \text{km/h}$
 Question 4

### Sofía's Restaurant offers the following choices: If the only dinner you can order must include one salad, one main course, and one dessert, how many different combinations of this dinner are possible?

 A $12$ B $60$ C $120$ D $144$ E $360$
Question 4 Explanation:
The correct answer is (B). This can be solved with the Counting Principle: If there are $m$ ways for one activity to occur and $n$ ways for a second activity to occur, then there are $m × n$ ways for both to occur.

To solve this problem multiply:

$4 \; (\text{the number of salads})$ $× 5 \; (\text{the number of main courses})$ $× 3 \; (\text{the number of desserts})$:

$4 × 5 × 3 = 60$ different combinations
 Question 5

### If $4x + 3x − 2(x + 5) = −9$, then $x =$ ?

 A $−\dfrac{1}{2}$ B $0$ C $\dfrac{1}{5}$ D $\dfrac{2}{3}$
Question 5 Explanation:
The correct answer is (C). Evaluate the expression by first distributing the $−2$ through the parentheses and then combining like terms:

$4x + 3x − 2x – 10 = −9$
$7x − 2x − 10 = −9$
$5x = 1$
$x = \dfrac{1}{5}$
 Question 6

### Li wants to buy as many bags of mulch as possible with his $\$305$, and he would like them to be delivered to his house. The cost is$\$7.50$ per bag and there is a $\$35.75$delivery charge. The mulch is only sold in full bags. How many bags can Li buy?  A$35$B$36$C$45$D$46$Question 6 Explanation: From Li’s initial amount of$\$305$, a flat $\$35.75$delivery charge is deducted:$\$305 − \$35.75 = \$269.25$.

We can then divide this amount by the cost per bag to find the total number of bags that Li can buy:

$\$269.25 ÷ \$7.50 = 35.9$ bags. However, the question states that the mulch can only be sold in full bags, so we must round our answer down to ensure that Li does not exceed his budget.
 Question 7

### Mason earns $\$8.10$per hour and worked$40$hours. Noah earns$\$10.80$ per hour. How many hours would Noah need to work to equal Mason’s earnings over $40$ hours?

 A $15$ B $25$ C $27$ D $28$ E $30$
Question 7 Explanation:
The correct answer is (E). Begin by calculating Mason’s total earnings after $40$ hours:

$40 \text{ hours} × \$8.10 \text{ per hour} = \$324$

Next, divide this total by Noah’s hourly rate to find the number of hours Noah would need to work:

$\$324 ÷ \$10.80 \text{ per hour} = 30$ hours
 Question 8

### Diego’s current age is five times Martina’s age ten years ago. If Martina is currently $m$ years old, what is Diego’s current age in terms of $m$?

 A $5m$ B $5m − 10$ C $5m − 50$ D $5m + (m − 10)$ E $10(m − 5)$
Question 8 Explanation:
The correct answer is (C). Martina’s age ten years ago is:
$m − 10$

So Diego’s age is:
$5(m − 10)$
$= 5m − 50$
 Question 9

### In a coordinate plane, triangle $ABC$ has coordinates: $(−2,7)$, $(−3,6)$, and $(4,5)$. If triangle $ABC$ is reflected over the $y$-axis, what are the coordinates of the new image?

 A $(−2,−7), (−3,−6), (−4,−5)$ B $(−2,−7), (−3,−6), (4,−5)$ C $(2,7), (3,6), (−4,5)$ D $(2,7), (3,6), (4,5)$
Question 9 Explanation:
The correct answer is (C). One way to solve this problem is to draw the figure and then count how many units each point is from the $y$-axis, and to then count the same number of units in the opposite direction to find each point’s reflection. A more efficient method is to recognize that reflecting over the $y$-axis causes the $x$-value to switch sign and does not influence the $y$-value of the point. Reflecting $(−2,7), (−3,6), (4,5)$ across the $y$-axis produces the points $(2,7), (3,6), (−4,5)$.
 Question 10

### Arrange the following fractions in order from least to greatest. $\dfrac{7}{5}, \dfrac{15}{4}, \dfrac{3}{2}, \dfrac{11}{4}, \dfrac{13}{3}$

 A $\dfrac{7}{5}, \dfrac{15}{4}, \dfrac{3}{2}, \dfrac{11}{4}, \dfrac{13}{3}$ B $\dfrac{7}{5}, \dfrac{3}{2}, \dfrac{15}{4}, \dfrac{11}{4}, \dfrac{13}{3}$ C $\dfrac{7}{5}, \dfrac{3}{2}, \dfrac{11}{4}, \dfrac{15}{4}, \dfrac{13}{3}$ D $\dfrac{7}{5}, \dfrac{15}{4}, \dfrac{11}{4}, \dfrac{3}{2}, \dfrac{13}{3}$
Question 10 Explanation:
There are two useful methods for directly comparing the values of fractions: converting each fraction to a decimal and rewriting each fraction with a common denominator. In this case, the least common denominator would be $60 (5 * 4 * 3)$, and as a result, the optimal approach entails converting each fraction to a decimal, instead:

$\frac{7}{5} = 1.4$, $\frac{15}{4} = 3.75$, $\frac{3}{2} = 1.5$, $\frac{11}{4} = 2.75$, $\frac{13}{3} = 1.33$

Now, order the decimals from least to greatest, and match the resulting list with its corresponding list of fractions: $1.4, 1.5, 2.75, 3.75$, and $4.33$ which corresponds to:

$\dfrac{7}{5}, \dfrac{3}{2}, \dfrac{11}{4}, \dfrac{15}{4}, \dfrac{13}{3}$
 Question 11

### This pie chart shows Al’s monthly expenses. If Al spent a total of $\$1,550$in one month, how much did he spend on clothes in that month?  A$\$77.50$ B $\$232.50$C$\$775$ D $\$7.75$E$\$1,627.50$
Question 11 Explanation:
The correct answer is (A). Remember that $5\%$ is equivalent to $\frac{5}{100}$ or $.05$:

$.05 × \$1,550 = \$77.50$
 Question 12

### Calculate the value of $x$ for the right triangle shown below. A $66$ B $89$ C $56$ D $65$ E $75$
Question 12 Explanation:
The correct answer is (D). The Pythagorean theorem can be used to find a missing segment of a right triangle if two side lengths are given. The theorem relates the legs of a right triangle to its hypotenuse as $a^2 + b^2 = c^2$, where $a$ and $b$ are the legs and $c$ is the hypotenuse. The hypotenuse is the longest side in a right triangle and is always opposite the right angle.

Substitute the known values into their appropriate places and solve for the unknown side length:
$33^2 + 56^2 = c^2$

Evaluate the squares:
$1,089 + 3,136 = c^2$
$4,225 = c^2$

Evaluate the square root of both sides:
$\sqrt{c^2} = \sqrt{4,225}$
$c = 65$
 Question 13

### Oscar purchased a new hat that was on sale for $\$5.06$. The original price was$\$9.20$. What percentage discount was the sale price?

 A $4.5\%$ B $41.4\%$ C $45\%$ D $55\%$
Question 13 Explanation:
The correct answer is (C). The percentage discount is the reduction in price divided by the original price. The difference between original price and sale price is:

$\$9.20 − \$5.06 = \$4.14$The percentage discount is this difference divided by the original price:$\$4.14 ÷ \$9.20 = 0.45$Convert the decimal to a percentage by multiplying by$100\%$:$0.45 × 100\% = 45\%$ Question 14 ### Anastasia needs to order liquid fertilizer for her landscaping company. She plans to keep the fertilizer in a large cylindrical storage tank, but isn’t sure how much it will hold. The tank is$10$feet tall and the circular base has a diameter of$10$feet. What is the volume of her storage tank?  A$78.5 \text{ ft}^3$B$157 \text{ ft}^3$C$3,140 \text{ ft}^3$D$1,570 \text{ ft}^3$E$785 \text{ ft}^3$Question 14 Explanation: The correct answer is (E). The following is the formula for a cylinder: Volume = (area of the base) (height) The base of a cylinder is a circle, so you will also need the formula for the area of a circle, which is also provided on the formula sheet:$\text{Area} = π(\text{radius})^2$Combining these formulas, we get:$V = πr^2hV$is volume,$r$is the radius, and$h$is the height of the cylinder. (Remember that the radius of a circle is half of the diameter.) Substitute the given values into the equation to solve for the volume:$V = π \cdot (5)^2 \cdot 10V = 250π$Recall that$π$(pi) is approximately equal to$3.14$:$250 \cdot 3.14 = 785$ Question 15 ### Kayla owns a house cleaning company and must give price quotes to potential customers. She determines her price by assuming a$\$25$ base charge and then adding $\$8$for each bathroom and$\$4$ for every other room. If she uses $P$ to represent the price, $B$ to represent the bathrooms, and $R$ to represent the other rooms, which of the following defines her price quote formula?

 A $P = 25 + 12(BR)$ B $P = 25(4R + 8B)$ C $P = 25 + 8B + 4R$ D $P = (4)(8)(R + B) + 25$ E $P = (25 + 8B) + (25 + 4R)$
Question 15 Explanation:
The correct answer is (C). The price quote, $P$, will be equal to the total amount of charges. It’s given that there is a $\$25$base charge, so we can begin by writing the price quote as$P = \$25$. To this base charge of $\$25$,$\$8$ per each bathroom, represented by the variable $B$, is added; our price quote is now $P = \$25 + \$8B$. Lastly, $\$4$per each other room,$R$, is added, and the actual price quote is$P = \$25 + \$8B + \$4R$, or $P = 25 + 8B + 4R$.
 Question 16

### Use the information below to answer the question that follows. A business recorded the number of customers who visited the store throughout the week. How many days had a number of visitors greater than the average number of visitors for the entire week?

 A $6$ days B $4$ days C $5$ days D $2$ days E $3$ days
Question 16 Explanation:
The correct answer is (E). To answer the question we need to determine the average number of visitors for the week:

$\dfrac{13 + 12 + 16 + 19 + 25 + 33 + 22}{7}$ $= \dfrac{140}{7} = 20$

How many days had more than $20$ visitors?

Fri $= 25$
Sat $= 33$
Sun $= 22$

$3$ days exceeded the $20$-visitor average.
 Question 17

### In a factory there are two separate containers of stress balls. In the first container are $80$ balls, and in the second are $90$ balls. $40\%$ of the balls in the first container are defective, and $20\%$ in the second container are defective. In total, how many balls in the two containers are defective?

 A $102$ B $50$ C $60$ D $51$
Question 17 Explanation:
The correct answer is (B). Calculate the number of defective balls in each container and then add these two amounts:

$40\%$ of $80 = 0.40 \cdot 80 = 32$
$20\%$ of $90 = 0.20 \cdot 90 = 18$

$32 + 18 = 50$
 Question 18

### Which of the following fractions is greater than $0.4$ and less than $0.5$?

 A $\dfrac{6}{11}$ B $\dfrac{3}{10}$ C $\dfrac{12}{23}$ D $\dfrac{9}{20}$
Question 18 Explanation:
The correct answer is (D). Remember that $0.5$ is equivalent to $\frac{1}{2}$. Our fraction needs to be less than $\frac{1}{2}$.

Since $5.5$ is half of $11$, $\frac{6}{11}$ is greater than $\frac{1}{2}$. No good.

Since $11.5$ is half of $23$, $\frac{12}{23}$ is greater than $\frac{1}{2}$. No good.

This means our answer is either $\frac{3}{10}$ or ${9}{20}$.

$\frac{3}{10}$ is equivalent to $0.3$, which is less than $0.4$. No good.

$\frac{9}{20}$ must be correct. You could also solve by recognizing that $0.4 = \frac{4}{10} = \frac{8}{20}$ and $0.5 = \frac{5}{10} = \frac{10}{20}$, so $\frac{9}{20}$ works.
 Question 19

### Use the menu below to answer the question that follows. Neil ordered the following for his family: two cheeseburgers, one hamburger, two large fries, one small fries, and three small sodas. If the total calories for the order is $3370$, what is the missing calorie information on the menu?

 A $1000$ B $400$ C $450$ D $480$ E $900$
Question 19 Explanation:
The correct answer is (C). Let's begin by totaling the calories for the items we are able to:

Two cheeseburgers $= 2 × 530 = 1,060$ calories
One hamburger $= 430$ calories
One small fries $= 230$ calories
Three small sodas $= 3 × 250 = 750$ calories

If we subtract each of the known calories from the total calories for the order we will know how many calories came from two large fries:

$3,370 − 1,060 − 430 − 230 − 750 = 900$

$900$ calories came from two orders of large fries. Each order of fries is $450$ calories.
 Question 20

### Aisha wants to paint the walls of a room. She knows that each can of paint contains one gallon. A half gallon will completely cover a $55$ square feet of wall. Each of the four walls of the room is $10$ feet high. Two of the walls are $10$ feet wide and two of the walls are $15$ feet wide. How many $1$-gallon buckets of paint does Aisha need to buy in order to fully paint the room?

 A $4$ B $5$ C $9$ D $10$ E $15$
Question 20 Explanation:
The correct answer is (B). The total number of buckets necessary will be the total area of the walls divided by the total area covered by each bucket. First, calculate the area of the walls Aisha wants to paint. Two of the walls are $10 × 10$ and two of the walls are $10 × 15$:

$2 (10 × 10) = 200$ sq. ft. $2 (10 × 15) = 300$ sq. ft. So the total square footage of the walls is $500$. If a half gallon of paint will cover $55$ square feet, then each gallon will cover $2 × 55 = 110$ square feet. Four gallons can only cover $440$ square feet. Five gallons will cover $550$ square feet, which will be enough for the entire area of the walls.
 Question 21

### The data in the table below shows the results of Tracy trying to train her dog Snowy. In what percentage of these trials did Snowy obey the command given?

 A $25\%$ B $37.5\%$ C $50\%$ D $62.5\%$ E $75\%$
Question 21 Explanation:
The correct answer is (C). Calculate the percentage of trials that Snowy obeyed by dividing the number of times Snowy obeyed by the total number of commands given. The number of times Snowy obeyed: trial $1$, trial $3$, trial $6$, trial $8$, which sums to $4$ trials. There are $8$ total trials so the percentage is:

$\dfrac{4}{8}$ $= \dfrac{1}{2} = 50\%$
 Question 22

### Max struggled with his math class early in the year, but he has been working hard to improve his scores. There is one test left, and he is hoping that his final average test score will be 75. What score will he need to get on Test 6 to finish the year with an average score of 75? A $75$ B $85$ C $92$ D $98$ E $100$
Question 22 Explanation:
The correct answer is (C). The average of a set of data points is the sum of the data points divided by the total number of data points. In this case, we are given $5$ out of $6$ data points, the number of data points, and the desired average. Substitute the given values into the formula for the average, using a variable to represent the unknown test score, and then solve for the variable:

Average = sum of data points ÷ number of data points

$75 = (50 + 52 + 77 + 88 + 91 + x) ÷ 6$

Now solve for $x$. Start with the addition:

$75 = (358 + x) ÷ 6$

Next, eliminate the denominator by multiplying both sides by $6$:

$450 = 358 + x$

The last step is to subtract $358$ from both sides:

$x = 92$
 Question 23

### A student is calculating the average of a list of seven numbers. After adding the seven numbers together, he accidentally multiplied by $7$ instead of dividing and got $392$. What could the student do to get the correct average without having to clear the calculator and start over?

 A Divide the incorrect result by $7$ B Divide the incorrect result by $49$ C Multiply the incorrect result by $7$ D Divide the incorrect result by $14$
Question 23 Explanation:
The correct answer is (B). Let $x$ equal the sum of the seven numbers. The student should have divided $x$ by $7$ to get the average, giving the correct result of $\frac{x}{7}$. Since the student accidentally multiplied by $7$, the calculator shows $7x$.

To get the correct answer, the student needs to divide by $7$ twice, which is the same as dividing by $49$:

$\dfrac{7x}{49} = \dfrac{x}{7}$
 Question 24

### List A consists of the numbers ${2, 9, 5, 1, 13}$, and list B consists of the numbers ${7, 4, 12, 15, 18}$. If the two lists are combined, what is the median of the combined list?

 A $5$ B $6$ C $7$ D $8$ E $9$
Question 24 Explanation:
The correct answer is (D). Recall that the median of a set of data is the value located in the middle of the data set. In the case of a data set that contains an even number of numbers, the median is the average of the two middle numbers. Combine the $2$ sets provided, and organize them in ascending order:

${1, 2, 4, 5, 7, 9, 12, 13, 15, 18}$

Since there are an even number of items in the resulting list, the median is the average of the two middle numbers.

Median $= (7 + 9) ÷ 2 = 8$
 Question 25

### Tavon's flight is $270$ minutes long. How many hours does the flight last?

 A $4$ hours B $4 \frac{1}{2}$ hours C $5$ hours D $5 \frac{1}{2}$ hours
Question 25 Explanation:
The correct answer is (B).

$\require{cancel} 270 \cancel{\text{minutes}} \cdot \dfrac{1 \text{ seconds}}{60 \cancel{\text{minutes}}}$ $= \dfrac{270}{60} \text{ hours}$ $= \dfrac{27}{6} \text{hours}$

$\dfrac{27}{6} = \dfrac{9}{2} = 4 \dfrac{1}{2}$
 Question 26

### $6.6 × 10^{−4}$

 A $0.000066$ B $0.00066$ C $0.0066$ D $0.066$ E $0.66$
Question 26 Explanation:
The correct answer is (B). Multiplying a decimal value by $10$ raised to a power is equivalent to moving the decimal point to the left or right the number of times indicated by the power.

In the case of a negative exponent, the decimal is moved to the left (this is the same as dividing by $10$ a number of times).

In the case of a positive exponent, the decimal is moved to the right (this is the same as multiplying by $10$ a number of times).

The negative exponent here, $−4$, indicates that the decimal point is to be moved to the left $4$ places:

$6.6 × 10^{−4} = 0.66 × 10^{−3}$
$= 0.066 × 10^{−2}$
$= 0.0066 × 10^{−1}$
$= 0.00066 × 10^{0}$
$0.00066$
 Question 27

### $ABC$ is a triangle with coordinates $(3, 1), (6, 1)$, and $(1, 3)$. It is translated to points $A′(−3, 3), B′( 0, 3)$, and $C′(−5, 5)$. Find the rule for the translation.

 A $6$ units right and $2$ units down B $2$ units left and $6$ units up C $6$ units down and $2$ units right D $6$ units left and $2$ units up
Question 27 Explanation:
The correct answer is (D). The rule of translation is the rule, which when applied to the first set of points, yields the second set of points. The difference between the $x$ values of the non-translated and translated points is:

$(x_2 − x_1) = (−3 − 3) = (0 − 6) = (−5 − 1) = −6$

This corresponds to a shift of $6$ units to the left.

Likewise, for the $y$ values:

$(y_2 − y_1) = (3 − 1) = (3 − 1) = (5 − 3) = 2$

This corresponds to a vertical shift of $2$ units.
 Question 28

### There are $5$ blue marbles, $4$ red marbles, and $3$ yellow marbles in a box. If Isabella randomly selects a marble from the box, what is the probability of her selecting a red or yellow marble?

 A $\dfrac{1}{4}$ B $\dfrac{1}{3}$ C $\dfrac{7}{12}$ D $\dfrac{3}{4}$ E $\dfrac{2}{3}$
Question 28 Explanation:
The correct answer is (C). A probability is the likelihood of a successful event occurring divided by the total number of events possible. In this case, a successful event is selecting either a red or a yellow marble and the total number of events possible is the total number of marbles.

Number of red and yellow marbles: $4 + 3 = 7$

Total number of marbles: $5 + 4 + 3 = 12$

Probability: $\frac{7}{12}$
 Question 29

### A river rafting guide offers group trips. She charges $\$250$for$2$people and$\$40$ more for each additional person. If $5$ friends share the cost of a trip equally, how much will each person pay?

 A $\$40$B$\$58$ C $\$74$D$\$90$ E $\$370$Question 29 Explanation: The correct answer is (C). First calculate the entire cost of the trip. Start with$\$250$ for the first two people and then add $\$40$for each of the other three:$250 + 40 + 40 + 40 = \$370$

Then divide this amount by $5$ to determine the amount that each person will pay:

$\$370 ÷ 5 = \$74$
 Question 30

### Consider the list: $2, 2, 3, 5, 9, 11, 17, 21$ If the number $23$ is added to the list, which measurement will NOT change?

 A Mean B Median C Mode D Range E Average
Question 30 Explanation:
The correct answer is (C). The mode will not change. The mode is the number that appears the most frequently; in this case it is $2$. The remaining measures will change:

The mean is the average, and in this case, it will increase because a number larger than the current average is added to the list.

The median is the number in the middle, and it will change from $7$ to $9$.

The range is the difference between the highest and lowest numbers and will change from $19$ to $21$ because a new maximum value is added to the list ($21 − 2 = 19$ and $23 − 2 = 21$).
 Question 31

### The points represented by the $(x, y)$ coordinate pairs in the table below all lie on line $k$. What is the slope of line $k$?

 A $−5$ B $−3$ C $−2$ D $2$ E $5$
Question 31 Explanation:
The correct answer is (C). We are told that all the points lie on line $k$. Therefore, we can calculate the slope as the change in $y$ divided by the change in $x$ using any two of the points. Using points $(1, 3)$ and $(5, −5)$ the slope is calculated as follows:

Slope $= \dfrac{y_2 − y_1}{x_2 − x_1}$

Slope $= \dfrac{−5 − 3}{5 − 1}$

Slope $= \dfrac{−8}{4}$

Slope $= −2$
 Question 32

### This pie chart shows Al’s monthly expenses. If Al spent $\$210$on transportation in one month, how much did he spend on food in that month?  A$\$126$ B $\$250$C$\$325$ D $\$350$E$\$325$
Question 32 Explanation:
The correct answer is (D). One method for solving this problem is to recognize that the ratio of the amount Al spends on food to the amount he spends on transportation will equal the ratio of the percentage of his income spent on food to the percentage spent on transportation. Expressed mathematically:

$\dfrac{\text{Amount spent on food}}{\text{Amount spent on transportation}}$

$= \dfrac{\text{Percentage spent on food}}{\text{Percentage spent on transportation}}$

Substitute the known values to then solve for the unknown amount spent on food:

$\dfrac{\text{Amount spent on food}}{\$210} = \dfrac{25}{15}$Cross multiply and simplify to solve: Amount spent on food$= \dfrac{\$210 \cdot 25}{15} = \$350\$
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