Below is our free PCAT Quantitative Reasoning practice test. On this section of the test there are 48 PCAT math questions that must be solved within 45 minutes. The topics covered on this section include basic math, algebra, probability, statistics, precalculus, and calculus. These quantitative reasoning practice problems are designed to be very challenging. Detailed explanations are provided after answering each question.

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Question 1 |

Jim has 10 apples and he gives 7 to Bill. Now Bill has 3 times as many apples as Jim had. How many apples did Bill have to begin with?

20 | |

17 | |

23 | |

2 |

Question 1 Explanation:

The correct answer is (C). Rephrase the question into an algebraic expression: 3 times Jim’s original amount is 3 * 10 = 30. This came after Jim gave 7 to Bill, so reduce this number by 7 to find Bill’s original amount: 30 − 7 = 23.

Question 2 |

If ƒ(x) = 3

*x*and*g*(*x*) = 2*x*², what is*g*(ƒ(−2))?24 | |

−72 | |

48 | |

72 |

Question 2 Explanation:

The correct answer is (D). First evaluate ƒ(−2):

3(−2)

= −6

Next evaluate

2(−6)²

=2(36)

= 72

3(−2)

= −6

Next evaluate

*g*(−6):2(−6)²

=2(36)

= 72

Question 3 |

What is the value of:

Question 3 Explanation:

The correct answer is (C). Simplify the expression using the rules of exponents:

Question 4 |

Which of the following is the same as log 85?

log 35 + log 50 | |

ln 1/85 | |

log 255 − log 3 | |

ln 85 |

Question 4 Explanation:

The correct answer is (C). Recall the rules for condensing and expanding logarithms:

log

and

log

Answer choice (C) can be rewritten as: log (225/3) which is equal to log 85.

log

*a*+ log*b*= log (*a***b*)and

log

*a*− log*b*= log (*a*/*b*)Answer choice (C) can be rewritten as: log (225/3) which is equal to log 85.

Question 5 |

A man went to the store and bought pencils, pens, and erasers in the ratio of 2:4:3.
If he bought 6 erasers then how many items total did he buy?

9 | |

18 | |

12 | |

10 |

Question 5 Explanation:

The correct answer is (B). The ratio can be expressed as: 2

*x*: 4*x*: 3*x*where 3*x*= 6 represents the number of erasers bought. Solving for*x*:*x*= 2, and plugging this value into the ratio: 2(2): 4(2): 3(2) → 4: 8: 6. Find the total number of items purchased by combining these numbers: 4 + 8 + 6 = 18.Question 6 |

How many solutions does the equation have?

3 | |

2 | |

1 | |

0 |

Question 6 Explanation:

The correct answer is (C). A solution to an equation is a value that when plugged in yields a true statement. Begin by isolating the square root and squaring both sides to eliminate it:

Move all of the terms to one side and factor to solve for

→ (

→

Check for extraneous solutions by plugging both solutions into the original equation:

The equation has only 1 solution.

Move all of the terms to one side and factor to solve for

*x*:*x*² −*x*− 2 = 0→ (

*x*+ 1) (*x*− 2) = 0→

*x*= −1,*x*= 2Check for extraneous solutions by plugging both solutions into the original equation:

The equation has only 1 solution.

Question 7 |

The following are the results of a class test:
65, 75, 87, 86, 78, 92, 77, 81, 82, 86, 21, 91, 3
Which of the following is the best to use to quantify the overall performance on the exam?

Mean | |

Median | |

Mode | |

Standard Deviation |

Question 7 Explanation:

The correct answer is (B). Since there are a few extreme outliers: 3 and 21, which differ from the norm, it is better to use the median, which is unaffected by outliers, rather than the mean, which is affected by outliers.

Question 8 |

What is the x-intercept of

*y*=*e*^{5x}+ 2?2 | |

0 | |

ln2 | |

no x-intercept |

Question 8 Explanation:

The correct answer is (D). The x-intercept is where the function crosses the x-axis and can be found by setting the function equal to 0 and solving for

0 =

→ −2 =

To isolate the variable, take the natural log of both sides; however, ln −2 is undefined, so this function does not cross the x-axis.

*x*:0 =

*e*^{5x}+ 2→ −2 =

*e*^{5x}To isolate the variable, take the natural log of both sides; however, ln −2 is undefined, so this function does not cross the x-axis.

Question 9 |

What is 0.25% of

^{1}/_{3}?0.83 | |

0.083 | |

0.0083 | |

0.00083 |

Question 9 Explanation:

The correct answer is (D). Convert 0.25% to a fraction and multiply it by

^{1}/_{3}:Question 10 |

What are the critical points of

*y*= 4*x*² + 3*x*?0 | |

^{−3}/_{8} | |

0 and ^{−3}/_{8} | |

0 and ^{3}/_{8} |

Question 10 Explanation:

The correct answer is (B). A critical point of a function is a point where the derivative of the function is equal to 0 or is undefined. Evaluate the derivative with respect to

0 = 8

→

*x*, set it equal to 0, and solve for*x*:0 = 8

*x*+ 3→

*x*=^{−3}/_{8}Question 11 |

ln

*e*^{7}= 1 −*x*, What is*x*?6 | |

ln 7 − e | |

−6 | |

−ln e − 7 |

Question 11 Explanation:

Recall that ln

7 * 1 = 1 −

→

*e*^{7}can be rewritten as 7 * ln*e*and that ln*e*= 1. Rewriting the original expression and solving for*x*:7 * 1 = 1 −

*x*→

*x*= −6Question 12 |

If a recipe calls for 3 parts sugar and 2 parts milk, and a total of 45 oz is made, how much milk is there?

18 oz | |

9 oz | |

27 oz | |

18 parts |

Question 12 Explanation:

The correct answer is (A). The ratio of sugar to milk is 3x:2x. Combining these and setting the amount equal to the given total:

5

→

Plugging this back in to find the total amount of milk:

2 * 9 = 18 oz

5

*x*= 45→

*x*= 9Plugging this back in to find the total amount of milk:

2 * 9 = 18 oz

Question 13 |

Question 13 Explanation:

The correct answer is (C). Recall that the anti-derivative of sin(

*x*) = −cos(*x*), because the derivative of cos(*x*) = −sin(*x*). So, the integral becomes −cos(*x*), evaluated from*π*to 2*π*= (−cos(2*π*)) − (−cos(*π*)) = −1 + (−1) = −2. Notice that all of the answer choices contain the integral of cos(*x*), which evaluates to sin(*x*). Calculate the definite integrals to find which evaluates to −2: answer choice (A), 2 * sin(*x*) from 0 to*π*/2 = (2 * sin(*π*/2)) − (2 * sin(0)) = 2 − 0 = 2. Answer choice (B) evaluates to 0. Answer choice (C) evaluates to −2 and is the correct answer.Question 14 |

Question 14 Explanation:

The correct answer is (D). Simplify the expression:

Question 15 |

John is running towards a finish line at 10 m/s and is 50m away. Bolt is running at 14 m/s but is 75m away. Assuming they keep their constant speed, who will reach the finish line first?

John | |

Bolt | |

It is a tie | |

Neither will finish |

Question 15 Explanation:

The correct answer is (A). Calculate how long it will take each runner to finish, the shorter time length indicates the first to finish. Recall that:

John finishes before Bolt.

John finishes before Bolt.

Question 16 |

What is the third derivative of −3sin(

*x*)?−3cos x | |

3cos x | |

−3sin x | |

3sin x |

Question 16 Explanation:

The correct answer is (B). Recall that the derivative of sin(

*x*) = cos(*x*) and the derivative of cos(*x*) = −sin(*x*). First derivative: −3 * cos(*x*). Second derivative: −3 * (−sin(*x*)) = 3sin(*x*). Third derivative: 3 * cos(*x*).Question 17 |

If a vector <

*x,*3*x*> is multiplied by 4, which statement is false?The second coordinate is 3 times the first. | |

The second coordinate is 8x greater than the first. | |

The second coordinate is 8x times the first. | |

After the multiplication, the second coordinate undergoes a greater overall change. |

Question 17 Explanation:

The correct answer is (C). Apply the scalar multiplication to the vector and evaluate the answer choices:

4 * <

= <4

4

4

4

4 * <

*x*,3*x*>= <4

*x*,12*x*>4

*x** 3 = 12*x*, so (A) is true.4

*x*+ 8*x*= 12*x*, so (B) is true.4

*x** 8*x*= 32*x*² ≠ 12*x*, so (C) is false and is the correct answer.Question 18 |

−∞ | |

0 | |

½ | |

∞ |

Question 18 Explanation:

The correct answer is (A). Begin by factoring out the largest power of

Cancel out the

*x*in the denominator from both the numerator and denominator:Cancel out the

*x*and evaluate the limits:Question 19 |

If ƒ(

*x*) = 2*x*^{2}−*x*, what is ƒ'(3)?^{3}15 | |

−9 | |

−15 | |

18 |

Question 19 Explanation:

The correct answer is (C). Recall the power rule:

If ƒ(

Here, ƒ'(

Evaluating the derivative at ƒ'(3) = 4(3) − 3(3)

= 12 − 27

= −15

If ƒ(

*x*) =*x*^{n}, ƒ'(*x*) =*nx*^{n − 1}Here, ƒ'(

*x*) = 2 * 2*x*^{1}− 3*x*^{2}Evaluating the derivative at ƒ'(3) = 4(3) − 3(3)

^{2}= 12 − 27

= −15

Question 20 |

What is the probability of rolling a prime number in each roll in 4 consecutive rolls of a 6 sided die?

^{1}/_{4} | |

^{1}/_{8} | |

^{1}/_{16} | |

^{1}/_{32} |

Question 20 Explanation:

The correct answer is (C). The prime numbers in the range are: 2, 3, 5. On any roll, the probability of rolling a prime number is:

The probability of rolling this 4 times in a row is:

The probability of rolling this 4 times in a row is:

Question 21 |

*p*= −3

*x*+ 5

*y*

*q*=

*x*+ 2

*y*What is the value of

*p*− 2

*q*?

5 x + y | |

−5 x + y | |

− y + 5x | |

5 + y |

Question 21 Explanation:

The correct answer is (B). Substitute the values of p and q into the expression:

(−3

Distribute and combine like terms to simplify:

= −3

= −5

(−3

*x*+ 5*y*) − 2(*x*+ 2*y*)Distribute and combine like terms to simplify:

= −3

*x*+ 5*y*− 2*x*− 4*y*= −5

*x*+*y*Question 22 |

A lab requires 0.76 L of a solution. How many 100 mL vials must be used in the lab?

7 | |

8 | |

6 | |

10 |

Question 22 Explanation:

The correct answer is (B). Recall that 100 mL = 0.1 L. Because the smaller vials only come in amounts of 100 mL, 8 vials will be needed to provide 0.76 L.

Question 23 |

Two rats are chosen for an experiment out of a group of 17 black rats and 11 white rats. The two rats must be of a different color for each experiment. What is the probability that the second rat is white?

Question 23 Explanation:

The correct answer is (D). In order to satisfy the condition of choosing a white rat second, the first rat chosen must be black. The probability of selecting a black rat is:

Question 24 |

At what time will an object be at rest if its position function is given by:
x(t) = t

^{3}− 9t^{2}+ 24tt = 0 | |

t = 3 | |

t = 4 | |

t = 8 |

Question 24 Explanation:

The correct answer is (C). Recall that the derivative of the position function is the velocity function and that an object is at rest when its derivative is equal to 0. Evaluate the derivative, set it equal to 0 and solve for t:

v(t) = x'(t) = 3t

→ 0 = t

Factor the expression to solve for t:

(t − 4) (t − 2) = 0

→ t = 2, t = 4

Only t = 4 is an answer choice.

v(t) = x'(t) = 3t

^{2}− 18t + 24→ 0 = t

^{2}− 6t + 8Factor the expression to solve for t:

(t − 4) (t − 2) = 0

→ t = 2, t = 4

Only t = 4 is an answer choice.

Question 25 |

Evaluate:

Question 25 Explanation:

The correct answer is (A). Evaluate the integral using a u-substitution:

*u*=*sinx, du = cosx dx*Question 26 |

The mean of 17 exam scores is 26. After one exam is dropped, the mean drops to 25.625. What is the value of the exam that was dropped?

33 | |

32 | |

30 | |

35 |

Question 26 Explanation:

The correct answer is (B). Recall that the formula for average is:

Rewriting this formula:

Find the difference between the sums of the 2 sets:

17 * 26 = 442

16 * 25.625 = 410

→ 442 − 410 = 32

Rewriting this formula:

*sum = average * number of data*Find the difference between the sums of the 2 sets:

17 * 26 = 442

16 * 25.625 = 410

→ 442 − 410 = 32

Question 27 |

What is:

Question 27 Explanation:

The correct answer is (A). Begin by simplifying the expression in the numerator:

Dividing the numerator by 4:

Dividing the numerator by 4:

Question 28 |

Which set contains the solutions for 6

*x*^{2}= −5*x*− 1?Question 28 Explanation:

The correct answer is (B). Rearrange the equation and solve for

*x*by factoring:Question 29 |

What is the radius of a wheel that covers 100 m in 50 rotations?

Question 29 Explanation:

The correct answer is (B). Divide the total distance covered by the number of rotations to find the distance covered in 1 rotation:

This indicates that the circumference of the wheel is 2 m, so:

This indicates that the circumference of the wheel is 2 m, so:

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