ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz: Evaluating Algebraic Expressions

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Evaluating Algebraic ExpressionsQuestion 1 of 20

What is the value of $|2m - n| - |n - 3m|$ when $m = -3$ and $n = 5$ ?

-25

-3

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ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz

ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz: Evaluating Algebraic Expressions

Practice Evaluating Algebraic Expressions in ACCUPLACER Quantitative Reasoning, Algebra & Statistics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Evaluating Algebraic Expressions, giving you a quick way to practice the rules, question types, and explanations that matter most for ACCUPLACER Quantitative Reasoning, Algebra & Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

What is the value of $|2m - n| - |n - 3m|$ when $m = -3$ and $n = 5$ ?

-25
-3 (correct answer)
7
25

Explanation: First, substitute

m = -3

and

n = 5

into the expression. For the first absolute value:

|2(-3) - 5| = |-6 - 5| = |-11| = 11

. For the second absolute value:

|5 - 3(-3)| = |5 - (-9)| = |5 + 9| = |14| = 14

. Finally, subtract the results:

11 - 14 = -3

A is incorrect because it results from ignoring the absolute value bars: $(2(-3) - 5) - (5 - 3(-3)) = (-11) - (14) = -25$ .
C is incorrect due to a sign error when evaluating the second absolute value term: $|5 - 9| = |-4| = 4$ , leading to $11 - 4 = 7$ .
D is incorrect because it results from adding the absolute values instead of subtracting: $11 + 14 = 25$ .

Question 2

What is the value of the expression $\frac{1}{2}x - \frac{3}{4}y$ for $x = -6$ and $y = 8$ ?

-9 (correct answer)
-3
3
9

Explanation: Substitute the values into the expression:

\frac{1}{2}(-6) - \frac{3}{4}(8)

. Calculate each term:

\frac{1}{2}(-6) = -3

\frac{3}{4}(8) = \frac{24}{4} = 6

. Now combine the terms:

-3 - 6 = -9

B is incorrect from calculating only the first term $\frac{1}{2}(-6) = -3$ and ignoring or miscalculating the second term.
C is incorrect due to a sign error. If the expression is calculated as $-3 - (-6) = -3 + 6 = 3$ , treating the subtraction incorrectly.
D is incorrect due to sign errors on both terms. If $\frac{1}{2}(-6)$ is calculated as $3$ and $-\frac{3}{4}(8)$ is calculated as $-6$ , then $3 - (-6) = 9$ .

Question 3

The volume of a sphere is given by the formula $V = \frac{4}{3}\pi r^3$ .

Using the formula $V = \frac{4}{3}\pi r^3$ , what is the volume of a sphere with a radius $r = 3$ , in terms of $\pi$ ?

$12\pi$
$27\pi$
$36\pi$ (correct answer)
$108\pi$

Explanation: Substitute

r=3

into the volume formula:

V = \frac{4}{3}\pi (3)^3

. First, calculate the exponent:

3^3 = 27

. The expression is now

V = \frac{4}{3}\pi (27)

. Multiply

\frac{4}{3}

by 27:

\frac{4 \times 27}{3} = 4 \times 9 = 36

. Therefore, the volume is

36\pi

A is incorrect because it results from squaring the radius instead of cubing it: $\frac{4}{3}\pi (3)^2 = \frac{4}{3}\pi (9) = 12\pi$ .
B is incorrect because it results from omitting the $\frac{4}{3}$ coefficient from the formula, calculating only $\pi r^3$ .
D is incorrect because it results from multiplying 4 by 27 before dividing by 3, but making an arithmetic mistake, or perhaps multiplying $4\pi$ by $r^3$ after multiplying the fraction's denominator into $r^3$ .

Question 4

If $p = -8$ and $q = 15$ , what is the value of $\sqrt{p^2 + q^2}$ ?

7
17 (correct answer)
23
161

Explanation: Substitute the values for

p

and

q

\sqrt{(-8)^2 + 15^2}

. Square each term inside the square root:

(-8)^2 = 64

and

15^2 = 225

. Add the results:

\sqrt{64 + 225} = \sqrt{289}

. The square root of 289 is 17. (This is a Pythagorean triple: 8-15-17).

A is incorrect because it results from the common mistake of calculating $p+q$ instead of $\sqrt{p^2+q^2}$ : $-8 + 15 = 7$ .
C is incorrect because it comes from the misconception that $\sqrt{p^2 + q^2}$ is equal to $|p| + |q|$ : $\sqrt{(-8)^2} + \sqrt{15^2} = 8 + 15 = 23$ .
D is incorrect and represents an intermediate calculation error, perhaps by subtracting instead of adding inside the radical ( $\sqrt{225-64}=\sqrt{161}$ ) or a simple arithmetic mistake.

Question 5

A savings account earns simple interest using the formula $I = Prt$ , where $I$ is the interest, $P$ is the principal amount, $r$ is the annual interest rate, and $t$ is the time in years.

How much interest is earned on a principal of $1,500 at an annual rate of 2.5% over 4 years?

$15
$150 (correct answer)
$1,500
$15,000

Explanation: The formula is

I = Prt

. The principal

P

is $1,500. The time

t

is 4 years. The annual rate

r

is 2.5%, which must be converted to a decimal by dividing by 100:

r = 0.025

. Now, substitute these values into the formula:

I = (1500)(0.025)(4)

. Multiplying

0.025 \times 4 = 0.1

. Then,

I = 1500 \times 0.1 = 150

. The interest earned is $150.

A is incorrect due to a decimal placement error, likely converting 2.5% to 0.0025.
C is incorrect due to a decimal placement error, converting 2.5% to 0.25 instead of 0.025.
D is incorrect from the common error of using the percentage value directly without converting to a decimal: $1500 \times 2.5 \times 4 = 15,000$ .

Question 6

A mathematical operation $\diamond$ is defined by $a \diamond b = (a-b)^2 - b$ .

What is the value of $2 \diamond (-5)$ ?

4
14
44
54 (correct answer)

Explanation: The definition of the operation is

a \diamond b = (a-b)^2 - b

. Substitute

a=2

and

b=-5

. Inside the parentheses:

(2 - (-5)) = (2 + 5) = 7

. Square the result:

7^2 = 49

. Subtract

b

49 - (-5) = 49 + 5 = 54

A is incorrect because of sign errors in both parts of the calculation: $(2-5)^2 - 5 = (-3)^2 - 5 = 9 - 5 = 4$ .
B is incorrect from a sign error inside the parentheses: $(2-5)^2 - (-5) = (-3)^2 + 5 = 9 + 5 = 14$ .
C is incorrect from a sign error on the final subtraction: $(2-(-5))^2 - 5 = 7^2 - 5 = 49 - 5 = 44$ .

Question 7

The cost $C$ , in dollars, to rent a car for a day is given by the expression $25 + 0.15m$ , where $m$ is the number of miles driven.

What is the cost if a person drives 120 miles?

$26.80
$43.00 (correct answer)
$123.75
$205.00

Explanation: Substitute

m = 120

into the cost expression:

C = 25 + 0.15(120)

. Following the order of operations, perform the multiplication first:

0.15 \times 120 = 18

. Then, perform the addition:

C = 25 + 18 = 43

. The cost is $43.00.

A is incorrect because of a decimal placement error in the multiplication, where $0.15 \times 120$ is incorrectly calculated as $1.8$ , leading to a cost of $25 + 1.80 = 26.80$ .
C is incorrect and results from confusing the roles of the numbers, for example calculating $120 + 0.15(25) = 120 + 3.75 = 123.75$ .
D is incorrect because of a decimal placement error in the multiplication, where $0.15 \times 120$ is incorrectly calculated as $180$ , leading to a cost of $25 + 180 = 205$ .

Question 8

The height $h$ in meters of a projectile after $t$ seconds is modeled by the expression $h = -5t^2 + 40t + 2$ .

What is the height of the projectile after 3 seconds?

75
77 (correct answer)
167
347

Explanation: To find the height after 3 seconds, substitute

t=3

into the expression:

h = -5(3)^2 + 40(3) + 2

. Following the order of operations, calculate the exponent first:

3^2 = 9

. The expression becomes

h = -5(9) + 40(3) + 2

. Next, perform the multiplications:

-5(9) = -45

and

40(3) = 120

. The expression is now

h = -45 + 120 + 2

. Finally, add the terms:

-45 + 120 = 75

, and

75 + 2 = 77

. The height is 77 meters.

A is incorrect because it omits the final $+2$ term in the calculation.
C is incorrect due to a sign error on the first term. If $-5(9)$ is incorrectly calculated as $45$ , the result is $45 + 120 + 2 = 167$ .
D is incorrect from an order of operations error, multiplying $-5 \times 3$ before squaring: $(-15)^2 + 120 + 2 = 225 + 122 = 347$ .

Question 9

The number of ways to arrange $n$ distinct items in a sequence is given by the expression $n(n-1)(n-2)...(1)$ . A company's setup cost is $10 times this number.

What is the setup cost for arranging $n=4$ items?

$40
$100
$240 (correct answer)
$1000

Explanation: First, evaluate the expression for the number of arrangements when

n=4

. The expression is

4(4-1)(4-2)(4-3)

, which is

4 \cdot 3 \cdot 2 \cdot 1 = 24

. The setup cost is $10 times this number. So, the cost is

10 \times 24 = 240

. The total setup cost is $240.

A is incorrect because it results from only calculating $10 \times n$ , or $10 \times 4 = 40$ , ignoring the rest of the arrangement expression.
B is incorrect and may result from an order of operations error, such as calculating $10 \times 4$ first and then multiplying by other terms, or some other miscalculation.
D is incorrect and may come from misinterpreting the expression as $10^n$ , resulting in $10^4 = 10000$ , or a similar conceptual error.

Question 10

What is the value of the expression $\frac{2k^2 - 1}{3k - 5}$ when $k = -2$ ?

-7
$-\frac{7}{11}$ (correct answer)
$\frac{9}{11}$
9

Explanation: Substitute

k=-2

into the numerator and the denominator. Numerator:

2(-2)^2 - 1 = 2(4) - 1 = 8 - 1 = 7

. Denominator:

3(-2) - 5 = -6 - 5 = -11

. The value of the expression is

\frac{7}{-11}

, which is

-\frac{7}{11}

A is incorrect because of a sign error in the denominator, calculating $3(-2) + 5 = -1$ , leading to $\frac{7}{-1} = -7$ .
C is incorrect because of an exponent error in the numerator, calculating $(-2)^2 = -4$ . This makes the numerator $2(-4) - 1 = -9$ , leading to $\frac{-9}{-11} = \frac{9}{11}$ .
D is incorrect because it results from making both the exponent error in the numerator (getting -9) and the sign error in the denominator (getting -1), leading to $\frac{-9}{-1} = 9$ .

Question 11

When $a = -1$ , $b = 4$ , and $c = -3$ , the expression $2a^2b - 3bc + a^3$ equals:

$45$
$43$ (correct answer)
$41$
$37$

Explanation: Substituting the given values:

2a^2b - 3bc + a^3 = 2(-1)^2(4) - 3(4)(-3) + (-1)^3 = 2(1)(4) - 3(4)(-3) + (-1) = 8 - (-36) + (-1) = 8 + 36 - 1 = 43

. Choice A (45) represents forgetting the

a^3 = -1

term entirely. Choice C (41) represents calculating

8 + 36 - 3 = 41

, making an error with the final term. Choice D (37) represents incorrectly calculating

(-1)^2 = -1

instead of

1

Question 12

What is the value of the expression $a^2 - 3ab$ when $a = -2$ and $b = 4$ ?

-28
-20
20
28 (correct answer)

Explanation: To evaluate the expression, substitute the given values for

a

and

b

(-2)^2 - 3(-2)(4)

. Following the order of operations, calculate the exponent first:

(-2)^2 = 4

. Then, perform the multiplication:

-3(-2)(4) = 6(4) = 24

. Finally, combine the terms:

4 - (-24) = 4 + 24 = 28

A is incorrect because of two sign errors. First, treating $(-2)^2$ as -4, and second, calculating $-3(-2)(4)$ as $-24$ , leading to $-4 - 24 = -28$ .
B is incorrect due to a sign error in multiplication. If $-3(-2)(4)$ is incorrectly calculated as $-24$ , the expression becomes $4 - 24 = -20$ .
C is incorrect due to an error in evaluating the exponent. If $(-2)^2$ is incorrectly calculated as $-4$ , the expression becomes $-4 - (-24) = -4 + 24 = 20$ .

Question 13

Evaluate the expression $\frac{x}{y} + \frac{1}{2}z$ for $x = 3$ , $y = 4$ , and $z = -5$ .

$-\frac{7}{4}$ (correct answer)
$-\frac{1}{2}$
$1$
$\frac{13}{4}$

Explanation: Substitute the values into the expression:

\frac{3}{4} + \frac{1}{2}(-5)

. This simplifies to

\frac{3}{4} - \frac{5}{2}

. To subtract the fractions, find a common denominator, which is 4. Convert

\frac{5}{2}

\frac{10}{4}

. The expression becomes

\frac{3}{4} - \frac{10}{4} = \frac{3-10}{4} = -\frac{7}{4}

B is incorrect because it results from incorrectly adding the numerators and denominators of the two terms before multiplication, as in $\frac{3+(-5)}{4+2} = \frac{-2}{6}$ , or a similar conceptual error.
C is incorrect and likely results from a combination of sign and fraction operation errors.
D is incorrect because it results from incorrectly adding $\frac{5}{2}$ instead of subtracting, due to a sign error: $\frac{3}{4} + \frac{10}{4} = \frac{13}{4}$ .

Question 14

What is the value of the expression $x - [y - (z - x)]$ for $x = -3$ , $y = 4$ , and $z = -1$ ?

-11
-9
-5 (correct answer)
-3

Explanation: Substitute the values and work from the innermost parentheses outward. The expression becomes

-3 - [4 - (-1 - (-3))]

. Innermost parentheses:

(-1 - (-3)) = -1 + 3 = 2

. The expression is now:

-3 - [4 - 2]

. Brackets:

[4 - 2] = 2

. The expression is now:

-3 - 2

. Final calculation:

-3 - 2 = -5

A is incorrect from a sign error in the innermost parentheses: $-1 - 3 = -4$ . Then $-3 - [4 - (-4)] = -3 - [8] = -11$ .
B is incorrect from an error in distributing the negative sign to the brackets: $-3 - 4 - 2 = -9$ .
D is incorrect from ignoring the parentheses and brackets: $x - y - z - x = -y - z = -4 - (-1) = -3$ .

Question 15

If $c = 5$ and $d = -3$ , what is the value of the expression $\frac{c^2 + d^2}{c + d}$ ?

2
8
16
17 (correct answer)

Explanation: Substitute the given values into the expression. Numerator:

c^2 + d^2 = (5)^2 + (-3)^2 = 25 + 9 = 34

. Denominator:

c + d = 5 + (-3) = 2

. Finally, divide the numerator by the denominator:

\frac{34}{2} = 17

A is incorrect and may result from incorrectly simplifying the expression to $c+d$ and then evaluating: $5 + (-3) = 2$ .
B is incorrect because of a common error in squaring a negative number: if $(-3)^2$ is calculated as $-9$ , the numerator becomes $25 - 9 = 16$ , and the result is $\frac{16}{2} = 8$ .
C is incorrect and likely results from an arithmetic error in the numerator, such as $25+9=32$ instead of 34, leading to $32/2 = 16$ .

Question 16

If $c = -4$ and $d = -2$ , what is the value of the expression $(c - 2d)^2$ ?

0 (correct answer)
4
16
64

Explanation: Substitute the values of

c

and

d

into the expression:

(-4 - 2(-2))^2

. First, calculate the product inside the parentheses:

2(-2) = -4

. The expression becomes

(-4 - (-4))^2

, which is

(-4 + 4)^2

. This simplifies to

(0)^2

, which is 0.

B is incorrect from a calculation error where the final squaring step is omitted or miscalculated.
C is incorrect from confusing the expression with $c^2$ , since $(-4)^2 = 16$ , ignoring the rest of the expression.
D is incorrect due to a sign error inside the parentheses. If $2(-2)$ is computed as $+4$ , the expression becomes $(-4 - 4)^2 = (-8)^2 = 64$ .

Question 17

What is the value of $5x^{-2} + 2y^0$ when $x = -1$ and $y = 10$ ?

-5
-3
5
7 (correct answer)

Explanation: This expression requires knowing the rules for negative and zero exponents. A negative exponent indicates a reciprocal:

x^{-2} = \frac{1}{x^2}

. Any non-zero number raised to the power of 0 is 1:

y^0 = 1

. Substitute the values:

5(-1)^{-2} + 2(10)^0 = 5(\frac{1}{(-1)^2}) + 2(1)

. Calculate the components:

(-1)^2 = 1

. So the expression becomes

5(\frac{1}{1}) + 2 = 5 + 2 = 7

A is incorrect because of errors on both terms: calculating $x^{-2}$ as $-x^2 = -1$ and $y^0$ as 0, leading to $5(-1) + 2(0) = -5$ .
B is incorrect from misinterpreting the negative exponent rule as $x^{-2} = -x^2 = -1$ , leading to $5(-1) + 2(1) = -3$ .
C is incorrect from misinterpreting the zero exponent rule as $y^0 = 0$ , leading to $5(1) + 2(0) = 5$ .

Question 18

What is the value of the expression $4 \cdot 3^k - k^2$ for $k=3$ ?

30
72
99 (correct answer)
1719

Explanation: Substitute

k=3

into the expression:

4 \cdot 3^3 - 3^2

. According to the order of operations (PEMDAS), evaluate exponents first:

3^3 = 27

and

3^2 = 9

. The expression becomes

4 \cdot 27 - 9

. Next, perform multiplication:

4 \cdot 27 = 108

. Finally, perform subtraction:

108 - 9 = 99

A is incorrect from misinterpreting exponents as multiplication: $4 \cdot (3 \cdot 3) - (2 \cdot 3) = 36 - 6 = 30$ .
B is incorrect from an order of operations error, subtracting before multiplying: $4 \cdot (27 - 9) = 4 \cdot 18 = 72$ .
D is incorrect from a significant order of operations error, multiplying $4 \cdot 3$ before applying the exponent: $(4 \cdot 3)^3 - 3^2 = 12^3 - 9 = 1728 - 9 = 1719$ .

Question 19

If $m = 2$ and $n = -5$ , what is the value of $(m+n)^2 - (m-n)^2$ ?

-58
-40 (correct answer)
0
50

Explanation: First, evaluate the expressions inside the parentheses.

m+n = 2 + (-5) = -3

m-n = 2 - (-5) = 2 + 5 = 7

. Now substitute these results back into the expression:

(-3)^2 - (7)^2

. Calculate the squares:

(-3)^2 = 9

and

7^2 = 49

. Finally, subtract:

9 - 49 = -40

A is incorrect because of an error squaring the negative number: if $(-3)^2$ is incorrectly calculated as $-9$ , the result is $-9 - 49 = -58$ .
C is incorrect from a sign error when calculating $m-n$ . If calculated as $2-5=-3$ , the expression becomes $(-3)^2 - (-3)^2 = 9-9=0$ .
D is incorrect and likely results from a misremembered algebraic identity or another calculation error.

Question 20

Evaluate the expression $8x^3 - 4x^2 + 2x$ for $x = -\frac{1}{2}$ .

-3 (correct answer)
-1
0
1

Explanation: Substitute

x = -\frac{1}{2}

into the expression:

8(-\frac{1}{2})^3 - 4(-\frac{1}{2})^2 + 2(-\frac{1}{2})

. Evaluate each term:

8(-\frac{1}{8})

since

(-\frac{1}{2})^3 = -\frac{1}{8}

. This term equals -1.

-4(\frac{1}{4})

since

(-\frac{1}{2})^2 = \frac{1}{4}

. This term equals -1.

2(-\frac{1}{2})

equals -1. Combine the terms:

-1 - 1 - 1 = -3

B is incorrect due to a sign error. This value is obtained if $(-\frac{1}{2})^3$ is incorrectly calculated as $\frac{1}{8}$ OR if $(-\frac{1}{2})^2$ is incorrectly calculated as $-\frac{1}{4}$ . Both errors lead to $-1$ .
C is incorrect and may result from an error in the last term, such as forgetting to substitute for $x$ and leaving it as +2, resulting in $-1 - 1 + 2 = 0$ .
D is incorrect from making sign errors on both the cubic and squared terms: $8(\frac{1}{8}) - 4(-\frac{1}{4}) - 1 = 1 + 1 - 1 = 1$ .