25 Quantitative Reasoning, Algebra & Statistics Practice Questions & Answers

Every Quantitative Reasoning, Algebra & Statistics practice question from the ACCUPLACER Practice Test, with the correct answer and a short explanation.

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1. Evaluate without a calculator: -3^2 - |5 - 12|
- A.-2
- B.-16✓ Answer
- C.16
- D.2
In -3^2 the exponent applies only to 3, so -3^2 = -(9) = -9, not (-3)^2 = 9. Next, |5 - 12| = |-7| = 7, and the minus sign in front stays outside the absolute value. So -9 - 7 = -16. Getting 2 comes from treating -3^2 as (-3)^2, getting -2 comes from adding the 7 instead of subtracting it, and getting 16 comes from treating -3^2 as 9 and adding the 7.
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2. Evaluate without a calculator: (-2/3) - (5/6) ÷ (-1/2)
- A.1✓ Answer
- B.3
- C.-1/4
- D.-7/3
Division comes before subtraction. (5/6) ÷ (-1/2) = (5/6)(-2) = -5/3. Then -2/3 - (-5/3) = -2/3 + 5/3 = 3/3 = 1. Working left to right first gives 3, losing the negative sign of the quotient gives -7/3, and multiplying by -1/2 instead of dividing gives -1/4.
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3. A paint mix uses blue paint and yellow paint in a ratio of 2 to 5. A painter wants to make 28 quarts of the mix in all. How many quarts of blue paint are needed?
- A.11.2 quarts
- B.5.6 quarts
- C.8 quarts✓ Answer
- D.20 quarts
The ratio 2:5 is part-to-part, so the whole has 2 + 5 = 7 parts and blue is 2/7 of the mix: (2/7)(28) = 8 quarts. Using 2/5 as if it were a part-to-whole fraction gives 11.2, 20 quarts is the yellow paint, and 5.6 comes from dividing 28 by the 5 yellow parts.
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4. A cyclist rides at a steady 6 miles per hour. What is this speed in feet per second? (1 mile = 5,280 feet)
- A.8.8 feet per second✓ Answer
- B.0.11 feet per second
- C.528 feet per second
- D.88 feet per second
Use dimensional analysis so the units cancel: (6 miles/1 hour)(5,280 feet/1 mile)(1 hour/3,600 seconds) = 31,680/3,600 = 8.8 feet per second. Dividing by 60 only once gives 528, which is feet per minute. A decimal slip gives 88, and inverting the conversion gives about 0.11.
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5. The distance d, in centimeters, that a spring stretches varies directly with the weight w, in pounds, hanging from it. A 4-pound weight stretches the spring 18 centimeters. Which equation models this relationship?
- A.d = 4.5w✓ Answer
- B.d = 72w
- C.d = w + 14
- D.d = (2/9)w
Direct variation has the form d = kw, where k = d/w. Here k = 18/4 = 4.5, so d = 4.5w. The equation d = w + 14 adds instead of multiplying, so it is not proportional. The constant 2/9 is w/d, which is the ratio flipped, and 72 comes from multiplying 18 by 4.
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6. After a 15% discount, a pair of boots costs $68. What was the price before the discount?
- A.$80.00✓ Answer
- B.$83.00
- C.$57.80
- D.$78.20
A 15% discount leaves 85% of the original price, so 0.85p = 68 and p = 68/0.85 = $80.00. Adding 15% of $68 gives $78.20, but the discount was a percent of the original price, not of $68. Adding $15 treats the percent as dollars, and $57.80 takes the discount a second time.
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7. Simplify without a calculator, assuming x is not 0: (2x^3)^2 · x^(-4) ÷ (4x)
- A.x✓ Answer
- B.1
- C.x/2
- D.x^9
By the power rule, (2x^3)^2 = 2^2 · x^6 = 4x^6, because the outer exponent applies to the 2 as well as to x^3. By the product rule, 4x^6 · x^(-4) = 4x^2. By the quotient rule, 4x^2 ÷ 4x = x. Not squaring the 2 gives x/2, adding 3 + 2 instead of multiplying gives 1, and treating x^(-4) as x^4 gives x^9.
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8. A bag holds 4 × 10^9 grains of sand, and each grain has a mass of 8 × 10^(-3) grams. What is the total mass of the sand, in scientific notation?
- A.3.2 × 10^12 g
- B.3.2 × 10^7 g✓ Answer
- C.1.2 × 10^7 g
- D.3.2 × 10^6 g
Multiply the coefficients and add the exponents: (4)(8) = 32 and 10^9 · 10^(-3) = 10^6, giving 32 × 10^6. Scientific notation needs a coefficient from 1 up to 10, so 32 × 10^6 = 3.2 × 10^7. Moving the decimal point without raising the exponent gives 3.2 × 10^6, subtracting the exponents gives 10^12, and adding 4 + 8 gives 1.2 × 10^7.
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9. Which expression represents "3 less than the quotient of a number n and 5"?
- A.5/n - 3
- B.(n - 3)/5
- C.n/5 - 3✓ Answer
- D.3 - n/5
"The quotient of n and 5" is n/5, with n first. "3 less than" a quantity means you subtract 3 from that quantity, so the expression is n/5 - 3. Writing 3 - n/5 reverses the order of "less than." Writing (n - 3)/5 subtracts before dividing, and 5/n flips the quotient.
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10. Which expression is equivalent to -2(3x - 5) + 4(x - 1)?
- A.-2x - 14
- B.-2x + 9
- C.10x + 6
- D.-2x + 6✓ Answer
Use the distributive property on each group: -2(3x - 5) = -6x + 10, because a negative times a negative is positive, and 4(x - 1) = 4x - 4. Then combine like terms: (-6x + 4x) + (10 - 4) = -2x + 6. Getting -2x - 14 means -2 times -5 was written as -10. Getting -2x + 9 means the 4 was not multiplied by the -1. Getting 10x + 6 means -2 times 3x was written as +6x.
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11. A phone plan charges $30 per month, which includes 200 text messages. Each text after the first 200 costs $0.10. Which expression gives the monthly charge, in dollars, for t texts when t is greater than 200?
- A.0.10(30 + t - 200)
- B.30 + 0.10(200 - t)
- C.30 + 0.10t - 200
- D.30 + 0.10(t - 200)✓ Answer
Only the texts beyond 200 cost extra, and there are t - 200 of them. Each costs $0.10, so the extra charge is 0.10(t - 200), added to the fixed $30. Leaving out the parentheses subtracts 200 dollars instead of 200 texts. Putting 0.10 in front of everything multiplies the $30 fee by 0.10, and 200 - t is negative when t is more than 200.
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12. Solve for x: (x + 1)/2 + x/3 = 8
- A.1
- B.19.5
- C.9.4
- D.9✓ Answer
Clear the fractions by multiplying every term on both sides by the least common denominator, 6: 3(x + 1) + 2x = 48. Then 3x + 3 + 2x = 48, so 5x = 45 and x = 9. Check: 10/2 + 9/3 = 5 + 3 = 8. Not multiplying the 8 by 6 gives 1, not distributing the 3 to the 1 gives 9.4, and adding the denominators to get 5 gives 19.5.
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13. Solve the inequality: 7 - 2(x + 4) ≥ 9
- A.x ≤ -5✓ Answer
- B.x ≥ 5
- C.x ≥ -5
- D.x ≤ 1
Distribute: 7 - 2x - 8 ≥ 9, so -2x - 1 ≥ 9 and -2x ≥ 10. Dividing both sides by a negative number reverses the inequality sign, so x ≤ -5. Keeping the sign as it was gives x ≥ -5. Writing -2(x + 4) as -2x + 4 gives x ≤ 1, and dividing by 2 instead of -2 gives x ≥ 5.
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14. The simple-interest formula A = P + Prt gives the total amount A after principal P earns interest at rate r for t years. Which equation correctly solves this formula for r?
- A.r = (A - P)/(Pt)✓ Answer
- B.r = (A - Pt)/P
- C.r = (A - P)/P - t
- D.r = (A - P)·Pt
Undo the operations around r in reverse order. Subtract P from both sides to get A - P = Prt. Then r is multiplied by Pt, so divide both sides by Pt: r = (A - P)/(Pt). Subtracting t undoes a multiplication by t the wrong way, and multiplying by Pt does the opposite of dividing.
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15. A plumber's charge C, in dollars, is a linear function of the number of hours h worked. A 2-hour job costs $155 and a 4-hour job costs $245. Which equation models the charge?
- A.C = 45h + 155
- B.C = 90h - 25
- C.C = 65h + 45
- D.C = 45h + 65✓ Answer
The slope is the rate per hour: (245 - 155)/(4 - 2) = 90/2 = 45 dollars per hour. Substitute one point to find the intercept: 155 = 45(2) + b, so b = 65, the fixed charge. Check: 45(4) + 65 = 245. Using 90 as the slope without dividing by the 2-hour change fits only one point. Swapping the slope and intercept gives 65h + 45, and using $155 as the starting value ignores that it already includes 2 hours of work.
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16. A candle burns down at a constant rate. Two hours after it is lit, it is 24 centimeters tall. Five hours after it is lit, it is 15 centimeters tall. How tall was the candle when it was lit?
- A.33 cm
- B.27 cm
- C.21 cm
- D.30 cm✓ Answer
The slope is (15 - 24)/(5 - 2) = -9/3 = -3 cm per hour, so the candle loses 3 cm each hour. To get from hour 2 back to hour 0, add 2 hours of burning: 24 + 2(3) = 30 cm. Going back only 1 hour gives 27, using the 9 cm drop as the hourly rate gives 33, and subtracting instead of adding gives 21.
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17. A graph of taxi fare (in dollars) versus distance (in miles) is a straight line that passes through the points (0, 3.50) and (10, 21.50). What does the taxi charge for each mile driven?
- A.$0.56
- B.$2.15
- C.$1.80✓ Answer
- D.$2.50
The charge per mile is the slope, rise over run: (21.50 - 3.50)/(10 - 0) = 18/10 = $1.80 per mile. The $3.50 at 0 miles is the starting fee, so it must be taken out first. Dividing 21.50 by 10 gives $2.15 because it leaves the starting fee in. Adding the fee instead of subtracting it gives $2.50, and run over rise gives about $0.56.
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18. In a club of 40 students, 18 play soccer, 15 play basketball, and 7 play both sports. If one student is chosen at random, what is the probability that the student plays neither sport?
- A.13/20
- B.7/20✓ Answer
- C.7/40
- D.33/40
The students who play at least one sport form the union: n(A or B) = 18 + 15 - 7 = 26. You subtract the 7 because they were counted once in each group. The rest, 40 - 26 = 14, play neither sport, so P = 14/40 = 7/20. Forgetting to subtract the overlap gives 7/40, 13/20 is the probability of playing at least one sport, and 33/40 is the probability of not playing both.
Ref QB-TVD55MSource: College Board Next-Generation ACCUPLACER Test Specifications — Quantitative Reasoning, Algebra, and Statistics: Probability and sets (union, complement, P(A or B) = P(A) + P(B) - P(A and B))Report a problem with this question
19. A company has 120 employees. Of the 50 who work remotely, 30 have children. Of the 70 who work in the office, 28 have children. An employee who has children is chosen at random. What is the probability that this employee works remotely?
- A.1/4
- B.3/5
- C.15/29✓ Answer
- D.29/60
The condition "has children" limits the sample space to the 30 + 28 = 58 employees with children. Of these, 30 work remotely, so P(remote | children) = 30/58 = 15/29. The value 3/5 = 30/50 answers the reverse question, P(children | remote). The value 1/4 = 30/120 is the joint probability, and 29/60 = 58/120 is P(children).
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20. A data set contains the values 14, 16, 17, 19, 20, and 22. The value 85 is then added to the set. Which statement describes how the mean and median change?
- A.The mean rises but the median stays the same
- B.The mean rises by more than the median does✓ Answer
- C.The median rises by more than the mean does
- D.The mean and median rise by the same amount
Before the change, the mean is 108/6 = 18 and the median is (17 + 19)/2 = 18. After 85 is added, the mean is 193/7, about 27.6, which is a rise of about 9.6. The median becomes the 4th of 7 values, 19, which is a rise of only 1. The mean uses the size of every value, so an outlier pulls it strongly. The median depends only on the middle position, so it barely moves, but it does change.
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21. A box plot of delivery times, in minutes, has a minimum of 10, a first quartile of 14, a median of 16, a third quartile of 18, and a maximum of 40. Which statement best describes the data?
- A.Skewed left; the mean is likely below 16
- B.Skewed right; the mean is likely above 16✓ Answer
- C.Symmetric; the mean is likely equal to 16
- D.Skewed right; the mean is likely below 16
The box is balanced (14 to 16 and 16 to 18 are both 2), but the right whisker runs from 18 to 40, which is 22 minutes, while the left whisker runs from 10 to 14, which is only 4. A long tail toward high values means the data are skewed right. The few very large values pull the mean toward the tail, so the mean is likely greater than the median of 16. A balanced box does not make the data symmetric when the whiskers are this uneven.
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22. A survey asked 20 households how many pets they own. The results were: 0 pets, 5 households; 1 pet, 8 households; 2 pets, 4 households; 3 pets, 3 households. What is the median number of pets per household?
- A.4.5 pets
- B.1.25 pets
- C.1 pet✓ Answer
- D.1.5 pets
With 20 values, the median is the average of the 10th and 11th values in order. By cumulative count, positions 1 to 5 are 0 pets and positions 6 to 13 are 1 pet, so both the 10th and 11th values are 1 and the median is 1. The value 1.5 is the middle of the categories 0 through 3 and ignores the frequencies. The value 4.5 is the median of the frequencies themselves, and 1.25 = 25/20 is the mean.
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23. A garden bed is shaped like a trapezoid. Its parallel sides are 9 feet and 15 feet long, the distance between them is 4 feet, and each slanted side is 5 feet long. What is the area of the garden bed?
- A.34 square feet
- B.48 square feet✓ Answer
- C.96 square feet
- D.60 square feet
The area of a trapezoid is A = (1/2)h(b1 + b2), where h is the perpendicular distance between the parallel sides: (1/2)(4)(9 + 15) = (1/2)(4)(24) = 48 square feet. The 5-foot slanted sides are not the height, so they are not used. Leaving out the 1/2 gives 96, using the longer base for both bases, (1/2)(4)(15 + 15), gives 60, and 34 feet is the perimeter, which measures length, not area.
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24. A rectangle has a length of (3x - 2) units and a width of (x + 5) units. Which expression represents the perimeter of the rectangle?
- A.8x + 14
- B.7x + 1
- C.8x + 6✓ Answer
- D.4x + 3
The perimeter of a rectangle is P = 2(length + width) = 2[(3x - 2) + (x + 5)] = 2(4x + 3) = 8x + 6. Adding just one length and one width gives 4x + 3. Doubling only the length gives 7x + 1, and treating -2 as +2 gives 8x + 14.
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25. A wooden doorstop is a prism 12 inches long whose two ends are right triangles. In each right triangle, one leg is 6 inches and the hypotenuse is 10 inches. What is the volume of the doorstop?
- A.360 cubic inches
- B.576 cubic inches
- C.720 cubic inches
- D.288 cubic inches✓ Answer
First find the missing leg with the Pythagorean theorem: 6^2 + b^2 = 10^2, so b^2 = 64 and b = 8 inches. The triangular base has area B = (1/2)(6)(8) = 24 square inches. For any prism, V = Bh = 24 × 12 = 288 cubic inches. Using the hypotenuse as a leg gives 360, leaving out the 1/2 gives 576, and multiplying 6 × 10 × 12 gives 720.
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1. Evaluate without a calculator: -3^2 - |5 - 12|

2. Evaluate without a calculator: (-2/3) - (5/6) ÷ (-1/2)

3. A paint mix uses blue paint and yellow paint in a ratio of 2 to 5. A painter wants to make 28 quarts of the mix in all. How many quarts of blue paint are needed?

4. A cyclist rides at a steady 6 miles per hour. What is this speed in feet per second? (1 mile = 5,280 feet)

5. The distance d, in centimeters, that a spring stretches varies directly with the weight w, in pounds, hanging from it. A 4-pound weight stretches the spring 18 centimeters. Which equation models this relationship?

6. After a 15% discount, a pair of boots costs $68. What was the price before the discount?

7. Simplify without a calculator, assuming x is not 0: (2x^3)^2 · x^(-4) ÷ (4x)

8. A bag holds 4 × 10^9 grains of sand, and each grain has a mass of 8 × 10^(-3) grams. What is the total mass of the sand, in scientific notation?

9. Which expression represents "3 less than the quotient of a number n and 5"?

10. Which expression is equivalent to -2(3x - 5) + 4(x - 1)?

11. A phone plan charges $30 per month, which includes 200 text messages. Each text after the first 200 costs $0.10. Which expression gives the monthly charge, in dollars, for t texts when t is greater than 200?

12. Solve for x: (x + 1)/2 + x/3 = 8

13. Solve the inequality: 7 - 2(x + 4) ≥ 9

14. The simple-interest formula A = P + Prt gives the total amount A after principal P earns interest at rate r for t years. Which equation correctly solves this formula for r?

15. A plumber's charge C, in dollars, is a linear function of the number of hours h worked. A 2-hour job costs $155 and a 4-hour job costs $245. Which equation models the charge?

16. A candle burns down at a constant rate. Two hours after it is lit, it is 24 centimeters tall. Five hours after it is lit, it is 15 centimeters tall. How tall was the candle when it was lit?

17. A graph of taxi fare (in dollars) versus distance (in miles) is a straight line that passes through the points (0, 3.50) and (10, 21.50). What does the taxi charge for each mile driven?

18. In a club of 40 students, 18 play soccer, 15 play basketball, and 7 play both sports. If one student is chosen at random, what is the probability that the student plays neither sport?

19. A company has 120 employees. Of the 50 who work remotely, 30 have children. Of the 70 who work in the office, 28 have children. An employee who has children is chosen at random. What is the probability that this employee works remotely?

20. A data set contains the values 14, 16, 17, 19, 20, and 22. The value 85 is then added to the set. Which statement describes how the mean and median change?

21. A box plot of delivery times, in minutes, has a minimum of 10, a first quartile of 14, a median of 16, a third quartile of 18, and a maximum of 40. Which statement best describes the data?

22. A survey asked 20 households how many pets they own. The results were: 0 pets, 5 households; 1 pet, 8 households; 2 pets, 4 households; 3 pets, 3 households. What is the median number of pets per household?

23. A garden bed is shaped like a trapezoid. Its parallel sides are 9 feet and 15 feet long, the distance between them is 4 feet, and each slanted side is 5 feet long. What is the area of the garden bed?

24. A rectangle has a length of (3x - 2) units and a width of (x + 5) units. Which expression represents the perimeter of the rectangle?

25. A wooden doorstop is a prism 12 inches long whose two ends are right triangles. In each right triangle, one leg is 6 inches and the hypotenuse is 10 inches. What is the volume of the doorstop?

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