ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz: Translating To Algebraic Expressions
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Translating To Algebraic ExpressionsQuestion 1 of 20

Which expression represents one-half of the number that is 10 more than the square of xx?

x22+10\frac{x^2}{2} + 10
x2+102\frac{x^2 + 10}{2}
(x+10)22\frac{(x+10)^2}{2}
x2−102\frac{x^2 - 10}{2}
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ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz

ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz: Translating To Algebraic Expressions

Practice Translating To 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 Translating To Algebraic Expressions, giving you a quick way to practice the rules, question types, and explanations that matter most for ACCUPLACER Quantitative Reasoning, Algebra & Statistics.

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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.

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Question 1

Which expression represents one-half of the number that is 10 more than the square of xx?

  1. x22+10\frac{x^2}{2} + 10
  2. x2+102\frac{x^2 + 10}{2} (correct answer)
  3. (x+10)22\frac{(x+10)^2}{2}
  4. x2−102\frac{x^2 - 10}{2}
Explanation: Working from the inside out, 'the square of xx' is x2x^2. '10 more than' that is x2+10x^2 + 10. 'One-half of' this entire quantity means dividing the whole expression by 2, which gives x2+102\frac{x^2 + 10}{2}. Answer A, x22+10\frac{x^2}{2} + 10, represents '10 more than one-half of the square of xx', an order-of-operations error. Answer C represents 'one-half of the square of the sum of xx and 10'. Answer D incorrectly uses subtraction for 'more than'.

Question 2

A number, zz, is equal to twice the difference between another number, yy, and 7. Which equation represents this relationship?

  1. z=2y−7z = 2y - 7
  2. z=2(7−y)z = 2(7 - y)
  3. z=2(y−7)z = 2(y - 7) (correct answer)
  4. z=y−2(7)z = y - 2(7)
Explanation: 'The difference between yy and 7' is y−7y - 7. 'Twice' this difference means multiplying the entire expression by 2, which requires parentheses: 2(y−7)2(y - 7). Setting zz equal to this gives z=2(y−7)z = 2(y - 7). Answer A, z=2y−7z = 2y - 7, represents 'twice yy, minus 7', which is a common grouping error. Answer B represents 'twice the difference between 7 and yy'. Answer D misinterprets what is being multiplied by 2.

Question 3

Which expression represents the difference between a number nn cubed and the cube of 5?

  1. n3−5n^3 - 5
  2. (n−5)3(n - 5)^3
  3. n3−53n^3 - 5^3 (correct answer)
  4. 53−n35^3 - n^3
Explanation: The phrase 'a number nn cubed' is n3n^3. The phrase 'the cube of 5' is 535^3. The 'difference between' these two quantities is n3−53n^3 - 5^3. Answer A fails to cube the 5. Answer B represents 'the cube of the difference between nn and 5'. Answer D reverses the order of subtraction.

Question 4

A shirt costs cc dollars. The price is first increased by 20%, and then this new price is decreased by 10%. Which expression represents the final price of the shirt?

  1. c+0.20c−0.10cc + 0.20c - 0.10c
  2. 0.90(c+0.20c)0.90(c + 0.20c) (correct answer)
  3. c+0.10c + 0.10
  4. (c+0.20)−0.10(c + 0.20) - 0.10
Explanation: The initial price is cc. An increase of 20% makes the new price c+0.20cc + 0.20c, or 1.20c1.20c. This new price is then decreased by 10%. The discount amount is 10% of the new price, which is 0.10(c+0.20c)0.10(c + 0.20c). The final price is the new price minus the discount: (c+0.20c)−0.10(c+0.20c)(c + 0.20c) - 0.10(c + 0.20c). Factoring out (c+0.20c)(c + 0.20c) gives (1−0.10)(c+0.20c)(1 - 0.10)(c + 0.20c), which is 0.90(c+0.20c)0.90(c + 0.20c). Answer A incorrectly applies the 10% discount to the original price cc. Answers C and D incorrectly treat percentages as flat amounts.

Question 5

The value of a car, initially VV dollars, decreases by a total of DD dollars over a period of YY years at a constant rate. Which expression represents the value of the car after tt years, where t<Yt < Y?

  1. V−t(DY)V - t(\frac{D}{Y}) (correct answer)
  2. V+t(DY)V + t(\frac{D}{Y})
  3. V−DtV - Dt
  4. V−tYV - \frac{t}{Y}
Explanation: First, find the rate of decrease per year. If the value decreases by DD dollars in YY years, the constant rate is DY\frac{D}{Y} dollars per year. The total decrease after tt years is the rate multiplied by the time, which is t(DY)t(\frac{D}{Y}). The value after tt years is the initial value VV minus this total decrease: V−t(DY)V - t(\frac{D}{Y}). Answer A forgets to divide by YY to find the yearly rate. Answer B incorrectly adds the decrease, representing appreciation. Answer D omits the total decrease amount DD.

Question 6

A plumber charges a flat fee of $50 for a house call, plus an hourly rate of rr dollars. For a job that takes hh hours, the plumber gives a 15% discount on the hourly charges only. Which expression represents the total cost?

  1. 0.85(50+rh)0.85(50 + rh)
  2. 50+rh−0.1550 + rh - 0.15
  3. 50+0.85rh50 + 0.85rh (correct answer)
  4. 50+1.15rh50 + 1.15rh
Explanation: The total hourly charge is rhrh. A 15% discount on this amount means the customer pays 85% of it, which is 0.85rh0.85rh. The flat fee of $50 is not discounted. The total cost is the sum of the flat fee and the discounted hourly charges: 50+0.85rh50 + 0.85rh. Answer A incorrectly applies the discount to the flat fee as well. Answer B incorrectly subtracts a flat 15 cents instead of 15 percent. Answer D incorrectly adds 15% instead of subtracting it.

Question 7

A person has qq quarters, dd dimes, and nn nickels. The number of dimes is twice the number of quarters. Which expression represents the total value of the coins in cents, in terms of qq and nn?

  1. 3q+n3q + n
  2. 0.45q+0.05n0.45q + 0.05n
  3. 30q+5n30q + 5n
  4. 45q+5n45q + 5n (correct answer)
Explanation: The value of the coins in cents is 25q+10d+5n25q + 10d + 5n. The problem states that the number of dimes is twice the number of quarters, so d=2qd = 2q. We substitute 2q2q for dd in the value expression: 25q+10(2q)+5n25q + 10(2q) + 5n. This simplifies to 25q+20q+5n25q + 20q + 5n, which equals 45q+5n45q + 5n. Answer A represents the number of coins, not their value. Answer B represents the value in dollars, not cents. Answer C results from incorrectly setting up the relationship as q=2dq = 2d.

Question 8

A container holds GG gallons of water. A portion of the water, pp percent, is poured out. Then, 5 gallons are added. Which expression represents the final amount of water in the container, in gallons?

  1. G(1−p100)+5G(1 - \frac{p}{100}) + 5 (correct answer)
  2. (G+5)(1−p100)(G+5)(1 - \frac{p}{100})
  3. G−p+5G - p + 5
  4. G−p100G−5G - \frac{p}{100}G - 5
Explanation: The initial amount is GG. The amount poured out is pp percent of GG, which is (p100)G(\frac{p}{100})G. The amount remaining is G−(p100)GG - (\frac{p}{100})G, which can be factored as G(1−p100)G(1 - \frac{p}{100}). Finally, 5 gallons are added, so the final amount is G(1−p100)+5G(1 - \frac{p}{100}) + 5. Answer A treats the percentage pp as a flat amount. Answer B incorrectly applies the percentage decrease after the 5 gallons are added. Answer D incorrectly subtracts 5 instead of adding it.

Question 9

At a concert, the number of adult tickets sold was 40 fewer than three-fifths of the number of student tickets sold. If ss represents the number of student tickets sold, which expression represents the number of adult tickets sold?

  1. 35s−40\frac{3}{5}s - 40 (correct answer)
  2. 40−35s40 - \frac{3}{5}s
  3. 35(s−40)\frac{3}{5}(s - 40)
  4. s−35(40)s - \frac{3}{5}(40)
Explanation: 'Three-fifths of the number of student tickets' translates to 35s\frac{3}{5}s. The phrase '40 fewer than' this quantity means we must subtract 40 from it, resulting in 35s−40\frac{3}{5}s - 40. Answer B is a common reversal error for 'fewer than'. Answer C incorrectly takes the fraction of the difference, s−40s-40. Answer D misinterprets the phrase and subtracts three-fifths of 40 from ss.

Question 10

A rectangular garden has a width that is 8 feet less than twice its length. If the perimeter of the garden exceeds 64 feet, which algebraic expression represents the minimum possible length of the garden?

  1. l>1313l > 13\frac{1}{3} (correct answer)
  2. l>12l > 12
  3. l>1023l > 10\frac{2}{3}
  4. l>16l > 16
Explanation: Let ll = length and ww = width. Given: w=2l−8w = 2l - 8. Perimeter formula: P=2l+2wP = 2l + 2w. Substituting: 2l+2(2l−8)>642l + 2(2l - 8) > 64. Simplifying: 2l+4l−16>642l + 4l - 16 > 64, so 6l>806l > 80, thus l>806=1313l > \frac{80}{6} = 13\frac{1}{3}. Choice B incorrectly uses 6l−16=646l - 16 = 64. Choice C uses P=64P = 64 instead of >64> 64. Choice D assumes width equals length minus 8.

Question 11

Three consecutive odd integers have the property that when the largest is divided by the smallest, the quotient is 13\frac{1}{3} more than when the middle integer is divided by the smallest. If nn represents the smallest odd integer, which equation correctly models this relationship?

  1. n+4n=n+2n+13\frac{n+4}{n} = \frac{n+2}{n} + \frac{1}{3}
  2. n+4n−n+2n=13\frac{n+4}{n} - \frac{n+2}{n} = \frac{1}{3} (correct answer)
  3. n+4n+2=n+2n+13\frac{n+4}{n+2} = \frac{n+2}{n} + \frac{1}{3}
  4. n+4n=n+2n×43\frac{n+4}{n} = \frac{n+2}{n} \times \frac{4}{3}
Explanation: Three consecutive odd integers: nn, n+2n+2, n+4n+4. The condition states: largestsmallest=middlesmallest+13\frac{\text{largest}}{\text{smallest}} = \frac{\text{middle}}{\text{smallest}} + \frac{1}{3}. This gives n+4n=n+2n+13\frac{n+4}{n} = \frac{n+2}{n} + \frac{1}{3}. Rearranging: n+4n−n+2n=13\frac{n+4}{n} - \frac{n+2}{n} = \frac{1}{3}. Choice A is the unrearranged form. Choice C uses wrong denominator for largest/middle. Choice D interprets 'more than' as multiplication.

Question 12

A parking garage charges a base fee plus an hourly rate. The total cost for parking 3 hours is $12, and for 7 hours is $20. If a customer parks for $hh $ hours and receives a 10% discount when parking exceeds 5 hours, which expression represents their total cost?

  1. {6+2hif h≤50.9(6+2h)if h>5\begin{cases} 6 + 2h & \text{if } h \leq 5 \\ 0.9(6 + 2h) & \text{if } h > 5 \end{cases} (correct answer)
  2. {6+2hif h≤56+2h−0.1(6+2h)if h>5\begin{cases} 6 + 2h & \text{if } h \leq 5 \\ 6 + 2h - 0.1(6 + 2h) & \text{if } h > 5 \end{cases}
  3. {6+2hif h≤56+1.8hif h>5\begin{cases} 6 + 2h & \text{if } h \leq 5 \\ 6 + 1.8h & \text{if } h > 5 \end{cases}
  4. {6+2hif h≤56+2h−0.1(2h)if h>5\begin{cases} 6 + 2h & \text{if } h \leq 5 \\ 6 + 2h - 0.1(2h) & \text{if } h > 5 \end{cases}
Explanation: Let base fee = bb and hourly rate = rr. From given info: b+3r=12b + 3r = 12 and b+7r=20b + 7r = 20. Solving: 4r=84r = 8, so r=2r = 2 and b=6b = 6. Normal cost = 6+2h6 + 2h. For h>5h > 5, 10% discount applies to total cost: 0.9(6+2h)0.9(6 + 2h). Choice B is equivalent but unnecessarily complex. Choice C applies discount only to hourly portion. Choice D applies discount only to hourly rate, not total.

Question 13

A rectangular swimming pool is surrounded by a walkway of uniform width. The pool dimensions are 20 feet by 30 feet. If the walkway width is ww feet and the total area of the pool plus walkway exceeds the pool area by more than 440 square feet, which inequality represents this constraint?

  1. 4w2+100w>4404w^2 + 100w > 440 (correct answer)
  2. (20+2w)(30+2w)−600>440(20 + 2w)(30 + 2w) - 600 > 440
  3. 4w2+100w>10404w^2 + 100w > 1040
  4. (20+w)(30+w)−600>440(20 + w)(30 + w) - 600 > 440
Explanation: Pool area = 20×30=60020 \times 30 = 600 sq ft. Total area including walkway = (20+2w)(30+2w)=600+40w+60w+4w2=600+100w+4w2(20 + 2w)(30 + 2w) = 600 + 40w + 60w + 4w^2 = 600 + 100w + 4w^2. Difference = 600+100w+4w2−600=4w2+100w600 + 100w + 4w^2 - 600 = 4w^2 + 100w. Given this exceeds 440: 4w2+100w>4404w^2 + 100w > 440. Choice B is unreduced form. Choice C uses wrong threshold. Choice D assumes walkway only on two sides.

Question 14

A water tank is being filled at a variable rate. During the first hour, it fills at rr gallons per minute. Each subsequent hour, the rate increases by 25% of the previous hour's rate. Which expression represents the total gallons added during the first nn hours?

  1. 60r×1.25n−11.25−160r \times \frac{1.25^n - 1}{1.25 - 1}
  2. 60r×1.25n−10.2560r \times \frac{1.25^n - 1}{0.25}
  3. 60r(1+1.25+1.252+…+1.25n−1)60r(1 + 1.25 + 1.25^2 + \ldots + 1.25^{n-1}) (correct answer)
  4. 60r×1.25n+1−1.250.2560r \times \frac{1.25^{n+1} - 1.25}{0.25}
Explanation: Hour 1: rate = rr gal/min, total = 60r60r gallons. Hour 2: rate = 1.25r1.25r, total = 60(1.25r)=60r×1.2560(1.25r) = 60r \times 1.25. Hour 3: rate = 1.252r1.25^2r, total = 60r×1.25260r \times 1.25^2. The pattern shows total = 60r(1+1.25+1.252+…+1.25n−1)60r(1 + 1.25 + 1.25^2 + \ldots + 1.25^{n-1}). Choices A and B use wrong geometric series formula. Choice D has incorrect exponents in the numerator.

Question 15

A manufacturer produces widgets where the cost per widget decreases as production volume increases. The cost per widget is $8 for the first 100 widgets, then decreases by $0.02 for each additional group of 50 widgets produced. If $xx representsthetotalnumberofwidgetsproduced(whererepresents the total number of widgets produced (where x>100x > 100 $), which expression represents the cost per widget?

  1. 8−0.02×(x−100)8 - 0.02 \times (x-100)
  2. 8−0.02×x−100508 - 0.02 \times \frac{x-100}{50}
  3. 8−0.02×⌊x50⌋8 - 0.02 \times \lfloor\frac{x}{50}\rfloor
  4. 8−0.02×⌊x−10050⌋8 - 0.02 \times \lfloor\frac{x-100}{50}\rfloor (correct answer)
Explanation: When you encounter piecewise pricing problems like this, you need to carefully track how the price changes occur and what triggers each change. Let's break down what happens: The first 100 widgets cost $8 each. After that, the cost decreases by $0.02 for each additional group of 50 widgets. The key insight is that the price reduction happens in discrete steps - only when you complete another full group of 50 widgets beyond the initial 100. For the correct answer D, $8−0.02×⌊x−10050⌋8 - 0.02 \times \lfloor\frac{x-100}{50}\rfloor ,let′stracethroughthelogic.Ifyouproduce175widgets,you′vemade75widgetsbeyondthefirst100.Dividing75by50gives1.5,butthefloorfunction, let's trace through the logic. If you produce 175 widgets, you've made 75 widgets beyond the first 100. Dividing 75 by 50 gives 1.5, but the floor function ⌊1.5⌋=1\lfloor 1.5 \rfloor = 1 meansyou′vecompletedexactly1fullgroupof50.Sothecostperwidgetismeans you've completed exactly 1 full group of 50. So the cost per widget is 8−0.02(1)=7.988 - 0.02(1) = 7.98 $. Answer A treats every single widget beyond 100 as triggering a $0.02 reduction, which contradicts the "group of 50" requirement. Answer B uses the exact division without the floor function, so it would give partial reductions (like $0.03 for 75 widgets) rather than step-wise changes. Answer C incorrectly includes the first 100 widgets in the division, which would apply discounts to the base production volume. Remember: When you see "group of" or "every X units" in pricing problems, look for the floor function $$\lfloor \rfloor$$ to handle the step-wise changes correctly.

Question 16

A cylindrical water tank has radius rr feet. Water is pumped in at 10 cubic feet per minute, but due to evaporation, water is lost at a rate proportional to the surface area, with a constant of proportionality kk. Which expression represents the rate of change of water volume when the water height is hh feet?

  1. 10−k×(2πr2+2πrh)10 - k \times (2\pi r^2 + 2\pi r h)
  2. 10−k×2πrh10 - k \times 2\pi r h
  3. 10−k×πr2h10 - k \times \pi r^2 h
  4. 10−k×πr210 - k \times \pi r^2 (correct answer)
Explanation: When you encounter rate problems involving geometry and physical processes, focus on identifying what's being gained versus what's being lost, and carefully determine which surfaces are involved in each process. The correct answer is D) 10−k×πr210 - k \times \pi r^2 because evaporation occurs only at the water's surface. In a cylindrical tank, the exposed water surface is a circle with area πr2\pi r^2. Since water is pumped in at 10 cubic feet per minute and lost through evaporation at rate kk times the surface area, the net rate of volume change is 10−k×πr210 - k \times \pi r^2. Choice A is wrong because 2πr2+2πrh2\pi r^2 + 2\pi r h represents the total surface area of a complete cylinder (including both circular ends plus the curved side). This would only apply if the entire tank surface were exposed to evaporation, which isn't the case. Choice B is incorrect because 2πrh2\pi r h is the curved side surface area of the cylinder. Water doesn't evaporate through the solid walls of the tank. Choice C makes the error of using πr2h\pi r^2 h, which represents the volume of water, not a surface area. Evaporation rates depend on exposed surface area, not volume. Study tip: In rate problems involving containers, always identify which surfaces are actually exposed to the process described. Evaporation happens at the air-water interface, not through solid walls or submerged surfaces. Draw a quick sketch to visualize which area is relevant.

Question 17

Which expression represents five times the sum of a number, nn, and its reciprocal?

  1. 5n+1n5n + \frac{1}{n}
  2. 5(n+1n)5\left(n + \frac{1}{n}\right) (correct answer)
  3. 5+n+1n5 + n + \frac{1}{n}
  4. 5n⋅1n5n \cdot \frac{1}{n}
Explanation: First, identify the components. 'A number, nn' is nn. 'Its reciprocal' is 1n\frac{1}{n}. 'The sum of' these two is n+1nn + \frac{1}{n}. The phrase 'five times the sum' means the entire sum must be multiplied by 5, which requires parentheses: 5(n+1n)5\left(n + \frac{1}{n}\right). Answer A, 5n+1n5n + \frac{1}{n}, is a common order-of-operations error, representing 'five times a number plus its reciprocal'. Answer C incorrectly translates 'times' as addition. Answer D incorrectly translates 'sum' as a product.

Question 18

Maria is mm years old. Her brother, David, is 3 years older than her. Which expression represents the sum of their ages 5 years from now?

  1. 2m+32m + 3
  2. 2m+72m + 7
  3. 2m+82m + 8
  4. 2m+132m + 13 (correct answer)
Explanation: Currently, Maria's age is mm and David's age is m+3m + 3. In 5 years, Maria's age will be m+5m + 5 and David's age will be (m+3)+5(m + 3) + 5, which is m+8m + 8. The sum of their ages in 5 years will be (m+5)+(m+8)(m + 5) + (m + 8). Combining like terms gives 2m+132m + 13. Answer A is the sum of their current ages. Answer C represents adding 5 years only once to their combined current age (2m+3+52m+3+5). Answer B would be correct if David were 3 years younger than Maria.

Question 19

The length of a rectangular garden is 5 feet less than twice its width. If the width is represented by ww, which expression represents the perimeter of the garden?

  1. w(2w−5)w(2w - 5)
  2. w+(2w−5)w + (2w - 5)
  3. 2w+2(5−2w)2w + 2(5 - 2w)
  4. 2w+2(2w−5)2w + 2(2w - 5) (correct answer)
Explanation: First, translate the length. 'Twice its width' is 2w2w. '5 feet less than' that is 2w−52w - 5. The perimeter of a rectangle is 2L+2W2L + 2W, where LL is length and WW is width. Substituting the expressions gives 2(2w−5)+2w2(2w - 5) + 2w. Answer A represents the area, not the perimeter. Answer B represents the sum of one width and one length (half the perimeter). Answer C makes a reversal error for 'less than', writing the length as 5−2w5 - 2w.

Question 20

Let kk be the smallest of three consecutive odd integers. Which expression represents the sum of these three integers?

  1. (3k)
  2. 3k+33k + 3
  3. 3k+63k + 6 (correct answer)
  4. 3k−63k - 6
Explanation: Consecutive odd integers differ by 2. If the smallest is kk, the next odd integer is k+2k + 2, and the one after that is k+4k + 4. The sum is k+(k+2)+(k+4)k + (k + 2) + (k + 4). Combining like terms gives 3k+63k + 6. Answer B, 3k+33k + 3, represents the sum of three consecutive integers (e.g., k,k+1,k+2k, k+1, k+2). Answer A, (3k), would be the sum if the integers were k−2,k,k+2k-2, k, k+2, but the stem specifies that kk is the smallest. Answer D, 3k−63k - 6, would be the sum if kk were the largest integer (k−4,k−2,kk-4, k-2, k).