GRE Quantitative Reasoning Practice: Arithmetic and Algebra Problem Solving
- Arithmetic and Algebra Problem Solving
- 15 questions
- 26 minutes
These multiple-choice questions cover the arithmetic and algebra topics listed for GRE Quantitative Reasoning: integers and remainders, exponents, percents, rates, ratios, equations, inequalities and counting. The real test provides a basic on-screen calculator, but many items are faster with a quick algebraic step.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
2x + 3 = 32x − 1.
What is the value of x?
Correct answer: A
Write 32 as 25: 2x + 3 = 25x − 5, so x + 3 = 5x − 5 and x = 2. Check: 25 = 321.
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Question 2
When the positive integer n is divided by 6, the remainder is 4.
What is the remainder when 2n is divided by 6?
Correct answer: A
Write n = 6q + 4. Then 2n = 12q + 8 = 6(2q + 1) + 2, so the remainder is 2. Doubling the remainder to 8 is not enough, because 8 is larger than 6.
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Question 3
The price of a jacket is increased by 20%, and the new price is then decreased by 25%.
What is the overall change from the original price?
Correct answer: B
Multiply the factors: 1.20 · 0.75 = 0.90, which is a 10% decrease. Simply subtracting the percents (20 − 25 = −5) ignores that the 25% is taken from a larger price.
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Question 4
Pump A alone can fill a tank in 4 hours, and pump B alone can fill it in 6 hours.
Working together at these rates, how long do the two pumps take to fill the empty tank?
Correct answer: A
Together they fill 1/4 + 1/6 = 5/12 of the tank per hour, so the time is 12/5 hours = 2.4 hours = 2 hours 24 minutes. Averaging the two times (5 hours) is a common mistake.
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Question 5
A class of 40 students has boys and girls in the ratio 3 : 5.
How many more girls must join the class to make the ratio of boys to girls 1 : 2?
Correct answer: B
There are 40 · 3/8 = 15 boys and 25 girls. A 1 : 2 ratio needs 30 girls, so 5 more girls must join.
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Question 6
x² − 5x − 14 = 0.
What is the sum of the squares of the two solutions?
Correct answer: B
Factoring gives (x − 7)(x + 2) = 0, so the solutions are 7 and −2. Their squares are 49 and 4, with sum 53; 45 comes from treating (−2)² as −4.
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Question 7
|2x − 3| < 7
How many integers x satisfy the inequality?
Correct answer: D
The inequality means −7 < 2x − 3 < 7, so −4 < 2x < 10 and −2 < x < 5. The integers −1, 0, 1, 2, 3, 4 give 6 values; the endpoints −2 and 5 are excluded.
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Question 8
The average (arithmetic mean) of five numbers is 18. When one number is removed, the average of the remaining four numbers is 16.
What number was removed?
Correct answer: C
The five numbers sum to 5 · 18 = 90 and the four remaining numbers sum to 4 · 16 = 64, so the removed number is 90 − 64 = 26. Subtracting the averages (18 − 16 = 2) ignores how many numbers there are.
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Question 9
What is the value of 45 · 82163?
Correct answer: D
Write everything as powers of 2: 45 = 210, 82 = 26 and 163 = 212. The value is 210 + 6 − 12 = 24 = 16.
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Question 10
$2,000 is invested at 5% annual interest, compounded once a year. No money is added or withdrawn.
What is the value of the investment after 3 years?
Correct answer: C
Compound interest multiplies by 1.05 each year: 2,000 · 1.05³ = $2,315.25. The value $2,300.00 is simple interest (3 · $100 added to $2,000).
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Question 11
In an arithmetic sequence, the first term is 7 and each term is 4 more than the previous term.
What is the sum of the first 20 terms?
Correct answer: C
The 20th term is 7 + 19 · 4 = 83, and the sum is 20 · (7 + 83)/2 = 900. The value 83 is only the last term, and 980 uses 20 steps of 4 instead of 19.
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Question 12
f(x) = x² − 3x for all x.
Which expression is equal to f(a + 1) − f(a)?
Correct answer: A
f(a + 1) = a² + 2a + 1 − 3a − 3 = a² − a − 2. Subtracting f(a) = a² − 3a leaves 2a − 2.
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Question 13
3x + 2y = 19 and x − y = 3.
What is the value of x + y?
Correct answer: D
From x = y + 3, substitution gives 3(y + 3) + 2y = 19, so 5y = 10, y = 2 and x = 5. Therefore x + y = 7; 5 is the value of x alone.
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Question 14
How many positive integers less than 100 are multiples of 3 or multiples of 5?
Correct answer: C
There are 33 multiples of 3 and 19 multiples of 5 below 100, but the 6 multiples of 15 are counted twice: 33 + 19 − 6 = 46. Forgetting to subtract the overlap gives 52.
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Question 15
A driver travels 120 miles at 40 miles per hour and returns along the same road at 60 miles per hour.
What is the average speed for the whole round trip?
Correct answer: B
The trip takes 3 hours out and 2 hours back, so 240 miles in 5 hours is 48 miles per hour. The value 50 is the average of the two speeds, which ignores that more time is spent at the slower speed.
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