GRE Mixed Quantitative Reasoning (Problem Solving) 1 — Questions and Answers

Question 1: A store offers a 30% discount on an item already marked down 20% from its original price. If the final price is $112, what was the original price? (A) $180 (B) $200 (C) $220 (D) $240

$180
$200 (Correct answer)
$220
$240

Correct answer: $200

Final price = Original × 0.80 × 0.70 = Original × 0.56. 112 = 0.56 × P → P = 200.

After 20% markdown: price = P × 0.80. After 30% discount: price = P × 0.80 × 0.70 = P × 0.56. Setting equal to $112: 0.56P = 112 → P = 112/0.56 = $200.

Question 2: If 3x − 7 = 2(x + 4), what is the value of x? (A) 13 (B) 14 (C) 15 (D) 16

13
14
15 (Correct answer)
16

Correct answer: 15

3x − 7 = 2x + 8 → x = 15.

3x − 7 = 2(x + 4). Expand: 3x − 7 = 2x + 8. Subtract 2x from both sides: x − 7 = 8. Add 7: x = 15.

Question 3: Two trains leave the same station at the same time, traveling in opposite directions. Train A travels at 60 mph and Train B at 80 mph. After how many hours are they 350 miles apart? (A) 2.0 hours (B) 2.5 hours (C) 3.0 hours (D) 3.5 hours

2.0 hours
2.5 hours (Correct answer)
3.0 hours
3.5 hours

Correct answer: 2.5 hours

Combined speed = 60 + 80 = 140 mph. Time = 350/140 = 2.5 hours.

Moving in opposite directions, the distance between them increases at rate = 60 + 80 = 140 mph. Time = distance ÷ rate = 350 ÷ 140 = 2.5 hours.

Question 4: A rectangular garden has a perimeter of 54 meters. If its length is twice its width, what is the area of the garden? (A) 81 m² (B) 108 m² (C) 162 m² (D) 180 m²

81 m²
108 m² (Correct answer)
162 m²
180 m²

Correct answer: 108 m²

Width = w, length = 2w. Perimeter: 2(w + 2w) = 54 → 6w = 54 → w = 9. Length = 18. Area = 9 × 18 = 162 m².

Let w = width, then length = 2w. Perimeter = 2(l + w) = 2(2w + w) = 2(3w) = 6w = 54. w = 9 meters. Length = 18 meters. Area = 9 × 18 = 162 m².

Question 5: A solution is 20% salt by weight. How many grams of pure salt must be added to 300 grams of this solution to make it 25% salt? (A) 15g (B) 18g (C) 20g (D) 24g

15g
18g
20g (Correct answer)
24g

Correct answer: 20g

Start: 60g salt in 300g. Add x grams of salt: (60+x)/(300+x) = 0.25. 60+x = 75+0.25x → 0.75x = 15 → x = 20.

Original salt = 0.20 × 300 = 60g. Adding x grams of pure salt: (60 + x)/(300 + x) = 0.25. Cross-multiplying: 60 + x = 0.25(300 + x) = 75 + 0.25x. 0.75x = 15. x = 20 grams.

Question 6: A car travels 120 miles at 40 mph, then another 120 miles at 60 mph. What is the average speed for the entire trip? (A) 48 mph (B) 50 mph (C) 52 mph (D) 55 mph

48 mph (Correct answer)
50 mph
52 mph
55 mph

Correct answer: 48 mph

Total distance = 240 miles. Time = 120/40 + 120/60 = 3 + 2 = 5 hours. Avg speed = 240/5 = 48 mph.

Leg 1: 120 miles ÷ 40 mph = 3 hours. Leg 2: 120 miles ÷ 60 mph = 2 hours. Total: 240 miles in 5 hours. Average speed = 240/5 = 48 mph. (Note: this is the harmonic mean of the two speeds, not the arithmetic mean of 50 mph.)

Question 7: A company's profits increased by 25% in 2022 and then decreased by 20% in 2023. What was the net percentage change over the two years? (A) 0% (B) +5% (C) −5% (D) +10%

0% (Correct answer)
+5%
−5%
+10%

Correct answer: 0%

1.25 × 0.80 = 1.00. The profit returned to exactly the original amount — 0% net change.

1.25 × 0.80 = 1.00. The profit is back to its original value. Net percentage change = (1.00 − 1) × 100% = 0%. This illustrates why a 25% gain followed by a 20% loss nets to zero — the percentage is applied to different bases.

Question 8: In a class of 36 students, 20 passed the math test, 18 passed the English test, and 8 passed both. How many students failed both tests? (A) 4 (B) 6 (C) 8 (D) 10

4
6 (Correct answer)
8
10

Correct answer: 6

Passed at least one = 20 + 18 − 8 = 30. Failed both = 36 − 30 = 6.

By inclusion-exclusion: students who passed at least one test = |M ∪ E| = |M| + |E| − |M ∩ E| = 20 + 18 − 8 = 30. Students who failed both = Total − passed at least one = 36 − 30 = 6.

Question 9: What is the slope of a line perpendicular to the line 3x − 4y = 12? (A) 3/4 (B) 4/3 (C) −4/3 (D) −3/4

3/4
4/3
−4/3 (Correct answer)
−3/4

Correct answer: −4/3

Rearrange: y = (3/4)x − 3. Slope = 3/4. Perpendicular slope = −1/(3/4) = −4/3.

3x − 4y = 12. Solving for y: −4y = −3x + 12 → y = (3/4)x − 3. Slope of original = 3/4. Slope of perpendicular line = −1 ÷ (3/4) = −4/3.

Question 10: A jar contains 5 red, 4 blue, and 3 green marbles. Two marbles are drawn without replacement. What is the probability that both are red? (A) 5/33 (B) 5/22 (C) 2/11 (D) 5/12

5/33 (Correct answer)
5/22
2/11
5/12

Correct answer: 5/33

P = (5/12) × (4/11) = 20/132 = 5/33.

Total marbles = 5 + 4 + 3 = 12. P(first red) = 5/12. After removing one red, 4 red remain among 11 total. P(second red) = 4/11. P(both red) = (5/12) × (4/11) = 20/132 = 5/33.

Question 11: If f(x) = 2x² − 3x + 1, what is f(−2)? (A) 11 (B) 13 (C) 15 (D) 17

11
13
15 (Correct answer)
17

Correct answer: 15

f(−2) = 2(−2)² − 3(−2) + 1 = 2(4) + 6 + 1 = 8 + 6 + 1 = 15.

f(−2) = 2(−2)² − 3(−2) + 1. (−2)² = 4. So: 2(4) − 3(−2) + 1 = 8 + 6 + 1 = 15.

Question 12: A bicycle wheel has a radius of 35 cm. How many complete revolutions does it make while traveling 1,100 meters? (Use π ≈ 22/7) (A) 400 (B) 500 (C) 550 (D) 600

400
500 (Correct answer)
550
600

Correct answer: 500

Circumference = 2π r = 2 × (22/7) × 35 = 220 cm = 2.2 m. Revolutions = 1100/2.2 = 500.

Circumference = 2 × (22/7) × 35 = 2 × 22 × 5 = 220 cm = 2.2 m. Distance = 1,100 m. Revolutions = 1,100 ÷ 2.2 = 500.

Question 13: 5 workers can complete a project in 12 days. How many days would it take 8 workers (working at the same rate) to complete the same project? (A) 6 days (B) 7.5 days (C) 8 days (D) 9 days

6 days
7.5 days (Correct answer)
8 days
9 days

Correct answer: 7.5 days

Total work = 5 × 12 = 60 worker-days. Time for 8 workers = 60/8 = 7.5 days.

Total work = 5 workers × 12 days = 60 worker-days. With 8 workers: time = 60 worker-days ÷ 8 workers = 7.5 days.

Question 14: In a geometric sequence, the first term is 3 and the common ratio is 2. What is the sum of the first 6 terms? (A) 186 (B) 189 (C) 192 (D) 195

186
189 (Correct answer)
192
195

Correct answer: 189

S₆ = 3(2⁶ − 1)/(2 − 1) = 3(64 − 1) = 3 × 63 = 189.

Sₙ = a(rⁿ − 1)/(r − 1). a = 3, r = 2, n = 6. S₆ = 3(2⁶ − 1)/(2 − 1) = 3(64 − 1)/1 = 3 × 63 = 189.

Question 15: A cylindrical tank has a radius of 3 meters and height of 7 meters. What is the volume of the tank? (Use π ≈ 22/7) (A) 154 m³ (B) 198 m³ (C) 186 m³ (D) 210 m³

154 m³
198 m³ (Correct answer)
186 m³
210 m³

Correct answer: 198 m³

V = πr²h = (22/7) × 9 × 7 = 22 × 9 = 198 m³.

V = π × r² × h = (22/7) × 3² × 7 = (22/7) × 9 × 7 = 22 × 9 = 198 m³. The factor of 7 in h cancels with the 7 in the denominator of π.

Question 16: If a > 0, b > 0, and (a + b)² = 100, ab = 20, what is the value of a² + b²? (A) 50 (B) 60 (C) 70 (D) 80

50
60 (Correct answer)
70
80

Correct answer: 60

(a+b)² = a² + 2ab + b² = 100. a² + b² = 100 − 2ab = 100 − 40 = 60.

(a + b)² = a² + 2ab + b² = 100. Since ab = 20, we have 2ab = 40. Therefore a² + b² = 100 − 40 = 60.

Question 17: The ratio of boys to girls in a school is 3:5. If there are 720 students in total, how many more girls are there than boys? (A) 90 (B) 150 (C) 180 (D) 270

90
150
180 (Correct answer)
270

Correct answer: 180

Boys = (3/8) × 720 = 270. Girls = (5/8) × 720 = 450. Difference = 450 − 270 = 180.

Total ratio parts = 3 + 5 = 8. One part = 720 ÷ 8 = 90. Boys = 3 × 90 = 270. Girls = 5 × 90 = 450. Difference = 450 − 270 = 180.

Question 18: If p and q are roots of the equation x² − 5x + 6 = 0, what is the value of p² + q²? (A) 11 (B) 13 (C) 15 (D) 17

11
13 (Correct answer)
15
17

Correct answer: 13

p + q = 5, pq = 6. p² + q² = (p+q)² − 2pq = 25 − 12 = 13.

From x² − 5x + 6 = 0: sum of roots = p + q = 5; product = pq = 6. p² + q² = (p + q)² − 2pq = 5² − 2(6) = 25 − 12 = 13.

Question 19: A tank is filled by pipe A in 6 hours and drained by pipe B in 10 hours. If both are open simultaneously, how long does it take to fill an empty tank? (A) 10 hours (B) 12 hours (C) 15 hours (D) 18 hours

10 hours
12 hours
15 hours (Correct answer)
18 hours

Correct answer: 15 hours

Net rate = 1/6 − 1/10 = 5/30 − 3/30 = 2/30 = 1/15. Time = 15 hours.

Pipe A fills 1/6 of tank per hour. Pipe B drains 1/10 per hour. Net rate = 1/6 − 1/10 = 5/30 − 3/30 = 2/30 = 1/15 tank per hour. Time = 1 ÷ (1/15) = 15 hours.

Question 20: A ladder 10 meters long leans against a wall. Its base is 6 meters from the wall. How high does the ladder reach up the wall? (A) 6 m (B) 7 m (C) 8 m (D) 9 m

6 m
7 m
8 m (Correct answer)
9 m

Correct answer: 8 m

By the Pythagorean theorem: h² + 6² = 10² → h² = 100 − 36 = 64 → h = 8.

Ladder = hypotenuse = 10 m. Base = 6 m. Height h: h² + 6² = 10². h² = 100 − 36 = 64. h = √64 = 8 meters.

Question 21: A store sells shirts at $24 each or 3 for $60. What is the discount per shirt when buying 3 at a time? (A) 12.5% (B) 15% (C) 16.7% (D) 20%

12.5%
15%
16.7% (Correct answer)
20%

Correct answer: 16.7%

Discounted price per shirt: 60/3 = $20. Discount = (24−20)/24 = 4/24 ≈ 16.7%.

3 shirts for $60 = $20 per shirt. Original price = $24. Discount = ($24 − $20)/$24 = $4/$24 = 1/6 ≈ 0.1667 = 16.7%.

Question 22: Solve: |2x − 5| = 11 (A) x = 3 or x = 8 (B) x = 8 or x = −3 (C) x = 8 only (D) x = −3 only

x = 3 or x = 8
x = 8 or x = −3 (Correct answer)
x = 8 only
x = −3 only

Correct answer: x = 8 or x = −3

Case 1: 2x − 5 = 11 → 2x = 16 → x = 8. Case 2: 2x − 5 = −11 → 2x = −6 → x = −3.

|2x − 5| = 11 splits into: Case 1: 2x − 5 = 11 → 2x = 16 → x = 8. Case 2: 2x − 5 = −11 → 2x = −6 → x = −3. Both solutions check: |2(8)−5| = |11| = 11 ✓; |2(−3)−5| = |−11| = 11 ✓.

Question 23: A sphere has a surface area of 616 cm². What is its radius? (Use π ≈ 22/7) (A) 5 cm (B) 7 cm (C) 8 cm (D) 10 cm

5 cm
7 cm (Correct answer)
8 cm
10 cm

Correct answer: 7 cm

Surface area = 4πr² = 616. 4 × (22/7) × r² = 616. r² = 616 × 7/(4 × 22) = 4312/88 = 49. r = 7.

4πr² = 616. 4 × (22/7) × r² = 616. (88/7) × r² = 616. r² = 616 × 7/88 = 4312/88 = 49. r = √49 = 7 cm.

Question 24: If a train travels at 90 km/h, how long does it take to travel 375 km? (A) 3.5 hours (B) 4.0 hours (C) 4.2 hours (D) 4.5 hours

3.5 hours
4.0 hours
4.2 hours (Correct answer)
4.5 hours

Correct answer: 4.2 hours

Time = distance/speed = 375/90 = 4.1̄ ≈ 4.2 hours.

Time = 375 ÷ 90 = 4.1666... hours ≈ 4.2 hours (or 4 hours 10 minutes).

Question 25: The sum of an arithmetic sequence is given by S = n/2(a₁ + aₙ). If S = 210, a₁ = 1, and n = 20, what is aₙ? (A) 19 (B) 20 (C) 21 (D) 22

19
20
21 (Correct answer)
22

Correct answer: 21

210 = 20/2 × (1 + aₙ) = 10(1 + aₙ). 1 + aₙ = 21. aₙ = 20.

S = (n/2)(a₁ + aₙ). 210 = (20/2)(1 + aₙ) = 10(1 + aₙ). 1 + aₙ = 21. aₙ = 20.

Question 26: If √(x + 5) = x − 1, what is the value of x? (A) x = 4 (B) x = 1 (C) x = 4 or x = 1 (D) No real solution

x = 4 (Correct answer)
x = 1
x = 4 or x = 1
No real solution

Correct answer: x = 4

Square both sides: x + 5 = (x−1)² = x² − 2x + 1. x² − 3x − 4 = 0 → (x−4)(x+1) = 0. x = 4 or x = −1. Check: x = −1 gives √4 = −2 (invalid). x = 4 gives √9 = 3 = 4−1 ✓.

√(x + 5) = x − 1. Squaring: x + 5 = x² − 2x + 1. Rearranging: x² − 3x − 4 = 0. Factoring: (x − 4)(x + 1) = 0. x = 4 or x = −1. Check x = 4: √9 = 3 = 4 − 1 ✓. Check x = −1: √4 = 2, but x − 1 = −2 ✗. Only x = 4 is valid.

Question 27: In a class, 40% of students study Spanish, 30% study French, and 15% study both. What percentage studies neither language? (A) 35% (B) 40% (C) 45% (D) 50%

35%
40%
45% (Correct answer)
50%

Correct answer: 45%

Study at least one = 40 + 30 − 15 = 55%. Neither = 100 − 55 = 45%.

P(Spanish or French) = 40% + 30% − 15% = 55%. P(neither) = 100% − 55% = 45%.

Question 28: A cone has a base radius of 5 cm and height of 12 cm. What is its volume? (Use π ≈ 22/7) (A) 300 cm³ (B) 314 cm³ (C) 3142/7 cm³ (D) approximately 314.3 cm³

300 cm³
314 cm³ (Correct answer)
3142/7 cm³
approximately 314.3 cm³

Correct answer: 314 cm³

V = (1/3)πr²h = (1/3)(22/7)(25)(12) = (1/3)(22/7)(300) = (22×300)/(21) = 6600/21 ≈ 314.3. Closest to 314.

V = (1/3) × (22/7) × 5² × 12 = (1/3) × (22/7) × 25 × 12 = (22 × 25 × 12)/(3 × 7) = 6600/21 ≈ 314.3 cm³.

Question 29: A student scored 85, 90, 78, 92, and 75 on five tests. What score is needed on the sixth test to achieve an average of exactly 85? (A) 88 (B) 90 (C) 92 (D) 94

88
90 (Correct answer)
92
94

Correct answer: 90

Sum needed = 85 × 6 = 510. Sum of five tests = 85+90+78+92+75 = 420. Need: 510 − 420 = 90.

Required total = 85 × 6 = 510. Current sum = 85 + 90 + 78 + 92 + 75 = 420. Needed score = 510 − 420 = 90.

Question 30: How many different 4-digit PINs can be created if the first digit cannot be 0 and digits may repeat? (A) 9,000 (B) 10,000 (C) 8,100 (D) 5,040

9,000 (Correct answer)
10,000
8,100
5,040

Correct answer: 9,000

First digit: 9 choices (1–9). Remaining three digits: 10 choices each. Total = 9 × 10 × 10 × 10 = 9,000.

Digit 1: cannot be 0, so 9 choices (1, 2, 3, 4, 5, 6, 7, 8, 9). Digits 2, 3, 4: 10 choices each (0–9). Total PINs = 9 × 10 × 10 × 10 = 9,000.

Question 31: If log₂(x) = 5, what is x? (A) 10 (B) 25 (C) 32 (D) 64

10
25
32 (Correct answer)
64

Correct answer: 32

log₂(x) = 5 means 2⁵ = x = 32.

log₂(x) = 5 is equivalent to 2⁵ = x. 2⁵ = 32.

Question 32: A rectangular box has dimensions 4 cm × 5 cm × 6 cm. What is the length of the longest diagonal inside the box? (A) √76 cm (B) √77 cm (C) √78 cm (D) √79 cm

√76 cm
√77 cm (Correct answer)
√78 cm
√79 cm

Correct answer: √77 cm

Space diagonal = √(l² + w² + h²) = √(16 + 25 + 36) = √77 cm.

Space diagonal = √(4² + 5² + 6²) = √(16 + 25 + 36) = √77 ≈ 8.77 cm.

Question 33: A population of bacteria doubles every 3 hours. Starting at 500, how many bacteria are there after 12 hours? (A) 4,000 (B) 6,000 (C) 8,000 (D) 16,000

4,000
6,000
8,000 (Correct answer)
16,000

Correct answer: 8,000

Number of doublings = 12/3 = 4. Population = 500 × 2⁴ = 500 × 16 = 8,000.

In 12 hours, there are 12 ÷ 3 = 4 doubling periods. Population = 500 × 2⁴ = 500 × 16 = 8,000.

Question 34: Two numbers are in the ratio 5:9. If their difference is 32, what is the sum? (A) 100 (B) 110 (C) 112 (D) 120

100
110
112 (Correct answer)
120

Correct answer: 112

Numbers: 5k and 9k. Difference: 9k − 5k = 4k = 32 → k = 8. Numbers: 40 and 72. Sum = 112.

Let the numbers be 5k and 9k. 9k − 5k = 4k = 32. k = 8. Numbers: 5(8) = 40 and 9(8) = 72. Sum = 40 + 72 = 112.

GRE Mixed Quantitative Reasoning (Problem Solving) 1

A store offers a 30% discount on an item already marked down 20% from its original price.
If the final price is $112, what was the original price?
(A) $180
(B) $200
(C) $220
(D) $240

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