GRE Mathematics Preparation: Advanced Quantitative Reasoning
Welcome to this advanced-level GRE Mathematics preparation quiz, part of the September 2026 Series. This set of 50 questions is designed to challenge your quantitative reasoning skills, data analysis capabilities, and mathematical problem-solving agility—all essential for achieving a top score on the GRE.
This quiz covers a broad spectrum of topics frequently tested on the exam, including:
- Algebra and equation manipulation
- Geometry and spatial reasoning
- Data Analysis and interpretation
- Number Properties and integer logic
- Word Problems and real-world applications
- Quantitative Comparisons and strategic elimination
Each question is crafted to mirror the style and difficulty of the actual GRE, requiring not just rote memorization but a deep understanding of underlying mathematical principles. You will encounter multi-step problems, cleverly disguised traps, and questions that test your ability to reason efficiently under time pressure.
We recommend attempting each question without a calculator to simulate test-day conditions. After selecting your answer, read the detailed explanation to understand the underlying logic and identify any alternative solution paths. This active learning approach is the most effective way to solidify your skills and boost your confidence.
Good luck, and let’s begin your journey toward GRE mastery!
Q1. If ##x## and ##y## are integers and ##x^2 - y^2 = 37##, what is the value of ##x^2 + y^2##?
Since ##37## is prime, the only integer factor pairs for ##(x-y)(x+y)## are ##1## and ##37##. Solving ###x-y=1,\qquad x+y=37### gives ##x=19## and ##y=18##. Therefore, ###x^2+y^2=19^2+18^2=361+324=685.###
Q2. In a geometric sequence, the second term is 6 and the fifth term is 162. What is the sum of the first 8 terms?
The common ratio ##r## satisfies ###6r^3=162### so ##r^3=27## and ##r=3##. The first term is ##a=\frac{6}{3}=2##. Using the geometric-series formula, ###S_8=a\frac{r^8-1}{r-1}=2\frac{3^8-1}{3-1}=2\frac{6561-1}{2}=6560.###
Q3. A circle is inscribed in a square with a perimeter of 32. What is the area of the region inside the square but outside the circle?
The square has side length ##8##, so its area is ##8^2=64##. The inscribed circle has diameter ##8##, hence radius ##r=4## and area ##\pi r^2=16\pi##. Therefore, the required area is ###64-16\pi.###
Q4. If the average (arithmetic mean) of 5 consecutive even integers is 24, what is the largest of these integers?
Let the integers be ##n,n+2,n+4,n+6,n+8##. Their average is ###\frac{n+(n+2)+(n+4)+(n+6)+(n+8)}{5}=n+4=24.### Hence ##n=20##, so the largest integer is ##20+8=28##.
Q5. A data set consists of 10 numbers. If the sum of the 4 smallest numbers is 40 and the average of the 6 largest numbers is 20, what is the average of all 10 numbers?
The sum of the four smallest numbers is ##40##. The sum of the six largest numbers is ##6\times20=120##, so the total is ##40+120=160##. Hence the average of all ten numbers is ###\frac{160}{10}=16.###
Q6. If ##f(x) = 2x^2 - 3x + 1##, what is the value of ##f(f(1))##?
First, ###f(1)=2(1)^2-3(1)+1=0.### Then, ###f(0)=2(0)^2-3(0)+1=1.### Thus ##f(f(1))=f(0)=1##. The correct answer is ##1##.
Q7. In a class of 30 students, 18 play soccer and 14 play basketball. If 6 students play neither, how many students play both sports?
Using inclusion-exclusion, ###30=18+14-\text{Both}+6.### Thus ###30=38-\text{Both}### and therefore ##\text{Both}=8##.
Q8. What is the remainder when ##3^87## is divided by 5?
The powers of ##3## modulo ##5## repeat with period ##4##: ##3,4,2,1##. Since ##87\equiv3\pmod{4}##, the remainder corresponds to the third term of the cycle. Therefore, ###3^{87}\equiv2\pmod{5}.###
Q9. If the median of a set of 9 distinct integers is 15, what is the maximum possible value of the largest integer if the smallest integer is 1?
A median of ##15## for nine distinct integers means the fifth smallest number is ##15##. There is no upper-bound condition on the integers greater than ##15##, so the largest integer can be arbitrarily large. Therefore, the maximum possible value is unbounded.
Q10. A rectangular box has dimensions 3, 4, and 12. What is the distance from one corner to the opposite corner?
The space diagonal ##d## of a rectangular box satisfies ###d=\sqrt{3^2+4^2+12^2}=\sqrt{9+16+144}=\sqrt{169}=13.###
Q11. If ##x > 0## and ##x^(1/2) + x^(-1/2) = 3##, what is the value of ##x + x^(-1)##?
Square both sides of the equation: ###\left(x^{1/2}+x^{-1/2}\right)^2=9.### Expanding gives ###x+2+x^{-1}=9.### Therefore, ###x+x^{-1}=7.###
Q12. A group of 5 friends decides to split a restaurant bill of $120 equally. If two friends forget their wallets and the remaining friends cover their share, how much more does each paying friend contribute?
Originally, each friend pays ##\frac{120}{5}=24## dollars. With only three friends paying, each pays ##\frac{120}{3}=40## dollars. The additional contribution per paying friend is ###40-24=16\text{ dollars}.###
Q13. For how many integer values of ##k## is the equation ##x^2 + kx + 12 = 0## have two distinct integer roots?
For integer roots, ##k## must be the negative of the sum of two integer factors of ##12##. The positive factor pairs are ##(1,12),(2,6),(3,4)##. Each pair can also occur with both factors negative, producing the opposite sign of ##k##. Thus the number of distinct integer values is ###3\times2=6.###
Q14. If the probability of event A is 0.4 and the probability of event B is 0.6, and the events are independent, what is the probability that neither A nor B occurs?
The probability that ##A## does not occur is ##1-0.4=0.6##, and the probability that ##B## does not occur is ##1-0.6=0.4##. Since the events are independent, ###P(A^c\cap B^c)=0.6\times0.4=0.24.###
Q15. A cylindrical tank has a radius of 3 feet and a height of 10 feet. If water is poured in at a rate of 2 cubic feet per minute, how long will it take to fill the tank?
The cylinder volume is ###V=\pi r^2h=\pi(3)^2(10)=90\pi\text{ ft}^3.### At ##2\text{ ft}^3/\text{min}##, the filling time is ###t=\frac{90\pi}{2}=45\pi\text{ minutes}.###
Q16. If ##a## and ##b## are positive integers such that ##a^b = 64##, what is the maximum possible value of ##a + b##?
The possible pairs ##(a,b)## satisfying ##a^b=64## are ##(2,6),(4,3),(8,2),(64,1)##. Their sums are ##8,7,10,65## respectively. Hence the maximum is ##65##.
Q17. In a survey, 70% of people like coffee and 60% like tea. If 10% like neither, what percentage likes both?
Using inclusion-exclusion, ###100\%=70\%+60\%-\text{Both}+10\%.### Hence ###100\%=140\%-\text{Both}###, so ##\text{Both}=40\%##.
Q18. What is the sum of all the even integers from 1 to 100, inclusive?
The even integers are ##2,4,\ldots,100##, an arithmetic sequence with ##50## terms. Therefore, ###S=\frac{50}{2}(2+100)=25\times102=2550.###
Q19. If the slope of a line is -2 and it passes through the point (3, 4), what is the y-intercept of the line?
Using point-slope form, ###y-4=-2(x-3).### Simplifying, ###y=-2x+10.### Therefore, the ##y##-intercept is ##10##.
Q20. A bag contains 4 red marbles, 5 blue marbles, and 6 green marbles. If two marbles are drawn without replacement, what is the probability that both are blue?
The probability of drawing a blue marble first is ##\frac{5}{15}=\frac13##. Without replacement, the probability of a second blue marble is ##\frac{4}{14}=\frac27##. Thus ###P=\frac13\times\frac27=\frac{2}{21}.###
Q21. If ##x## and ##y## are positive integers and ##x + y = 10##, what is the maximum value of ##xy##?
For a fixed sum, the product is maximized when the two positive integers are as close as possible. Here they are ##5## and ##5##, so ###xy=5\times5=25.###
Q22. A train travels at a speed of 60 miles per hour. How many seconds does it take to travel 1 mile?
At ##60## miles per hour, the train travels ##1## mile per minute because ##60\text{ miles}/60\text{ minutes}=1\text{ mile/minute}##. Therefore, it takes ##60## seconds to travel ##1## mile.
Q23. What is the value of ##log_2(32) + log_3(81)##?
We have ##\log_2(32)=5## because ##2^5=32##, and ##\log_3(81)=4## because ##3^4=81##. Therefore, ###\log_2(32)+\log_3(81)=5+4=9.###
Q24. If the function ##g(x) = 3x - 2## and ##g(a) = 10##, what is the value of ##a##?
Set ###3a-2=10.### Adding ##2## to both sides gives ##3a=12##, and dividing by ##3## gives ###a=4.###
Q25. A circle has a circumference of 12π. What is the area of a sector with a central angle of 60 degrees?
From ##C=2\pi r=12\pi##, the radius is ##r=6##. The circle area is ##\pi r^2=36\pi##. A ##60^\circ## sector is ##\frac{60}{360}=\frac16## of the circle, so ###A=\frac16(36\pi)=6\pi.###
Q26. If the average of ##x## and ##y## is 12 and the average of ##y## and ##z## is 18, what is the average of ##x## and ##z##?
The given averages imply ###x+y=24,\qquad y+z=36.### Subtracting gives ##z-x=12##. This determines only the difference between ##x## and ##z##, not their sum ##x+z##. Therefore, their average ##\frac{x+z}{2}## cannot be determined.
Q27. How many different 3-digit numbers can be formed using the digits 1, 2, 3, 4, and 5 if repetition is not allowed?
There are ##5## choices for the hundreds digit, then ##4## choices for the tens digit, and ##3## choices for the units digit. Hence ###5\times4\times3=60### different three-digit numbers can be formed.
Q28. If ##x^2 + 6x + 9 = 0##, what is the value of ##x^3##?
The equation factors as ###(x+3)^2=0,### so ##x=-3##. Therefore, ###x^3=(-3)^3=-27.###
Q29. A rectangular garden has a length that is 3 meters more than twice its width. If the perimeter is 42 meters, what is the area of the garden?
Let ##w## be the width. Then the length is ##2w+3##. From the perimeter, ###2\bigl(w+(2w+3)\bigr)=42.### Thus ##3w+3=21## and ##w=6##. The length is ##2(6)+3=15##, so the area is ###A=6\times15=90\text{ m}^2.###
Q30. If the sum of the first ##n## terms of an arithmetic sequence is given by ##S_n = 2n^2 + 3n##, what is the 10th term?
The ##n##th term is ##a_n=S_n-S_{n-1}##. Hence ###a_{10}=S_{10}-S_9=\bigl(2(10)^2+3(10)\bigr)-\bigl(2(9)^2+3(9)\bigr)=230-189=41.###
Q31. A solution is 20% alcohol. How many liters of pure alcohol must be added to 10 liters of the solution to make a solution that is 50% alcohol?
The original solution contains ##0.20\times10=2## liters of alcohol. Let ##x## liters of pure alcohol be added. Then ###\frac{2+x}{10+x}=0.5.### Solving, ##2+x=5+0.5x##, so ##0.5x=3## and ###x=6\text{ liters}.###
Q32. If ##x## and ##y## are integers and ##x^2 + y^2 = 100##, what is the maximum possible value of ##x + y##?
Integer solutions to ##x^2+y^2=100## include ##(6,8),(8,6),(0,10),(10,0)## and their sign variants. The largest sum occurs for ##(6,8)## or ##(8,6)##, giving ###x+y=14.###
Q33. In a certain town, the population increases by 10% each year. If the current population is 10,000, what will the population be in 3 years?
After three years of ##10\%## annual growth, ###P=10000(1.1)^3=10000\times1.331=13310.### Therefore, the population will be ##13{,}310##.
Q34. What is the area of a triangle with vertices at (0,0), (4,0), and (0,3)?
The triangle is right-angled with perpendicular legs ##4## and ##3##. Therefore, ###A=\frac12\times4\times3=6.###
Q35. If ##f(x) = x^3 - 2x##, what is the value of ##f(-1)##?
Substitute ##x=-1## into the function: ###f(-1)=(-1)^3-2(-1)=-1+2=1.###
Q36. A car travels 120 miles at a speed of 40 miles per hour and then travels another 120 miles at a speed of 60 miles per hour. What is the average speed for the entire trip?
The first leg takes ##\frac{120}{40}=3## hours, and the second takes ##\frac{120}{60}=2## hours. Thus the total distance is ##240## miles and the total time is ##5## hours. Hence ###v_{\text{avg}}=\frac{240}{5}=48\text{ mph}.###
Q37. If the product of two consecutive positive integers is 156, what is the sum of these integers?
Let the consecutive positive integers be ##n## and ##n+1##. Then ###n(n+1)=156\quad\Rightarrow\quad n^2+n-156=0.### Factoring gives ##(n-12)(n+13)=0##, so the positive solution is ##n=12##. The integers are ##12## and ##13##, with sum ##25##.
Q38. A standard deck of 52 cards is shuffled. What is the probability of drawing a heart or a queen?
There are ##13## hearts and ##4## queens, but the queen of hearts is counted in both sets. Therefore, the number of favorable outcomes is ##13+4-1=16##, and ###P=\frac{16}{52}=\frac{4}{13}.###
Q39. If ##x > 1## and ##x^(1/3) = 2##, what is the value of ##x^(3/2)##?
From ##x^{1/3}=2##, we get ##x=2^3=8##. Therefore, ###x^{3/2}=8^{3/2}=(2^3)^{3/2}=2^{9/2}=2^4\sqrt2=16\sqrt2.###
Q40. A line passes through the points (2, 5) and (6, 9). What is the equation of the line in slope-intercept form?
The slope is ###m=\frac{9-5}{6-2}=\frac44=1.### Using point-slope form through ##(2,5)##, ###y-5=1(x-2),### which simplifies to ###y=x+3.###
Q41. If the sum of the interior angles of a polygon is 1080 degrees, how many sides does the polygon have?
For an ##n##-sided polygon, the sum of interior angles is ##(n-2)180^\circ##. Thus ###(n-2)180=1080.### Hence ##n-2=6## and ##n=8##.
Q42. A number is increased by 20% and then decreased by 20%. What is the net percentage change?
Let the original number be ##100##. After a ##20\%## increase it becomes ##120##, and after a ##20\%## decrease it becomes ###120(0.8)=96.### The net change is ##96-100=-4##, i.e. a ##4\%## decrease.
Q43. If ##x## and ##y## are positive integers such that ##x^y = 81##, what is the maximum possible value of ##x + y##?
The possible pairs ##(x,y)## satisfying ##x^y=81## are ##(3,4),(9,2),(81,1)##. Their sums are ##7,11,82## respectively. Therefore, the maximum is ##82##.
Q44. A cylindrical can has a volume of 100π cubic centimeters and a height of 4 centimeters. What is the radius of the can?
Using ##V=\pi r^2h##, ###100\pi=\pi r^2(4).### Dividing by ##4\pi## gives ##r^2=25##. Since a radius is positive, ###r=5\text{ cm}.###
Q45. If the average of 6 numbers is 15 and one of the numbers is 25, what is the average of the remaining 5 numbers?
The total sum of the six numbers is ##6\times15=90##. Removing ##25## leaves ##90-25=65## for the remaining five numbers. Their average is ###\frac{65}{5}=13.###
Q46. A bag contains 3 red balls and 2 blue balls. If two balls are drawn with replacement, what is the probability of drawing a red ball and then a blue ball?
With replacement, the two draws are independent. Thus ###P(\text{red then blue})=\frac35\times\frac25=\frac{6}{25}.###
Q47. If ##x^2 - 5x + 6 = 0##, what is the sum of the roots of the equation?
For ##ax^2+bx+c=0##, the sum of the roots is ##-\frac{b}{a}##. Here ##a=1## and ##b=-5##, so ###-\frac{-5}{1}=5.###
Q48. A right triangle has legs of length 5 and 12. What is the length of the altitude to the hypotenuse?
The hypotenuse is ###c=\sqrt{5^2+12^2}=\sqrt{169}=13.### The triangle area is ##\frac12(5)(12)=30##. If ##h## is the altitude to the hypotenuse, then ###\frac12(13)h=30,### so ###h=\frac{60}{13}.###
Q49. If the ratio of the areas of two similar triangles is 9:16, what is the ratio of their corresponding perimeters?
For similar figures, the ratio of areas equals the square of the ratio of corresponding lengths. Therefore, the perimeter ratio is ###\sqrt{\frac{9}{16}}=\frac34.###
Q50. A sequence is defined by ##a_1 = 2## and ##a_n = 3a_(n-1) + 1## for ##n > 1##. What is the value of ##a_4##?
Using ##a_1=2## and ##a_n=3a_{n-1}+1##, ###a_2=3(2)+1=7,\qquad a_3=3(7)+1=22,\qquad a_4=3(22)+1=67.###