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. The questions cover a broad spectrum of topics frequently tested on the GRE, including algebra, geometry, number properties, probability, statistics, and data interpretation.
Each question is crafted to mirror the style and difficulty of the most challenging problems you will encounter on the exam. You will need to apply multi-step reasoning, recognize subtle patterns, and avoid common traps. Use this quiz to identify your strengths, pinpoint areas needing further review, and build the stamina required for the quantitative sections of the GRE.
Read each question carefully, consider all answer choices, and select the best possible answer. Detailed explanations are provided for every question to help you understand the underlying concepts and refine your test-taking strategies. Good luck!
Q1. If the average (arithmetic mean) of 11 consecutive integers is 12, what is the sum of the smallest and largest integers in the set?
For 11 consecutive integers, the average is the middle, or 6th, integer. Thus the 6th integer is ##12##. The smallest integer is ##12-5=7## and the largest is ##12+5=17##. Therefore, their sum is ##7+17=24##.
Q2. In a certain sequence, each term after the first is obtained by multiplying the previous term by 3 and then subtracting 1. If the 3rd term is 23, what is the 1st term?
Let the first term be ##x##. Then the second term is ##3x-1##, and the third term is ##3(3x-1)-1=9x-4##. Since the third term is 23, ##9x-4=23##. Thus ##9x=27## and ##x=3##.
Q3. A rectangular box has a volume of 144 cubic inches. If the length is 2 inches more than the width, and the height is 3 inches less than the width, what is the width of the box?
Let the width be ##w## inches. Then the length is ##w+2## and the height is ##w-3##. The volume condition gives ##w(w+2)(w-3)=144##. Testing ##w=6## gives ##6(8)(3)=144##. Therefore, the width is ##6## inches.
Q4. If ##x## and ##y## are positive integers such that ##x+y=10## and ##x^2+y^2=68##, what is the value of ##xy##?
Using ##(x+y)^2=x^2+y^2+2xy##, we have ##10^2=68+2xy##. Thus ##100=68+2xy##, so ##2xy=32## and therefore ##xy=16##.
Q5. A circle is inscribed in a square with a perimeter of 32. What is the area of the circle?
The square has side length ##32/4=8##. The diameter of the inscribed circle is therefore ##8##, so its radius is ##r=4##. Hence the area is ##A=\pi r^2=\pi(4)^2=16\pi##.
Q6. If the probability of event A occurring is 0.4 and the probability of event B occurring is 0.6, and the events are independent, what is the probability that both A and B occur?
For independent events, ##P(A\cap B)=P(A)P(B)##. Therefore, ##P(A\cap B)=0.4\times0.6=0.24##.
Q7. A data set consists of 5 numbers. The mean is 10 and the median is 12. If the smallest number is 4 and the largest number is 18, what is the sum of the two remaining numbers?
The sum of all 5 numbers is ##5\times10=50##. The smallest number, median, and largest number have sum ##4+12+18=34##. Therefore, the two remaining numbers have sum ##50-34=16##.
Q8. 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##. Therefore, ##f(f(1))=f(0)=2(0)^2-3(0)+1=1##.
Q9. In a right triangle, the length of one leg is 5 and the length of the hypotenuse is 13. What is the length of the other leg?
By the Pythagorean theorem, ##5^2+b^2=13^2##. Hence ##25+b^2=169##, so ##b^2=144## and ##b=12##.
Q10. If ##3^{x+2}=81##, what is the value of ##x##?
Since ##81=3^4##, the equation becomes ##3^{x+2}=3^4##. Therefore, ##x+2=4## and ##x=2##.
Q11. A group of 6 friends is going to sit in a row of 6 seats. If two specific friends must sit next to each other, how many different seating arrangements are possible?
Treat the two specified friends as one block. There are then 5 units to arrange, giving ##5!=120## arrangements. The two friends can switch places within their block in ##2!=2## ways. Therefore, the total is ##5!\times2!=120\times2=240##.
Q12. If the ratio of the areas of two similar triangles is 9:16, what is the ratio of their corresponding perimeters?
For similar triangles, the ratio of areas is the square of the ratio of corresponding lengths. Thus the perimeter ratio is ##\sqrt{\frac{9}{16}}=\frac34##, or ##3:4##.
Q13. What is the units digit of ##7^{2026}##?
The units digits of powers of 7 repeat in the cycle ##7,9,3,1##, which has period 4. Since ##2026\equiv2\pmod4##, the required units digit is the second number in the cycle, namely ##9##.
Q14. If ##x## is a positive integer and ##x^2## is divisible by 12, what is the smallest possible value of ##x##?
Since ##12=2^2\cdot3##, the square ##x^2## must contain at least ##2^2## and a factor of 3. Thus ##x## must contain both a factor of 2 and a factor of 3. The smallest possibility is ##x=6##, and indeed ##6^2=36## is divisible by 12.
Q15. A train travels at a speed of 60 miles per hour. How many feet does it travel in 5 seconds? (1 mile = 5280 feet)
Convert the speed to feet per hour: ##60\times5280=316800## feet per hour. Since one hour contains ##3600## seconds, the speed is ##316800/3600=88## feet per second. In 5 seconds, the train travels ##88\times5=440## feet.
Q16. If the average of ##x## and ##y## is 15, and the average of ##y## and ##z## is 20, what is the average of ##x## and ##z##?
The conditions give ##\frac{x+y}{2}=15## and ##\frac{y+z}{2}=20##, so ##x+y=30## and ##y+z=40##. Subtracting gives ##z-x=10##, but this does not determine ##x+z##. Therefore, the average ##\frac{x+z}{2}## cannot be determined.
Q17. In a coordinate plane, what is the distance between the points ##(3,-2)## and ##(-1,4)##?
Using the distance formula, ##d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}##. Thus ##d=\sqrt{(-1-3)^2+(4-(-2))^2}=\sqrt{(-4)^2+6^2}=\sqrt{16+36}=\sqrt{52}##.
Q18. If ##a## and ##b## are integers such that ##a/b=0.125##, what is the value of ##a+b##?
Since ##0.125=\frac18##, we have ##\frac{a}{b}=\frac18##. Thus ##a=k## and ##b=8k## for some nonzero integer ##k##. Therefore, ##a+b=9k##, which depends on ##k## and cannot be uniquely determined.
Q19. A cylindrical tank has a radius of 3 feet and a height of 10 feet. If the tank is half full, what is the volume of the water in the tank?
The full volume is ##V=\pi r^2h=\pi(3)^2(10)=90\pi## cubic feet. When the tank is half full, the volume of water is ##\frac{90\pi}{2}=45\pi## cubic feet.
Q20. 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 nth term is ##a_n=S_n-S_{n-1}##. Thus ##a_{10}=S_{10}-S_9##. We have ##S_{10}=2(10)^2+3(10)=230## and ##S_9=2(9)^2+3(9)=189##. Therefore, ##a_{10}=230-189=41##.
Q21. If ##x## is an integer and ##|x-3|<2##, what is the sum of all possible values of ##x##?
The inequality ##|x-3|<2## is equivalent to ##-2<x-3<2##. Adding 3 gives ##1<x<5##. Thus ##x=2,3,4##, whose sum is ##2+3+4=9##.
Q22. A bag contains 4 red marbles, 6 blue marbles, and 10 green marbles. If two marbles are drawn at random without replacement, what is the probability that both are blue?
There are 20 marbles in total. The probability of drawing two blue marbles is ##\frac{6}{20}\times\frac{5}{19}=\frac{30}{380}=\frac{3}{38}##.
Q23. If the function ##g(x)=x^3-x##, what is the value of ##g(-2)##?
Substitute ##x=-2##: ##g(-2)=(-2)^3-(-2)=-8+2=-6##.
Q24. In a triangle, the measures of the angles are in the ratio 2:3:4. What is the measure of the largest angle?
The ratio contains ##2+3+4=9## equal parts. Since the angles of a triangle sum to ##180^\circ##, each part measures ##180^\circ/9=20^\circ##. The largest angle is ##4\times20^\circ=80^\circ##.
Q25. If ##2x+3y=12## and ##4x-y=10##, what is the value of ##x+y##?
From ##4x-y=10##, we get ##y=4x-10##. Substitute into ##2x+3y=12##: ##2x+3(4x-10)=12##. Thus ##14x-30=12##, so ##14x=42## and ##x=3##. Then ##y=2##, giving ##x+y=5##.
Q26. A certain number when divided by 7 leaves a remainder of 5. What is the remainder when twice the number is divided by 7?
Let the number be ##n=7k+5##. Then ##2n=14k+10=7(2k+1)+3##. Therefore, the remainder is ##3##.
Q27. If the area of an equilateral triangle is ##9\sqrt{3}##, what is the length of its side?
The area of an equilateral triangle with side ##s## is ##A=\frac{\sqrt3}{4}s^2##. Thus ##\frac{\sqrt3}{4}s^2=9\sqrt3##. Dividing by ##\sqrt3## gives ##\frac{s^2}{4}=9##, so ##s^2=36## and ##s=6##.
Q28. If ##\log_{10}(x)=2.5##, what is the value of ##\log_{10}(x^2)##?
Using ##\log_{10}(x^2)=2\log_{10}(x)##, we obtain ##\log_{10}(x^2)=2(2.5)=5.0##.
Q29. A car travels 120 miles at a speed of 40 mph and then travels another 120 miles at a speed of 60 mph. What is the average speed for the entire trip?
The first 120 miles take ##120/40=3## hours, and the second 120 miles take ##120/60=2## hours. The total distance is ##240## miles and the total time is ##5## hours. Thus the average speed is ##240/5=48## mph.
Q30. If ##x^2+y^2=25## and ##xy=12##, what is the value of ##(x+y)^2##?
Using ##(x+y)^2=x^2+y^2+2xy##, we obtain ##(x+y)^2=25+2(12)=25+24=49##.
Q31. In a class of 30 students, 18 play basketball and 15 play soccer. If 5 students play neither, how many students play both?
The number who play at least one sport is ##30-5=25##. By inclusion-exclusion, ##|B\cup S|=|B|+|S|-|B\cap S|##. Hence ##25=18+15-|B\cap S|##, so ##|B\cap S|=8##.
Q32. What is the slope of a line that is perpendicular to the line ##3x-2y=6##?
Rewrite the equation as ##y=\frac32x-3##, so the given line has slope ##\frac32##. A perpendicular line has slope equal to the negative reciprocal, ##-\frac23##.
Q33. If the product of two consecutive positive odd integers is 143, what is the sum of the two integers?
Let the integers be ##n## and ##n+2##. Then ##n(n+2)=143##, so ##n^2+2n-143=0##. Factoring gives ##(n-11)(n+13)=0##. Since ##n## is positive, ##n=11##. The integers are ##11## and ##13##, whose sum is ##24##.
Q34. A standard die is rolled twice. What is the probability that the sum of the two rolls is 7?
There are ##6\times6=36## equally likely ordered outcomes. Six give a sum of 7: ##(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)##. Thus the probability is ##\frac{6}{36}=\frac16##.
Q35. If the volume of a cube is 64 cubic centimeters, what is the total surface area of the cube?
If the cube has side length ##s##, then ##s^3=64##, so ##s=4## cm. Its total surface area is ##6s^2=6(4^2)=6(16)=96## square cm.
Q36. If ##f(x)=3x-2## and ##g(x)=x^2+1##, what is the value of ##f(g(2))##?
First, ##g(2)=2^2+1=5##. Then ##f(g(2))=f(5)=3(5)-2=15-2=13##.
Q37. If ##x## is a positive integer such that ##x!## is divisible by ##2^{10}##, what is the smallest possible value of ##x##?
The exponent of 2 in ##x!## is ##\left\lfloor\frac{x}{2}\right\rfloor+\left\lfloor\frac{x}{4}\right\rfloor+\left\lfloor\frac{x}{8}\right\rfloor+\cdots##. For ##x=11##, the exponent is ##5+2+1=8##. For ##x=12##, it is ##6+3+1=10##. Hence ##12!## is divisible by ##2^{10}##, and 12 is the smallest possible value.
Q38. In a geometric sequence, the 2nd term is 6 and the 5th term is 162. What is the 3rd term?
Let the first term be ##a## and common ratio be ##r##. Then ##ar=6## and ##ar^4=162##. Dividing gives ##r^3=27##, so ##r=3##. Therefore, the third term is ##ar^2=(ar)r=6(3)=18##.
Q39. If the average of 5 numbers is 20, and one of the numbers is removed, the average of the remaining 4 numbers is 18. What is the number that was removed?
The original sum is ##5\times20=100##. The remaining four numbers have sum ##4\times18=72##. Therefore, the removed number is ##100-72=28##.
Q40. What is the value of ##(\sqrt2+\sqrt8)^2##?
Since ##\sqrt8=2\sqrt2##, we have ##(\sqrt2+\sqrt8)^2=(3\sqrt2)^2=9\cdot2=18##.
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 the interior angles is ##(n-2)180^\circ##. Thus ##(n-2)180^\circ=1080^\circ##, so ##n-2=6## and ##n=8##.
Q42. If ##x## and ##y## are integers such that ##x^2-y^2=45##, what is the maximum possible value of ##x+y##?
Factor the equation as ##(x-y)(x+y)=45##. To maximize ##x+y##, choose ##x-y=1## and ##x+y=45##. These give ##x=23## and ##y=22##, so the maximum possible value of ##x+y## is ##45##.
Q43. A circle has a circumference of ##12\pi##. What is the area of a sector with a central angle of 60 degrees?
From ##2\pi r=12\pi##, we get ##r=6##. The full circle has area ##\pi r^2=36\pi##. A ##60^\circ## sector is ##\frac{60}{360}=\frac16## of the circle, so its area is ##\frac16(36\pi)=6\pi##.
Q44. If ##5^x=125##, what is the value of ##2^{2x}##?
Since ##125=5^3##, we have ##x=3##. Therefore, ##2^{2x}=2^{2(3)}=2^6=64##.
Q45. A set of data has a mean of 50 and a standard deviation of 5. If each data point is multiplied by 2 and then 10 is added, what is the new standard deviation?
Multiplying every data value by 2 multiplies the standard deviation by ##|2|=2##. Adding 10 to every value does not change the standard deviation. Thus the new standard deviation is ##5\times2=10##.
Q46. If the probability of an event is 0.15, what are the odds in favor of the event?
The probability that the event does not occur is ##1-0.15=0.85##. Therefore, the odds in favor are ##0.15:0.85=15:85=3:17##.
Q47. If ##x+\frac1x=5##, what is the value of ##x^2+\frac{1}{x^2}##?
Squaring ##x+\frac1x=5## gives ##x^2+2+\frac{1}{x^2}=25##. Therefore, ##x^2+\frac{1}{x^2}=25-2=23##.
Q48. A rectangular garden has a length that is 3 meters more than twice its width. If the area of the garden is 65 square meters, what is the width?
Let the width be ##w##. Then the length is ##2w+3##. The area condition gives ##w(2w+3)=65##, or ##2w^2+3w-65=0##. Factoring, ##(2w+13)(w-5)=0##. Since the width is positive, ##w=5## meters.
Q49. If the median of a set of 9 consecutive integers is 15, what is the average of the set?
For an odd number of consecutive integers, the middle integer is both the median and the arithmetic mean. Since the median is ##15##, the average is also ##15##.
Q50. If ##a## and ##b## are positive integers and ##a^b=64##, what is the maximum possible value of ##a+b##?
The positive-integer pairs satisfying ##a^b=64## include ##(64,1)##, ##(8,2)##, ##(4,3)##, and ##(2,6)##. Their sums are ##65,10,7,8## respectively. Therefore, the maximum possible value is ##65##.