15 Hardest GRE Math Questions with Answers
The Quantitative Reasoning section of the GRE, also known as GRE math, assesses your problem-solving skills and data analysis abilities. This GRE quant section covers a broad spectrum of topics, ranging from arithmetic and algebra to geometry and data interpretation. While the GRE math section is undeniably challenging, with a strategic approach and focused practice, you can achieve your target score.
This blog is your guide to acing the GRE math. We’ve compiled 15 of the most difficult GRE math practice questions. These challenging questions are drawn from our comprehensive GRE prep resources, including the GRE Premium Prep book and our online GRE prep course materials. Each question comes with a detailed explanation, guiding you through the thought process needed to solve even the trickiest problems.
15 Hardest GRE Math Questions Solved
Question 1: If set A contains only odd integers and the median of set A is 11, then, compared to the mean (average) of set A, is the median greater than, equal to, or less than the mean?
Answer: Less than
Explanation: The median is the “middle” number in a set when ordered from least to greatest. Since the set only contains odd numbers, the mean (average) must also be odd. However, the mean is influenced by all the numbers in the set, whereas the median is not necessarily the average of the two middle numbers (if there are even middle numbers). Because of this, the mean will tend to be slightly larger than the median in an odd set, so the median is less than the mean.
Question 2: If x is an integer and 3x2 + 11x - 4 = 0, what is the value of x + 4?
Answer: -1
Explanation: This question requires solving a quadratic equation. We could use the quadratic formula, but a quicker approach might be to try to guess which integer value of x would satisfy the equation.
If we plug in x = -1, we get 3(-1)2 + 11(-1) - 4 = 3 - 11 - 4 = -12. Since the opposite of -12 is 12, and we need the value of x + 4, then x + 4 = -1 + 4 = 3.
Question 3: If set B contains only even integers and the range of set B is 10, what is the minimum number of elements in set B?
Answer: 5
Explanation: The range is the difference between the highest and lowest values in a set. Since the set only contains even numbers, the difference between any two consecutive even numbers is 2. Therefore, to get a range of 10, we would need at least 5 even numbers (i.e., a difference of 2 between each number, resulting in a total difference of 10).
Question 4: If 2a + 5b = 21 and a + 3b = 12, what is the value of a - b?
Answer: -3
Explanation: This question requires solving a system of linear equations. We can eliminate a by adding the equations together when the coefficients have opposite signs.
Adding the equations together gives us 3a + 8b = 33. Dividing both sides by 3 gives a + (8/3)b = 11. We can now subtract the second equation (a + 3b = 12) from this equation to isolate b. Subtracting the equations gives us (8/3)b - 3b = 11 - 12, which simplifies to (-5/3)b = -1. Therefore, b = 3/5. We can now plug this value back into either original equation to solve for a. Using the first equation, 2a + 5(3/5) = 21. Solving for a gives us a = 6. Finally, we can calculate a - b = 6 - (3/5) = 3 - (3/5) = -3/5.
Question 5: Two trains leave a station at the same time, travelling in opposite directions. Train A travels at a speed of 60 mph, and Train B travels 20 mph slower than Train A. If, after 3 hours, they are 420 miles apart, what was the original speed of Train B?
- 60 mph
- 40 mph
- 80 mph
- 120 mph
Answer: (2) 40 mph
Explanation: Let x be the speed of Train B. The relative speed between the two trains is the sum of their individual speeds (since they are travelling in opposite directions), which is 60 + x mph. The total distance covered after 3 hours is the product of relative speed and time: (60 + x) * 3 hours. We are given that this distance is 420 miles. Converting this to an equation, we get (60 + x) * 3 = 420. Simplifying the left side gives 180 + 3x = 420. Solving for x (subtracting 180 from both sides and then dividing both sides by 3) gives x = 80 mph. However, the question asks for the original speed of Train B, which is 20 mph slower than Train A (60 mph), so the answer is 60 mph - 20 mph = 40 mph.
Question 6: A store advertises a 20% discount on a jacket with an original price of p dollars. What is the discounted price of the jacket?
Answer: 0.8p
Explanation: A discount rate of 20% means the customer pays 100% - 20% = 80% of the original price. To find the discounted price, multiply the original price (p) by 80%, which is the same as multiplying by 0.8: 0.8p.
Question 7: A bag contains 5 red marbles and 7 blue marbles. If you draw two marbles without replacement (meaning you don’t put the first marble back in after drawing it), what is the probability of drawing two red marbles?
Answer: 5/63
Explanation: Probability is the likelihood of an event happening. Since we are drawing without replacement, the number of marbles available for the second draw will depend on what you drew in the first draw. Without a replacement, there are two possibilities:
- First draw red: There are 5 red marbles initially, and after drawing one, there will be 4 left. There are still 7 blue marbles remaining. So, the probability of drawing a red marble first and then another red marble is (5/12) * (4/11).
- First draw blue: There are 7 blue marbles initially, and after drawing one, there will be 6 left. There are still 5 red marbles remaining. So, the probability of drawing a blue marble first, then a red marble (which doesn’t contribute to getting two red marbles) is irrelevant to this question because we only care about the probability of getting two red marbles.
Since these are two independent events (the outcome of the first draw doesn’t affect the second draw), to get the overall probability, we need to add the probabilities of these two scenarios: (5/12) * (4/11) + (probability of an irrelevant scenario, which is 0). This simplifies to 5/63.
Question 8: Let a, b, and c be integers such that (a + b)(a + c) = 11. What is the value of a2 + b2 + c2 ?
- 24
- 26
- 28
- 30
Answer: (1) 24
Explanation: We can try to factor 11 to see if it can be expressed as the sum of two integers. Since 11 = 1 + 10 or 11 = -1 - 10, we can see that a + b = 1 and a + c = 10, or a + b = -1 and a + c = -10. Squaring both equations in each scenario and adding them will result in a2 + b2 + c2 = 24.
Question 9: A rectangular picture frame has a perimeter of 34 cm. If the length is 3 cm longer than the width, what is the width of the picture frame?
- 5 cm
- 8 cm
- 7 cm
- 9 cm
Answer: (3) 7 cm
Explanation: Let x represent the width of the frame. The length would then be x + 3 cm. The perimeter is the total length of all the sides of the shape added together. In this case, perimeter = 2(width) + 2(length) = 2x + 2(x + 3) = 34. Simplifying the equation, we get 4x = 28, and x = 7 cm.
Question 10: A chemist needs to create a 40% alcohol solution by mixing a 90% alcohol solution and a 10% alcohol solution. How many litres of the 90% solution should be mixed with 3 litres of the 10% solution to create the desired mixture?
- 3 litres
- 2 litres
- 4 litres
- 2.5 litres
Answer: (2) 2 litres
Explanation: Let x represent the number of litres of the 90% solution needed. The total amount of alcohol in the final mixture will come from both solutions. The 90% solution will contribute 0.9x litres of alcohol, and the 10% solution will contribute 0.1(3) = 0.3 litres of alcohol. Since the final solution needs to be 40% alcohol and the total volume will be x + 3 litres, we can set up the equation: 0.9x + 0.3 = 0.4(x + 3). Solving for x, we find that x = 2 litres.
Question 11: The price of a used car decreases by 15%. If the original price was $18,000, what is the new price of the car?
- $15,300
- $16,200
- $1,530
- $2,700
- $12,600
Answer: (1) $15,300
Explanation: A 15% decrease means the new price is 100% - 15% = 85% of the original price. To find the new price, multiply the original price ($18,000) by 85%, which is the same as multiplying by 0.85: $18,000 x 0.85 = $15,300.
Question 12: A square has a side length of x. What is the area of the square?
- X
- x2
- 2x
- 4x
- x + 4
Answer: x2
Explanation: The area of a square is calculated by squaring the side length. Area = x2. In this case, the side length is x, so the area of the square is x2.
Question 13: The table below shows the data for two delivery companies, Speedy and Reliable, delivering packages over different distances. At what distance (approximately) will Speedy take the same amount of time as Reliable to deliver a package?
- 122 miles
- 125 miles
- 124 miles
- 123 miles
Distance (miles) |
Speedy (hours) |
Reliable (hours) |
---|---|---|
50 |
2 |
3 |
100 |
4 |
5 |
150 |
6 |
7.5 |
Answer: (2) 125 miles
Explanation: Analyse the table and look for trends. You can see that Speedy is generally faster than Reliable for all distances listed. To find the distance where they take the same time, you can estimate the rate of change (speed) for each company. Since the table shows constant rates of change (linear relationship), graphing the data points might be helpful. Plot distance on the x-axis and time on the y-axis. Although not entirely necessary, by eyeing the graph (which you can sketch yourself), you’ll see the lines for Speedy and Reliable will likely intersect at some point. The answer choices provide a range of distances.
Pick a value in the middle, like 125 miles. Calculate the time it would take Speedy and Reliable to travel 125 miles using their respective rates of change from the table (Speedy: 6 hours for 150 miles, so for 125 miles it would be less than 6 hours; Reliable: 7.5 hours for 150 miles, so for 125 miles it would be a bit less than 7.5 hours). If the estimated times are close to that range, then 125 miles is a reasonable answer.
Question 14: If f(x) = x3 – 3x2 – 4x + 12, what is the sum of the roots of the equation f(x) = 0?
Answer: --31 = 3
Explanation: To find the sum of the roots, we can use Vieta’s formulas, which state that the sum of the roots of a cubic equation ax3 + bx2 + cx + d = 0 is -ba. In this case, a = 1, b = -3, c = -4, and d = 12, so the sum of the roots is --31 = 3.
Question 15: A machine can package 10 boxes in 2 hours. If the speed is increased by 20%, how many boxes can be packaged in 4 hours at the new speed?
- 20
- 24
- 30
- 36
- 40
Answer: (2) 24
Explanation: The original rate is 10 boxes / 2 hours = 5 boxes/hour. The new rate is 1.2 * 5 boxes/hour = 6 boxes/hour. In 4 hours at the new speed, the machine can package 6 boxes/hour * 4 hours = 24 boxes.
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Suggested: Importance of Knowing Formulas for the GRE Quant Section
From the Desk of Yocket
While the GRE math section stays away from super-advanced topics, it can sometimes combine seemingly straightforward concepts in a way that requires a multi-step approach or a creative leap in logic. These questions might involve manipulating equations in innovative ways, using GRE probability in non-standard scenarios, or recognising subtle connections between seemingly separate pieces of information.
Some questions might be considered “hard” because they tap into areas not explicitly covered in most prep materials. While the core concepts tested on the GRE are well-defined, there might be occasional questions that require you to think outside the box and apply your general problem-solving skills in a new way.
Another factor is the time pressure of the GRE exam. Straightforward questions can become difficult if they require a lot of calculations or if they test a less familiar concept that you need to reason through under pressure. In these cases, strong estimation skills and the ability to identify efficient solution methods become crucial. Additionally, yocket GRE prep offers the latest GRE practice questions with answers to enhance your GRE preparation.