1 September 2024

8 minutes read

# 10 Most Asked GRE Math Practice Questions with Explanations

### Key Takeaways

- Focus on practicing the most frequently asked GRE math questions to boost your confidence.
- Strengthen your fundamentals to tackle diverse GRE math problems effectively.
- Develop a strategic approach to problem-solving instead of relying on memorization.
- Manage your time wisely during the test to avoid unnecessary stress and errors.

Answering the quantitative reasoning section of the GRE exam – correctly is a make-or-break moment for many aspiring graduate school candidates. However, with the right GRE prep strategy, you can approach test day with confidence. One proven technique is to focus your practice on the most frequently asked GRE math questions, covering everything from quantitative comparison to data interpretation.

In this easy guide, we’ll cover the top 10 most asked GRE math problem practice questions, complete with detailed explanations that go beyond simply revealing the correct answer.

We aim to equip you with a deep understanding of the underlying quantitative concepts, problem-solving techniques, and thought processes required to tackle questions like finding that x satisfies the equation or comparing the two quantities. With this knowledge, you’ll be ready to excel on the GRE prep course and the actual test, setting you on the path to your dream graduate school.

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## What to Expect in GRE Math Practice Questions?

When it comes to the GRE math questions, you can expect a diverse range of challenges that will test your quantitative skills to the core. From grappling with numeric entry questions and great quantitative comparison conundrums to navigating the intricacies of math problems and question types, the GRE quant section is designed to push your problem-solving abilities to new heights.

To truly improve your GRE score and be ready for the GRE, it becomes important to familiarise yourself with these types of questions through practice questions with answers and detailed explanations. Investing in high-quality GRE prep course materials, such as practice tests and sample questions, can be invaluable in your GRE preparation journey.

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## Expert’s Tips To Answer GRE Math Questions?

The quantitative reasoning section of the GRE is a true test of your problem-solving abilities and mathematical prowess. With a wide range of question types, from data interpretation to quantitative comparison, it’s essential to be prepared with a well-rounded skillset and a calm, focused mindset.

Now, let’s explore some expert tips to help you conquer the GRE math questions:

### Strengthen Your Fundamentals

Ensure you have a solid grasp of the foundational mathematical concepts covered on the GRE, including arithmetic, algebra, geometry, and data analysis. Revisiting these basics can help you approach questions with confidence and clarity.

### Practice with Authentic GRE Questions

Incorporate official GRE practice questions and tests into your preparation routine. These resources will familiarize you with the unique phrasing, format, and difficulty level of the actual GRE questions. It will help you develop the necessary critical thinking and problem-solving skills.

### Develop a Strategic Approach

Instead of relying solely on memorization, cultivate a strategic mindset. Learn to identify the core concepts being tested in each question and apply relevant problem-solving techniques, such as plugging in values, working backwards, or visualizing the problem.

### Manage Your Time Effectively

Time management skills are crucial in the GRE. Practice pacing yourself and learn to identify questions that may require more time, allowing you to prioritize and allocate your time accordingly during the actual test.

### Check Your Work

While working under time pressure, it’s easy to make careless mistakes. Develop a habit of reviewing your work, double-checking calculations, and ensuring your answer aligns with the question being asked.

### Stay Calm and Focused

The GRE can be mentally taxing, but maintaining a positive mindset and staying focused is essential. Take breaks when needed, and remember to breathe deeply and stay centred during the test.

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## 5 GRE Quantitative Comparison Questions

Practicing GRE quantitative comparison questions is invaluable for your preparation. These questions challenge your ability to analyse and compare two quantities, a skill that is essential for success on the quantitative reasoning section.

Here are 5 mostly asked GRE quantitative practice questions:

### If x and y are positive integers, is x^2 > y^3

To determine if x^2 > y^3 for positive integers x and y, we need to consider different cases:

- If x = 1, then x^2 = 1. For any positive y, y^3 > 1, so x^2 is not greater than y^3.
- If y = 1, then y^3 = 1. For any x > 1, x^2 > 1, so x^2 is greater than y^3.
- If both x and y are greater than 1, the cube function (y^3) grows faster than the square function (x^2). As x and y increase, y^3 will eventually become greater than x^2.

Therefore, without specific values of x and y, the relationship between x^2 and y^3 cannot be determined for all positive integers x and y. It depends on the values assigned to x and y.

### If a and b are integers, is a^2 + b^2 > 2ab?

To determine if a^2 + b^2 > 2ab for integers a and b, we can consider the following cases:

- If a = b = 0, then a^2 + b^2 = 0 and 2ab = 0. Therefore, a^2 + b^2 is not greater than 2ab.
- If a = 0 and b β 0 (or b = 0 and a β 0), then a^2 + b^2 = b^2 (or a^2) and 2ab = 0. In this case, a^2 + b^2 is greater than 2ab.
- If a and b are both non-zero, then a^2 + b^2 > 2ab because (a + b)^2 = a^2 + 2ab + b^2 > a^2 + b^2. This means that a^2 + b^2 is always greater than 2ab when a and b are non-zero integers.

Therefore, for integers a and b, a^2 + b^2 > 2ab is true, except when a = b = 0, in which case a^2 + b^2 = 2ab.

### If x and y are positive integers, is x + y > xy?

To determine if x + y > xy for positive integers x and y, we can consider the following cases:

- If x = 1 or y = 1, then xy = x or xy = y, respectively. In both cases, x + y is always greater than xy.
- If both x and y are greater than 1, we can analyze the inequality by substituting some values:
- For x = 2 and y = 2, we have 2 + 2 = 4 > 2 Γ 2 = 4
- For x = 3 and y = 3, we have 3 + 3 = 6 > 3 Γ 3 = 9

In general, for x, y > 1, the product xy will be greater than the sum x + y.

Therefore, for positive integers x and y, the inequality x + y > xy is true only when at least one of x or y is equal to 1. If both x and y are greater than 1, then x + y is not greater than xy.

In summary, x + y > xy is true if x = 1 or y = 1, but false if both x and y are greater than 1.

### If a and b are positive integers, is a^2 + b^2 > 2ab?

To determine if a^2 + b^2 > 2ab for positive integers a and b, we can consider the following cases:

- If a = b = 1, then a^2 + b^2 = 2 and 2ab = 2. So, a^2 + b^2 is not greater than 2ab.
- If either a or b is 1, say a = 1 and b > 1, then a^2 + b^2 = 1 + b^2 and 2ab = 2b. Since b^2 > 2b for b > 1, we have a^2 + b^2 > 2ab.
- If both a and b are greater than 1, then (a + b)^2 = a^2 + 2ab + b^2 > a^2 + b^2. This means a^2 + b^2 > 2ab.

Therefore, for positive integers a and b, the inequality a^2 + b^2 > 2ab holds true except when a = b = 1, in which case a^2 + b^2 = 2ab.

### If x and y are positive integers, is x^2 + y^2 > 2xy?

To determine if x^2 + y^2 > 2xy for positive integers x and y, we need to consider different cases:

- If x = 1 or y = 1, then one of the terms on the left-hand side (x^2 or y^2) becomes 1. In this case, x^2 + y^2 > 2xy always holds true.

For example, if x = 1 and y = 2, we have 1 + 2^2 = 5 > 2(1)(2) = 4.

- If both x and y are greater than 1, we can use the algebraic identity (x + y)^2 = x^2 + 2xy + y^2. Rearranging, we get x^2 + y^2 = (x + y)^2 – 2xy > 2xy.

Therefore, for x, y > 1, we have x^2 + y^2 > 2xy.

In summary, the inequality x^2 + y^2 > 2xy holds true for all positive integers x and y, regardless of their values.

The key points are:

- If x = 1 or y = 1, the inequality is automatically satisfied.
- If x, y > 1, we can use the algebraic identity (x + y)^2 = x^2 + 2xy + y^2 to prove the inequality.

So, for positive integers x and y, x^2 + y^2 is always greater than 2xy.

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## 5 GRE Numeric Entry Questions

Practicing GRE numeric entry questions is invaluable because these questions test your ability to perform precise calculations and provide an exact numerical answer, a skill that is essential in quantitative fields and research-based disciplines.

Here are five examples of GRE numeric entry questions to help you develop this crucial skill:

### If x + 3y = 12 and x – y = 4, what is the value of x?

To find the value of x given the equations:

- π₯+3π¦=12x+3y=12
- π₯βπ¦=4xβy=4

Follow these steps:

- Solve the second equation for π₯x:

π₯=π¦+4x=y+4

- Substitute π₯x in the first equation:

(π¦+4)+3π¦=12(y+4)+3y=12

- Simplify and solve for π¦y:

π¦+4+3π¦=12y+4+3y=12 4π¦+4=124y+4=12 4π¦=84y=8 π¦=2y=2

- Substitute π¦y back into π₯=π¦+4x=y+4:

π₯=2+4x=2+4 π₯=6x=6

So, the value of x is 6.

### The sum of three consecutive even integers is 42. What is the smallest of these integers?

Let’s denote the three consecutive even integers as π₯x, π₯+2x+2, and π₯+4x+4.

The sum of these integers is given by:

π₯+(π₯+2)+(π₯+4)=42x+(x+2)+(x+4)=42

Simplify the equation:

π₯+π₯+2+π₯+4=42x+x+2+x+4=42 3π₯+6=423x+6=42

Subtract 6 from both sides:

3π₯=363x=36

Divide by 3:

π₯=12x=12

So, the smallest of these integers is 1212.

### A company’s profit is calculated as 25% of its revenue. If the company’s revenue last year was $1,200,000, what was its profit? (Enter your answer as a decimal, rounded to the nearest whole number.)

To calculate the company’s profit, which is 25% of its revenue, follow these steps:

- Determine the percentage as a decimal: 25%=0.2525%=0.25
- Multiply the revenue by the percentage: Profit=0.25Γ$1,200,000Profit=0.25Γ$1,200,000
- Perform the multiplication: Profit=$300,000Profit=$300,000

Thus, the company’s profit last year was $300,000.

### A rectangular garden has an area of 48 square meters and a length of 8 meters. What is the perimeter of the garden in meters?

To find the perimeter of the rectangular garden, follow these steps:

- Calculate the width of the garden using the area and length.

Given: Area=48 square metersArea=48 square meters Length=8 metersLength=8 meters

Width=Area/LengthWidth=Area/Length Width=48/8Width=48/8 Width=6 metersWidth=6 meters

- Calculate the perimeter using the formula for the perimeter of a rectangle: Perimeter=2(Length+Width)

Perimeter=2(8+6)

Perimeter=2Γ14 Perimeter=28 meters

So, the perimeter of the garden is 28 meters.

So, the perimeter of the garden is 28 meters.

### If 2x^2 + 3x – 5 = 0, what is the value of x? (Enter your answer as a decimal, rounded to two decimal places.)

To solve the quadratic equation 2π₯2+3π₯β5=02×2+3xβ5=0, we can use the quadratic formula:

π₯=βπΒ±π2β4ππ2πx=2aβbΒ±b2β4acββ

In this equation, π=2a=2, π=3b=3, and π=β5c=β5.

- Calculate the discriminant: π2β4ππ=32β4(2)(β5)b2β4ac=32β4(2)(β5) =9+40=9+40 =49=49
- Substitute the values into the quadratic formula: π₯=β3Β±492Γ2x=2Γ2β3Β±49ββ π₯=β3Β±74x=4β3Β±7β

This gives us two possible solutions:

- Calculate the first solution: π₯=β3+74x=4β3+7β π₯=44x=44β π₯=1x=1
- Calculate the second solution: π₯=β3β74x=4β3β7β π₯=β104x=4β10β π₯=β2.5x=β2.5

So, the solutions are π₯=1x=1 and π₯=β2.5x=β2.5.

The values of π₯x rounded to two decimal places are: π₯=1.00 and π₯=β2.50x=1.00 and x=β2.50

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## Conclusion

Preparing for the GRE math section doesn’t have to be overwhelming. By focusing on the most frequently asked questions and employing strategic problem-solving techniques, you can boost your confidence and your score. Remember, consistent practice and a solid understanding of fundamental concepts are key to success.

As you practice, take time to reflect on your mistakes. Understanding why you got a question wrong is just as important as knowing the right answer. This reflection helps you identify patterns in your thinking and strengthens your problem-solving skills, ensuring you’re better prepared for test day.

Navigate your path to GRE success with Ambitio’s precision preparation. Designed for ambitious students, our platform provides a strategic approach to the GRE, offering in-depth content review, practice tests, and personalized feedback to optimize your study time and results.

## FAQs

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