GRE Quantitative Reasoning Guide

Building Your Math Foundation

Mastering GRE math is like building a house. Your Arithmetic is the foundation—it must be solid to support everything else. Algebra provides the framing and structure, allowing you to solve complex problems using rules. Geometry is the layout and design of the space, and Data Analysis is the final inspection, where you interpret all the various components to ensure everything fits together accurately.

1. Arithmetic

The Foundation of Your Math Skills

Integers and Prime Factorization

Integers include positive and negative whole numbers and zero. A prime number is an integer greater than 1 with only two divisors: 1 and itself. Every integer greater than 1 can be uniquely expressed as a prime factorization.

Example:

The prime factorization of 800 is:

800 = (25)(52)

This breaks down as: 2 × 2 × 2 × 2 × 2 × 5 × 5 = 800

Division and Remainders

When an integer c is divided by a positive integer d, the result is a quotient q and a remainder r, where:

r = c - qd

Example:

100 divided by 45 is 2 remainder 10, because 2 × 45 = 90, which is 10 less than 100.

Percent Change

To find a percent increase, divide the amount of increase by the original base amount.

Example:

If a quantity increases from 600 to 750:

  • The increase is: 750 - 600 = 150
  • The percent increase is: 150 ÷ 600 = 0.25
  • Answer: 25%

2. Algebra

The Framework and Structure

Linear Equations in Two Variables

These can be solved using substitution or elimination.

Example (Substitution Method):

Solve the system:

4x + 3y = 13

x + 2y = 2

Solution:

  1. From the second equation: x = 2 - 2y
  2. Substitute into the first equation: 4(2 - 2y) + 3y = 13
  3. Simplify: 8 - 8y + 3y = 13
  4. Solve: -5y = 5, so y = -1
  5. Find x: x = 2 - 2(-1) = 4
  6. Answer: x = 4, y = -1

Quadratic Equations

These follow the form ax2 + bx + c = 0 and can be solved using the quadratic formula:

x = (-b ± √(b2 - 4ac)) / 2a

Example:

For x2 + 4x + 4 = 0:

  • a = 1, b = 4, c = 4
  • x = (-4 ± √(16 - 16)) / 2
  • x = (-4 ± 0) / 2 = -2
  • Answer: Only one real solution, x = -2

Work Problems

These involve combining individual production rates to find a total time.

Example:

Machine A takes 3 hours to produce a batch. Machine B takes 2 hours. How long do they take working together?

  • Machine A rate: 1/3 batch per hour
  • Machine B rate: 1/2 batch per hour
  • Combined rate: 1/3 + 1/2 = 5/6 batch per hour
  • Time to complete 1 batch: 6/5 hours
  • Answer: 1 hour and 12 minutes

Coordinate Geometry

The slope (m) of a line through two points (x1, y1) and (x2, y2) is:

m = (y2 - y1) / (x2 - x1)

Example:

For points Q(-2, -3) and R(4, 1.5):

  • m = (1.5 - (-3)) / (4 - (-2))
  • m = 4.5 / 6
  • Answer: m = 0.75

3. Geometry

The Layout and Design

Polygon Interior Angles

The sum of the interior angles of an n-sided polygon is:

(n - 2) × 180°

Example:

A regular octagon (8 sides):

  • Sum of interior angles: (8 - 2) × 180 = 1,080°
  • Each angle in a regular octagon: 1,080 ÷ 8
  • Answer: 135°

Special Right Triangles

The GRE frequently tests the 30°-60°-90° triangle, where the side lengths are in a ratio of:

1 : √3 : 2

Example:

In a 30°-60°-90° triangle:

  • If the hypotenuse is 2x
  • The side opposite the 30° angle is x
  • The side opposite the 60° angle is x√3

Circles and Arcs

The length of an arc is proportional to the degree measure of its central angle relative to 360°.

Arc Length = (θ/360°) × Circumference

Example:

In a circle with radius 5:

  • Circumference = 2πr = 10π
  • For an arc with a 50° central angle:
  • Arc length = (50/360) × 10π
  • Answer: ≈ 4.4

4. Data Analysis

The Final Inspection

Standard Deviation

This measures how much data values differ from the mean.

Example:

For the data: 0, 7, 8, 10, 10

  1. Mean = (0 + 7 + 8 + 10 + 10) / 5 = 7
  2. Squared differences from mean: 49, 0, 1, 9, 9
  3. Average of squared differences: (49 + 0 + 1 + 9 + 9) / 5 = 13.6
  4. Standard deviation = √13.6
  5. Answer: ≈ 3.7

Counting Methods (Combinations)

Use combinations when the order does not matter. The formula for "n choose k" is:

C(n,k) = n! / (k!(n-k)!)

Example:

To select a 3-person committee from 9 students:

  • C(9,3) = 9! / (3!(6!))
  • = (9 × 8 × 7) / (3 × 2 × 1)
  • = 504 / 6
  • Answer: 84 ways

Probability of Independent Events

If two events E and F are independent, the probability of both occurring is:

P(E and F) = P(E) × P(F)

Example:

If a fair die is rolled twice, what's the probability of rolling a 3 both times?

  • P(rolling a 3) = 1/6
  • P(3 both times) = 1/6 × 1/6
  • Answer: 1/36

Normal Distribution

In a perfectly symmetric bell curve, the mean, median, and mode are equal.

Key Fact:

About two-thirds (68%) of the data in a normal distribution falls within 1 standard deviation of the mean.

Ready to Practice?

Now that you've reviewed these core concepts, it's time to put your knowledge to the test with practice questions.

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