This GRE Quantitative Reasoning formula sheet organizes the math rules most useful for review, shows how to apply them in context, and gives you a repeatable checklist for testing formulas under time pressure. Use it as a reference during study, then return to it to mark rules you still miss.
Overview
GRE Quantitative Reasoning rewards more than memorizing formulas. You also need to recognize which relationship a problem is testing, translate words into equations, estimate efficiently, and check whether an answer is reasonable. A compact formula sheet helps with the first step, but it should support problem solving rather than replace it.
The main GRE quantitative reasoning topics include arithmetic, algebra, geometry, data analysis, ratios, percentages, probability, and statistics. The formulas below are grouped by use so you can find a rule quickly during review. Some questions can be solved without a formula by comparing quantities, estimating, or testing simple values. When a formula is available, choose the method that is both reliable and efficient.
Arithmetic and number properties
- Order of operations: Evaluate grouping symbols, exponents, multiplication and division, then addition and subtraction.
- Percent: Percent means “per 100.” A percent written as a decimal is divided by 100; for example, 18% = 0.18.
- Percent change: (new value − original value) ÷ original value. Multiply by 100 to express the result as a percentage.
- Average: Average = sum of values ÷ number of values.
- Distance, rate, and time: Distance = rate × time. Therefore, rate = distance ÷ time and time = distance ÷ rate.
- Simple interest: Interest = principal × rate × time, when the rate and time are expressed in compatible units.
- Divisibility: A number is even if it is divisible by 2. A number is divisible by 5 if it ends in 0 or 5. For larger divisibility rules, test the actual values carefully rather than relying on memory alone.
Algebra and equations
- Distributive property: a(b + c) = ab + ac.
- Difference of squares: a² − b² = (a − b)(a + b).
- Exponent rules: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ when a is nonzero; (aᵐ)ⁿ = aᵐⁿ.
- Quadratic formula: For ax² + bx + c = 0, x = [−b ± √(b² − 4ac)] ÷ 2a.
- Slope: m = (y₂ − y₁) ÷ (x₂ − x₁). In y = mx + b, m is the slope and b is the y-intercept.
- Arithmetic sequence: The difference between consecutive terms is constant. The nth term is aₙ = a₁ + (n − 1)d.
- Geometric sequence: The ratio between consecutive terms is constant. The nth term is aₙ = a₁rⁿ⁻¹.
Geometry and measurement
- Rectangle: Area = length × width; perimeter = 2(length + width).
- Triangle: Area = ½ × base × height.
- Parallelogram: Area = base × height.
- Circle: Circumference = 2πr; area = πr².
- Rectangular solid: Volume = length × width × height.
- Cylinder: Volume = πr²h.
- Pythagorean theorem: In a right triangle, a² + b² = c², where c is the hypotenuse.
- Triangle angle sum: The interior angles of a triangle add to 180 degrees. A straight line measures 180 degrees, and a full rotation measures 360 degrees.
Ratios, probability, and data
- Proportion: If a ÷ b = c ÷ d, then ad = bc, provided the denominators are nonzero.
- Probability: Probability of an event = favorable outcomes ÷ total possible outcomes when outcomes are equally likely.
- Complement: P(not A) = 1 − P(A).
- Independent events: P(A and B) = P(A) × P(B), when the occurrence of one event does not affect the other.
- Range: Range = maximum value − minimum value.
- Weighted average: Multiply each value by its weight, add the products, then divide by the sum of the weights.
How to estimate
Use the formula sheet as a decision tool. Before calculating, identify the quantities given, the quantity requested, and the units involved. Then ask whether the problem calls for an exact calculation, an approximation, or a comparison.
- Translate the wording. For example, “20% greater than x” means 1.20x, while “20 percentage points greater” describes a difference between percentages.
- Choose a representation. Draw a quick diagram for geometry, make a table for a sequence, or write an equation for a rate problem.
- Estimate before solving. If the answer should be near 50, an answer near 500 suggests a setup or arithmetic error.
- Use convenient values when permitted. In some comparison questions, assigning a simple value to a variable can reveal the relationship without lengthy algebra. Check that the chosen value satisfies every stated condition.
- Check units and restrictions. A probability must fall between 0 and 1, a length cannot be negative, and a denominator cannot equal zero.
The on-screen calculator can be useful for arithmetic, roots, and decimals, but entering an expression does not correct an incorrect setup. Practice deciding what to calculate before reaching for it. For broader pacing practice, review this section-by-section timing and pacing guide; the habit of setting checkpoints transfers well to quantitative test sections.
Inputs and assumptions
Formulas are only as dependable as the assumptions behind them. Keep these checks beside your GRE quant notes:
- Convert units before applying a rate, area, or volume formula.
- Confirm whether a percentage is based on the original value or the new value.
- Do not treat an average as the middle value unless the problem specifically asks for a median.
- For geometry, distinguish a slanted height from a perpendicular height.
- For probability, determine whether outcomes are equally likely and whether events are independent.
- For algebra, record domain restrictions created by division, square roots, or logarithmic expressions.
- Read graph labels, scales, and units before interpreting a data display.
Mark each formula with one of three labels: automatic if you can recall and apply it, slow if you know it but need time, or uncertain if you regularly confuse it with another rule. Your study plan should focus first on formulas that are both uncertain and connected to common problem types.
Worked examples
Percent change
A quantity rises from 80 to 92. The change is 12, and the original value is 80. Percent change = 12 ÷ 80 = 0.15, or 15%. The important assumption is that the denominator is the original amount.
Rate and time
A cyclist travels 42 miles at 14 miles per hour. Using time = distance ÷ rate, the travel time is 42 ÷ 14 = 3 hours. If a problem combines different speeds over different distances, calculate each segment separately before combining the times.
Right-triangle geometry
A right triangle has legs of 6 and 8. The hypotenuse is √(6² + 8²) = √100 = 10. Recognizing familiar relationships can save time, but still verify which side is opposite the right angle before applying the theorem.
Weighted average
Suppose a course grade consists of a project worth 40% and an exam worth 60%. If the scores are 75 and 85, the weighted result is (0.40 × 75) + (0.60 × 85) = 30 + 51 = 81. Adding the two scores and dividing by two would incorrectly give both components equal weight.
When to recalculate
Revisit this GRE quant formula sheet whenever a practice set exposes a recurring error, when you begin a new study phase, and before a full-length practice test. Recalculate your priorities after each review rather than studying every formula equally.
- Complete a timed quantitative set without looking at the sheet.
- Record every missed, guessed, or unusually slow question.
- Classify the issue: formula recall, translation, setup, arithmetic, timing, or careless reading.
- Write one corrected example beside the relevant formula.
- Repeat a short timed set several days later and check whether the same error returns.
Use a 12-week test-prep planner to schedule formula review, mixed practice, and full-test analysis. You can also create concise recall cards with this flashcard guide. The goal is not to memorize an oversized list. It is to recognize the right relationship, apply it accurately, and verify the result under timed conditions.
