Study guide
The Quantitative Reasoning section of the GMAT contains 21 Problem Solving questions in 45 minutes, and every question is standard five-choice multiple choice with no calculator allowed. The current exam format, introduced in 2023 as the GMAT Focus Edition and officially renamed simply the GMAT Exam in July 2024, removed geometry and moved Data Sufficiency into the new Data Insights section, so this chapter concentrates entirely on arithmetic and elementary algebra: percents, ratios, number properties, statistics, equations, inequalities, exponents, functions, and basic coordinate-plane work. Because you cannot use a calculator, the section rewards clean mental arithmetic and efficient setups over brute-force computation.
Percents, Ratios, and Rates
Percent problems on the GMAT almost always hinge on correctly identifying the base, the number that 100 percent refers to. A price that rises from 80 to 100 has increased by 20/80 = 25 percent, not 20 percent, because the base is the original price of 80, not the new price. When a quantity changes twice in sequence, such as a 10 percent increase followed by a 10 percent decrease, do not average the percentages; multiply the successive factors instead: 1.10 times 0.90 equals 0.99, a net 1 percent decrease, not zero. Ratio problems are best handled by introducing a single unknown multiplier. If the ratio of Priya's marbles to Tomas's marbles is 3 to 5, write their counts as 3k and 5k, so that any additional condition, such as a total of 96 marbles, converts directly into 8k = 96 and k = 12. Rate problems, especially work-rate and distance-rate-time items, are usually easiest when you convert everything into a common unit of work or distance per unit time and add or subtract rates rather than trying to average speeds. For instance, if Ana can paint a fence in 6 hours and Ben can paint the same fence in 3 hours, their combined rate is 1/6 + 1/3 = 1/2 of the fence per hour, so together they finish in 2 hours. Setting up a rate as a fraction of the job per unit time, rather than reasoning in whole fences, avoids the most common rate-problem errors.
Number Properties
Number-properties questions test reasoning about integers, odd and even values, primes, factors, multiples, and remainders, usually without requiring heavy calculation once the underlying rule is recognized. A useful habit is testing small representative numbers rather than trying to prove a claim algebraically under time pressure. To check whether the sum of two odd integers is always even, test 3 + 5 = 8 and 7 + 9 = 16; both are even, which is consistent with the general rule that odd plus odd equals even. Divisibility rules save time: a number is divisible by 3 if its digit sum is divisible by 3, and divisible by 9 under the same digit-sum test using 9. Prime factorization is the backbone of many harder number-properties questions, including those asking for the greatest common factor or least common multiple of two numbers; writing 84 as 2^2 x 3 x 7 and 90 as 2 x 3^2 x 5 lets you read off the GCF (2 x 3 = 6) and LCM (2^2 x 3^2 x 5 x 7 = 1260) directly from the shared and combined prime powers. Remainder questions often ask what remains when an expression like n^2 is divided by a fixed number given a condition on n; picking two or three values of n that satisfy the stated condition and computing the remainder directly is usually faster and safer than trying to reason abstractly about modular arithmetic under time pressure.
Statistics: Mean, Median, and Standard Deviation
The arithmetic mean of a data set is the sum of the values divided by the count of values, and the GMAT frequently tests how the mean shifts when a value is added, removed, or changed. If a class of 9 students has an average score of 80, the total of all scores is 9 x 80 = 720; adding a tenth student who scores 90 changes the total to 810 and the new average to 810/10 = 81. The median is the middle value in a data set sorted from least to greatest, or the average of the two middle values when the count is even; unlike the mean, the median is not pulled toward outliers, so a data set with one extreme value can have a mean quite different from its median. Standard deviation measures how spread out the values in a set are around the mean; you will rarely need to compute it exactly on the GMAT, but you should be able to reason about which of two data sets has a larger standard deviation by comparing how far their values sit from their respective means. A set clustered tightly around its mean, such as 48, 50, 52, has a small standard deviation, while a set with the same mean but wider spread, such as 20, 50, 80, has a much larger one. Questions frequently combine these ideas, for example asking how the median changes when the smallest value in a set is replaced by an even smaller one, which typically changes the mean but leaves the median unchanged if the set is large enough.
Equations, Inequalities, and Exponents
Linear equations with one variable are solved by isolating that variable through balanced operations on both sides, and systems of two equations with two variables are usually solved fastest by substitution when one equation already isolates a variable, or by elimination when the coefficients align conveniently after multiplication. Inequalities follow the same algebraic rules as equations with one critical exception: multiplying or dividing both sides by a negative number reverses the inequality sign. Given -2x < 8, dividing both sides by -2 gives x > -4, not x < -4. Compound inequalities and absolute-value inequalities often require splitting into cases; |x - 3| < 5 means -5 < x - 3 < 5, which simplifies to -2 < x < 8. Exponent rules are tested constantly: a^m times a^n equals a^(m+n), a^m divided by a^n equals a^(m-n), and (a^m)^n equals a^(mn). A negative exponent such as a^(-2) equals 1/a^2, and any nonzero base raised to the 0 power equals 1. Functions on the GMAT are typically defined by an explicit rule, such as f(x) = 2x^2 - 3, and questions ask you to evaluate the function at a given input, solve for an input that produces a given output, or compare f at two different inputs; treat the function notation as a direct substitution instruction rather than something conceptually mysterious.
Coordinate-Plane Basics
The current GMAT classifies basic coordinate-plane questions as algebra rather than geometry, and they typically involve lines, slopes, and intercepts rather than shapes or angles. The slope of a line through two points (x1, y1) and (x2, y2) is (y2 - y1)/(x2 - x1), and the slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept, the point where the line crosses the y-axis at x = 0. Two lines are parallel if they have the same slope and distinct intercepts, and perpendicular if the product of their slopes equals -1. A common question type gives two points and asks for the equation of the line through them, or gives a line's equation and asks whether a specific point lies on it, above it, or below it; to check, substitute the point's coordinates into the equation and compare. Word problems can also be modeled on the coordinate plane, such as a scenario where x represents time in hours and y represents distance traveled, with the y-intercept representing a starting distance and the slope representing a constant speed. Recognizing that a linear relationship between two changing quantities corresponds to a straight-line graph, and that the rate of change is the slope, connects this topic directly back to the rate and ratio reasoning used elsewhere in the Quantitative Reasoning section.
Key terms
- Base (in percent problems)
- — The original or reference quantity that a percent change or percent value is measured against.
- Ratio multiplier
- — A single unknown, such as k, introduced so that quantities in a given ratio (e.g., 3k and 5k) can be solved using one additional condition.
- Prime factorization
- — Expressing an integer as a product of prime numbers raised to whole-number powers, used to find greatest common factors and least common multiples.
- Greatest common factor (GCF)
- — The largest integer that divides evenly into two or more given integers.
- Least common multiple (LCM)
- — The smallest positive integer that is a multiple of two or more given integers.
- Mean
- — The arithmetic average of a data set: the sum of all values divided by the number of values.
- Median
- — The middle value of a data set sorted in order, or the average of the two middle values when the set has an even count.
- Standard deviation
- — A measure of how spread out the values in a data set are around the mean; larger spread means larger standard deviation.
- Slope
- — The rate of change of a line, calculated as the change in y divided by the change in x between two points on the line.
- Slope-intercept form
- — The equation of a line written as y = mx + b, where m is the slope and b is the y-intercept.
- Function notation
- — A rule, such as f(x) = 2x^2 - 3, that assigns an output value to each input value x, evaluated by direct substitution.
Exam tips
- Since no calculator is allowed, practice mental shortcuts such as converting percents to fractions (25% = 1/4) rather than doing long decimal arithmetic.
- When a percent problem involves two successive changes, multiply the decimal factors together instead of adding or averaging the percentages.
- For number-properties claims, test two or three small numbers first; if a pattern holds across varied examples it is usually the intended rule.
- Remember to flip an inequality's direction whenever you multiply or divide both sides by a negative number.
- On coordinate-plane questions, substitute given points directly into a candidate equation rather than trying to visualize the graph from memory.