GMAT® Fractions: A Complete Guide

You've been using fractions since school, mostly without the label. Halving a recipe, splitting a dinner check, reading a price per ounce: each one is fraction work, done so often it doesn't need a name. So it can make sense to file the topic under "no prep needed" and spend the study time on material that looks more exam-specific.

The Quant section tests fractions anyway, and it does so without a calculator. Data Insights has one; Quant doesn't. Fraction questions in Quant run through a short list of conventions that school rarely made explicit, like which denominators you can add across and which terms you can cancel. This guide covers what a fraction is, all four operations done by hand, and the cancellation rule that several of the exam's wrong answers are built on.

What a fraction is

A fraction is division, written with a bar. The number being divided sits on top, the number doing the dividing sits below the bar, and the bar itself groups the top into one quantity:

$\frac{6}{2} = 3$

The top is the numerator, which is the dividend from long division wearing a new name. The bottom is the denominator, which is the divisor. Those two words show up in question stems, so they're worth knowing cold rather than translating from "top" and "bottom" under time pressure.

Fractions matter on the exam because some values don't fit into whole numbers. A shop that wants to charge more than $1 but less than $2 lands somewhere in between, at $1.50, and needs numbers that can live between two integers. Division produces exactly those values: 3 divided by 2 is $1.5$, and the fraction $\frac{3}{2}$ is the same value in costume.

Multiplying fractions

Multiplication is the simplest operation fractions allow: multiply the tops by the tops and the bottoms by the bottoms.

$\frac{3}{4} \times \frac{7}{16} = \frac{21}{64}$

Three times 7 on top, four times 16 on the bottom, and the question is done. When more than two fractions multiply together, the same rule stretches: all the tops multiply into one product, all the bottoms multiply into another.

Answers often arrive in a form that can be written more simply. Six twelfths and one half are the same value, and simplifying is just dividing the top and the bottom by the same number, which leaves the value untouched. Which numbers divide the top and bottom is a divisibility question, and our divisibility rules guide covers the rules worth memorizing.

Adding and subtracting fractions

You can only add tops when the bottoms match. With $\frac{3}{4} + \frac{7}{16}$, the denominators are different, so the first job is making them the same, and the tool for that is multiplying by 1 in a clever form. Since $\frac{4}{4}$ equals 1, multiplying $\frac{3}{4}$ by it changes nothing about the value while giving the 16 the question wants:

$\frac{3}{4} \times \frac{4}{4} = \frac{12}{16}$

Now the bottoms match, and the addition runs in one step:

$\frac{12}{16} + \frac{7}{16} = \frac{19}{16}$

Add the tops and keep the bottom, which gives $\frac{19}{16}$. Subtraction runs the same route with a minus sign in place of the plus.

The wrong answer worth naming is the one you get by adding tops and bottoms together. In the classic setup, $\frac{1}{2} + \frac{1}{3}$, that error produces $\frac{2}{5}$, and it shows up among the answer choices often enough to matter. The error is tempting under time pressure because every number on the page got used in some operation, but a sum only works when the bottoms agree first. The cheapest common denominator to reach for is usually the least common multiple of the two bottoms, which our LCM guide covers alongside the two ways to compute it, and any common denominator still reaches the answer, since a bigger one adds only an extra simplifying step.

Dividing fractions

When you divide fractions by hand, no division happens. The work flips the fraction doing the dividing and multiplies by the flipped version instead:

$\frac{2}{3} \div \frac{3}{4} = \frac{2}{3} \times \frac{4}{3} = \frac{8}{9}$

The flipped version has a name, the reciprocal, and it's a term the exam uses directly. The reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$: the numerator and denominator trade places. Algebra explains why flipping works, and that derivation can be a fun rabbit hole, but it's overkill for this exam, so we'd suggest memorizing the move instead: flip the divisor, then multiply.

Two slips cause a lot of the lost points here. The first is flipping the wrong fraction, which is easy to do when the division reads left to right, and the answer choices include the product you'd get from flipping the first fraction. The second is flipping the divisor and then dividing again anyway, which leaves the flip unused. A question that hands you a reciprocal by name is asking for the same flip in vocabulary form.

The fraction bar groups the whole top

Everything above the bar is one quantity, and everything below it is one quantity. That sentence sounds obvious until the top contains an addition, which is where the exam likes to place it. The expression $\frac{20^2 + 20}{20}$ means the whole top, $20^2 + 20$, divided by 20, and not just the first term divided while the second term waits alongside.

Writing the top in parentheses when you copy the expression to your scratch pad costs one moment and keeps the grouping visible:

$\frac{20^2 + 20}{20} = \frac{400 + 20}{20} = \frac{420}{20} = 21$

The bar is doing the work that parentheses do elsewhere, which is why a fraction sometimes reads as one stacked line on the screen but behaves like a fully parenthesized expression. Our equations guide covers the same habit in algebra, where parentheses around a plugged-in value prevent a near-identical slip.

Cancel across multiplication, never across addition

Canceling divides the top and the bottom by the same factor, and that division reaches whatever it faces. A factor multiplied into the whole top cancels cleanly. A factor facing a sum has to divide both terms, and that restriction is where the exam's fraction traps tend to live.

A problem from the 11th edition Official Guide asks for the value of $\frac{29^2 + 29}{29}$, and it has two clean routes.

Route one splits the single fraction into two, since the denominator is already common to both pieces:

$\frac{29^2}{29} + \frac{29}{29} = 29 + 1 = 30$

The split deserves a second look, because school math trains the combining direction so heavily that the splitting direction can feel illegal. It isn't. A fraction with a sum on top splits into two fractions with the same bottom, and on this exam the split is often the move that unlocks the question.

Route two factors the top first, so the 29 becomes multiplication that the denominator can cancel against:

$\frac{29(29 + 1)}{29} = 29 + 1 = 30$

Both routes land on 30. The tempting third route cancels the 29 below the bar against only the $29^2$ above it, which leaves $29 + 29$, and 58 is one of the wrong choices. Both routes above exist to make sure the 29 reaches both terms.

Try this one

Try this one before reading on.

Splitting the top into two fractions finishes the setup, since the denominator is already common to both pieces:

$\frac{11^2}{11} + \frac{11}{11} = 11 + 1 = 12$

The answer is (B). Factoring runs the same route in one line, since $\frac{11(11 + 1)}{11}$ cancels down to $11 + 1$ as well.

Choice (C) is the cancel-across-addition error: canceling the 11 below the bar against only the $11^2$ above it leaves $11 + 11 = 22$. Choice (A) subtracts the second term instead of adding, (D) is $11^2$ before any division reaches it, and (E) is the factored top with the cancel never run.

FAQ

What is a fraction on the GMAT®?

Division written with a bar. The numerator on top is the dividend, the denominator on the bottom is the divisor, and the bar groups the entire top as one quantity. $\frac{6}{2}$ and $6 \div 2$ are the same value.

How do you multiply fractions?

Multiply the tops by the tops and the bottoms by the bottoms: $\frac{3}{4} \times \frac{7}{16} = \frac{21}{64}$. Simplify afterward by dividing the top and bottom by a shared factor.

How do you add fractions with different denominators?

Rewrite one or both fractions by multiplying by 1 in a useful form, such as $\frac{4}{4}$, until the bottoms match, then add the tops and keep the bottom. $\frac{3}{4} + \frac{7}{16}$ becomes $\frac{12}{16} + \frac{7}{16} = \frac{19}{16}$.

How do you divide fractions?

Flip the divisor and multiply. $\frac{2}{3} \div \frac{3}{4}$ becomes $\frac{2}{3} \times \frac{4}{3} = \frac{8}{9}$. The flipped fraction is called the reciprocal.

What is a reciprocal?

The flipped version of a fraction, with the numerator and denominator trading places. The reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$, and a nonzero fraction times its reciprocal equals 1.

Can you cancel across addition?

No. Canceling divides the top and bottom by the same factor, and a sum on top only divides evenly if every term gets the division. Split the fraction into two, or factor the top so the shared factor becomes multiplication.

Do fractions show up in Quant or Data Insights?

Both. Quant Problem Solving uses them with no calculator allowed, and Data Sufficiency in the Data Insights section leans on fraction algebra such as clearing denominators from equations.

Want to learn even more?

Related reading:

• GMAT® Number Properties: A Complete Guide
• GMAT® Divisibility Rules: A Complete Guide
• GMAT® Least Common Multiples: A Complete Guide
• GMAT® Terminating and Repeating Decimals: A Complete Guide
• GMAT® Data Sufficiency: How to Approach DS Questions

The Math Basics series rebuilds all four operations from the ground up, including "GMAT® Focus Edition Math Basics: Fractions," which walks every step of every operation by hand. The cancellation problem comes from Episode 12 of our Real GMAT® Problems podcast series, "Fractions, Exponents, and Algebra." You can hear everything on Spotify, Apple Podcasts, or YouTube.