Fractions

How to Divide Fractions

How are fractions divided? You flip the second fraction and multiply. This guide explains why that works, covers dividing by whole numbers and by mixed numbers, and shows how to cancel common factors before you multiply so the answer arrives already simplified.

How are fractions divided?

Keep the first fraction as it is, change the division sign to a multiplication sign, and flip the second fraction upside down. Then multiply straight across: numerator times numerator, denominator times denominator. Simplify what you get. That is the entire method, and it handles every fraction division problem there is.

The flipped fraction has a proper name. 5/2 is the reciprocal of 2/5, and every number except zero has one. To find one, swap the top and the bottom: the reciprocal of 7/9 is 9/7, and the reciprocal of 6 is 1/6 because 6 is really 6/1. Zero has no reciprocal, which is another way of saying you cannot divide by zero.

The defining property of a reciprocal is that a number multiplied by its own reciprocal equals exactly 1: 2/5 × 5/2 = 10/10 = 1. That single fact is the hinge the whole method turns on, and the next section shows how.

Why multiplying by the reciprocal works

Keep-change-flip is usually taught as a rule to memorize. It is not magic, and there are three ways to see why it has to be true. Any one of them is enough.

1. Division asks how many fit

The question 3 ÷ 1/4 means “how many quarters fit into 3?” Four quarters fit into each whole, and there are three wholes, so twelve quarters fit. Notice what you just did to get there: you multiplied 3 by 4. Multiplying by 4 is exactly what flipping 1/4 produces.

2. Multiply the top and bottom by the same thing

Write the division as one fraction stacked on another. You are always allowed to multiply the top and the bottom of a fraction by the same number without changing its value, so multiply both by the reciprocal of the bottom.

The bottom collapses to 1 because a fraction times its reciprocal is 1. The top is left holding the multiplication you wanted. Nothing was assumed and nothing was memorized.

3. Division undoes multiplication

Asking for 3/4 ÷ 2/5 is asking: what number, multiplied by 2/5, gives 3/4? To undo a multiplication by 2/5 you multiply by 5/2, because those two cancel to 1. So the missing number is 3/4 × 5/2. It is the same reason dividing by 5 matches multiplying by 0.2, and the reason the ordinary division calculator agrees with a multiplication by a reciprocal.

The four shapes of a division problem

Every fraction division you will be handed is one of four shapes. The method never changes; only the tidying-up before it does.

Fraction ÷ fraction

Nothing to prepare. Flip the second one and multiply. 5/6 ÷ 5/6 = 5/6 × 6/5 = 30/30 = 1, which is the right answer for any number divided by itself.

Fraction ÷ whole number

A whole number is a fraction with a denominator of 1. Write 4 as 4/1, flip it to 1/4, and multiply.

There is a shortcut worth knowing: dividing a fraction by a whole number just multiplies the denominator by that number. 2/3 ÷ 4 = 2/(3×4) = 2/12 = 1/6. It makes sense in real terms. If two thirds of a pizza is shared between four people, each person gets a sixth of a pizza.

Whole number ÷ fraction

Same preparation, opposite order. Put the whole number over 1 and flip the fraction that follows the division sign.

Answers here are usually larger than the number you started with, and often come out as whole numbers.

Mixed number ÷ mixed number

Convert both to improper fractions first. You cannot flip a mixed number; there is no single top and bottom to swap. 2 1/4 becomes 9/4 because 2 × 4 + 1 = 9, and 1 1/2 becomes 3/2.

The improper fraction calculator does that first conversion and shows the arithmetic, and the mixed number calculator takes it back the other way once you have an answer like 18/12 that you would rather read as 1 1/2.

All four shapes side by side

Read down the third column and the pattern is plain. The only thing that ever changes is what you had to do before the flip.

Cancel common factors before you multiply

Once the problem is a multiplication, you can cancel any factor shared by a top number and a bottom number, even across the two fractions. Doing it first keeps the numbers small and usually means the answer needs no reducing at the end.

Cancelling only works across a multiplication sign. Never cancel a number that sits on one side of a plus or a minus, and never cancel before you have flipped — 8/9 ÷ 4/3 is not the same problem as 8/9 × 4/3, and cancelling the 8 against the 4 while the division sign is still there gives the wrong answer.

How to reduce fractions at the end

If you did not cancel first, reduce afterward. Divide the numerator and the denominator by their greatest common factor — the largest whole number that goes into both exactly.

If spotting the greatest common factor is hard, chip away in stages instead. Halve both while both are even, then try 3, then 5. 36/48 → 18/24 → 9/12 → 3/4. The destination is the same; you just take more steps. A fraction is in lowest terms when the only whole number dividing both parts is 1.

How to do cross multiplication

Cross multiplication means multiplying the top of one fraction by the bottom of the other, diagonally. It solves three different problems, and mixing them up is a common source of wrong answers.

Solving a proportion

When two fractions are set equal and one part is unknown, cross multiply to get a simple equation.

Check it by putting 9 back in: 9/12 reduces to 3/4. This is how you scale a recipe or find a missing side in similar shapes.

Comparing two fractions

Which is bigger, 5/8 or 7/12? Cross multiply and compare the two products, each keeping the side of its own numerator.

In decimals that is 0.625 against about 0.583, which agrees. A fraction to decimal chart settles comparisons like this at a glance when the numbers are common ones.

Dividing in a single line

Keep-change-flip has a cross-multiplied form: a/b ÷ c/d = (a × d) / (b × c). The numerator of the first meets the denominator of the second, and the two remaining numbers pair off underneath.

It is the same operation written more compactly. Use whichever version you can carry out without losing track of which number went where.

How to multiply a fraction by a fraction, a whole number or a mixed number

Division turns into multiplication after the flip, so multiplying is the operation you actually spend your time doing. It is the easiest of the four, because there is no common denominator to find.

Fraction times fraction

Multiply the tops, multiply the bottoms, simplify.

The word of is a multiplication sign in disguise. Two thirds of four fifths is 8/15, and three quarters of 20 is 3/4 × 20 = 60/4 = 15.

Fraction times whole number

Put the whole number over 1 and carry on. Cancelling first keeps the numbers manageable.

Without cancelling you would get 36/8, which reduces to the same 9/2.

How to multiply mixed numbers

Convert to improper fractions first, every time.

Multiplying by a proper fraction shrinks a number instead of growing it: 2 1/2 × 3/5 = 5/2 × 3/5 = 15/10 = 1 1/2.

Why dividing by a fraction makes the answer bigger

Most people meet division as something that shrinks a number, so an answer larger than the number you started with feels wrong. It is not. Watch what happens as the divisor gets smaller.

Each divisor is half the one above it, and each answer doubles. Nothing changes direction at the point where the divisor stops being a whole number. Smaller pieces means more pieces.

The dividing line is 1, not the boundary between fractions and whole numbers:

• Divide by something less than 1 and the answer grows. 3/4 ÷ 1/8 = 3/4 × 8 = 6.
• Divide by exactly 1 and nothing changes. 3/4 ÷ 1 = 3/4.
• Divide by something greater than 1 and the answer shrinks. 3/4 ÷ 3/2 = 3/4 × 2/3 = 6/12 = 1/2.

A mixed number like 1 1/2 counts as greater than 1, so dividing by it shrinks the result even though a fraction is involved. This gives you a free sanity check on every answer. Before you write it down, ask whether the divisor was above or below 1, and confirm the answer moved the way it should have.

Subtraction behaves nothing like this, which is worth keeping straight. Taking 1/8 away from 3/4 gives 5/8, a smaller number, while dividing by 1/8 gives 6. Same two fractions, wildly different questions.

Common mistakes

• Flipping the wrong fraction. Only the divisor — the one after the division sign — gets flipped. 3/4 ÷ 2/5 becomes 3/4 × 5/2, never 4/3 × 2/5.
• Flipping both fractions. That inverts the whole answer. It gives 8/15 here instead of 15/8, and the two are reciprocals of each other, which is a useful tell that this is the error you made.
• Hunting for a common denominator. You need one to add or subtract. Multiplying and dividing never require one.
• Leaving a mixed number unconverted. 2 1/4 has to become 9/4 before anything gets flipped.
• Forgetting a whole number is n/1. The reciprocal of 4 is 1/4, not 4. Writing it as 4/1 first makes the flip automatic.
• Cancelling in the wrong place. Cancelling is only valid across a multiplication sign, so flip first and then look for shared factors. It is never valid across a plus or minus: in (3 + 6)/6 the sixes do not cancel, and the value is 9/6, or 3/2.
• Stopping before the answer is simplified. 18/12 is correct but unfinished. Reduce it to 3/2, then write it as 1 1/2 if a mixed number is what was asked for.
• Rejecting an answer for being too big. Dividing by anything smaller than 1 is supposed to produce a larger result.

Frequently asked questions

Calculators for this

Every one of these shows the working, so you can check the method as well as the answer.

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