Dividing Fractions

Turning division into multiplication

Dividing fractions sounds tricky, but there is a reliable method that turns every division into a multiplication you already know how to do. The whole trick is to multiply by the reciprocal, often remembered as Keep, Change, Flip. By the end of this lesson you will divide any fractions, including whole numbers and mixed numbers, with confidence.

This lesson builds directly on Multiplying Fractions, so make sure you are comfortable multiplying tops and bottoms first. A quick reminder of the basics is in our Introduction to Fractions lesson.

What division of fractions asks

Before the method, understand the question division asks. 6 ÷ 2 = 3 asks "how many 2s fit into 6?" Fraction division is the same. 1/2 ÷ 1/4 asks "how many quarters fit into one half?" Picture half a pizza cut into quarter slices — there are 2 quarters in a half. So 1/2 ÷ 1/4 = 2. Keeping this picture in mind explains a surprising fact: dividing by a small fraction gives a big answer, because many tiny pieces fit inside.

The reciprocal

The reciprocal of a fraction is what you get when you swap the numerator and denominator — you flip it upside down.

Fraction	Reciprocal
3/4	4/3
2/5	5/2
1/6	6/1 = 6
7 (= 7/1)	1/7

A fraction multiplied by its reciprocal always equals 1 (for example, 3/4 × 4/3 = 12/12 = 1). That property is the secret behind the method.

Keep, Change, Flip

Rule: To divide by a fraction, Keep the first fraction, Change the ÷ sign to ×, and Flip the second fraction to its reciprocal. Then multiply as usual and simplify.

Example 1 — Work out 1/2 ÷ 1/4.

1. Keep the first fraction: 1/2.
2. Change ÷ to ×.
3. Flip the second fraction: 1/4 becomes 4/1.
4. Multiply: 1/2 × 4/1 = (1×4)/(2×1) = 4/2.
5. Simplify: 4/2 = 2.

This matches the pizza picture: there are 2 quarters in a half.

Example 2 — Work out 3/5 ÷ 2/3.

1. Keep 3/5, change to ×, flip 2/3 to 3/2.
2. The problem is now 3/5 × 3/2.
3. Multiply: (3×3)/(5×2) = 9/10.
4. 9 and 10 share no common factor, so 9/10 is the final answer.

Example 3 — Work out 4/9 ÷ 2/3.

1. Keep 4/9, change to ×, flip 2/3 to 3/2: 4/9 × 3/2.
2. Cancel before multiplying: 4 and 2 share a factor of 2 (giving 2 and 1); 9 and 3 share a factor of 3 (giving 3 and 1). This leaves 2/3 × 1/1.
3. Multiply: 2/3.

Dividing with whole numbers

Write any whole number as a fraction over 1, then use Keep-Change-Flip.

Example 4 — Work out 6 ÷ 1/2.

1. Write 6 as 6/1.
2. Keep 6/1, change to ×, flip 1/2 to 2/1: 6/1 × 2/1.
3. Multiply: (6×2)/(1×1) = 12.

Twelve halves fit into 6 — the answer is bigger than 6, exactly as the "how many fit" idea predicts.

Example 5 — Work out 3/4 ÷ 6.

1. Write 6 as 6/1.
2. Keep 3/4, change to ×, flip 6/1 to 1/6: 3/4 × 1/6.
3. Multiply: (3×1)/(4×6) = 3/24.
4. Simplify: 3/24 = 1/8.

Here the answer is smaller, because we are dividing into 6 parts.

Dividing mixed numbers

Convert mixed numbers to improper fractions first, then apply Keep-Change-Flip.

Example 6 — Work out 2 and 1/2 ÷ 1 and 1/4.

1. Convert each to an improper fraction:
2. 2 and 1/2 = (2×2 + 1)/2 = 5/2
3. 1 and 1/4 = (1×4 + 1)/4 = 5/4
4. Keep 5/2, change to ×, flip 5/4 to 4/5: 5/2 × 4/5.
5. Cancel: the two 5s cancel, and 4 and 2 share a factor of 2 (giving 2 and 1): 1/1 × 2/1.
6. Multiply: 2.

So 2 and 1/2 ÷ 1 and 1/4 = 2.

A reference table

Problem	Keep–Change–Flip	Multiply / simplify	Answer
1/2 ÷ 1/4	1/2 × 4/1	4/2	2
3/5 ÷ 2/3	3/5 × 3/2	9/10	9/10
4/9 ÷ 2/3	4/9 × 3/2	12/18	2/3
6 ÷ 1/2	6/1 × 2/1	12	12
3/4 ÷ 6	3/4 × 1/6	3/24	1/8
2 and 1/2 ÷ 1 and 1/4	5/2 × 4/5	20/10	2

Why flipping works

The method can feel like a magic trick, so here is the why. Dividing by a number means asking "how many of it fit?" — and that is the same as multiplying by its reciprocal. Think about whole numbers first: dividing by 2 is identical to multiplying by 1/2 (both halve the number). The same is true for fractions: dividing by 3/4 is the same as multiplying by 4/3.

Here is one way to see it. The reciprocal has the special property that 3/4 × 4/3 = 1. So when you write a division as a fraction, like (3/5) ÷ (2/3), you can multiply the top and bottom by the reciprocal 3/2 to turn the bottom into 1, leaving exactly 3/5 × 3/2 on top. Dividing by something is the inverse of multiplying by it, and the reciprocal is precisely the number that undoes the multiplication. That is why "flip and multiply" always works.

A practice activity

Use a measuring cup or just draw bars.

1. Imagine you have 3 cups of flour and each batch of biscuits needs 1/4 cup. How many batches can you make? This is 3 ÷ 1/4.
2. Keep-Change-Flip: 3/1 × 4/1 = 12 batches. Check by drawing 3 bars, each split into 4 quarters — you can count 12 quarter-pieces.
3. Now try the reverse: you have 1/2 a cup and a batch needs 1/4 cup. That is 1/2 ÷ 1/4 = 2 batches.

Then try these on paper (answers below): (a) 2/3 ÷ 1/6, (b) 5/8 ÷ 1/2, (c) 4 ÷ 2/3, (d) 1 and 1/2 ÷ 3/4.

Answers: (a) 2/3 × 6/1 = 12/3 = 4, (b) 5/8 × 2/1 = 10/8 = 5/4 = 1 and 1/4, (c) 4/1 × 3/2 = 12/2 = 6, (d) 3/2 × 4/3 = 12/6 = 2.

Where this leads

Dividing fractions completes the four operations and prepares you for ratios, rates and algebra, where reciprocals appear constantly. Pair it with Multiplying Fractions and the single rule to remember is Keep, Change, Flip — every fraction division becomes a multiplication you can already do.