Dividing fractions

3 ÷ 1/4 = 12: three whole pizzas cut into quarters make twelve slices.

Division asks how many fit

One meaning of 12 ÷ 3 is how many 3s fit into 12. The same question works for fractions. 3 ÷ 1/4 asks how many quarters fit into 3. Each whole holds 4 quarters, so 3 wholes hold 12: 3 ÷ 1/4 = 12. Dividing by a fraction smaller than 1 gives an answer bigger than the number you started with, because small pieces fit many times.

Now try 2 ÷ 2/3. How many jumps of 2/3 fit into 2? In thirds, 2 is 6/3 and each jump is 2 thirds, so 6 ÷ 2 = 3 jumps. 2 ÷ 2/3 = 3. Notice what happened: the 2 was multiplied by 3 to get thirds, then divided by 2. That is the same as multiplying 2 by 3/2.

The reciprocal

The reciprocal of a fraction is the fraction turned upside down. The reciprocal of 3/5 is 5/3, the reciprocal of 7 (which is 7/1) is 1/7, and the reciprocal of 1/4 is 4. A number times its reciprocal is always 1: 3/5 × 5/3 = 15/15 = 1. Zero has no reciprocal, since nothing times 0 gives 1.

The rule for division is: to divide by a fraction, multiply by its reciprocal. a/b ÷ c/d = a/b × d/c. Many people learn it as "flip and multiply" without being told why it works. Here are three explanations. Any one of them is enough, and each shows the rule from a different side.

Why it works 1: a common denominator

When two fractions have the same denominator, dividing them is just dividing the numerators. 6/8 ÷ 1/8 asks how many eighths fit into 6 eighths: 6. So 3/4 ÷ 1/8 = 6/8 ÷ 1/8 = 6. In general, rewrite a/b ÷ c/d with the denominator b × d: (a × d)/(b × d) ÷ (c × b)/(b × d). The pieces are the same size, so the answer is (a × d) ÷ (c × b), which is (a × d)/(b × c). That is exactly a/b × d/c.

Why it works 2: division undoes multiplication

Saying 12 ÷ 3 = 4 is the same as saying 4 × 3 = 12. So the answer to 5/6 ÷ 2/3 is the number that, multiplied by 2/3, gives 5/6. Try 5/6 × 3/2: multiplying that by 2/3 gives 5/6 × 3/2 × 2/3, and 3/2 × 2/3 = 1, so the result is 5/6. It works. Multiplying by the reciprocal is the move that undoes multiplying by the fraction, and undoing multiplication is what division is.

Why it works 3: make the divisor 1

Write the division as one fraction stacked on another: (3/5) over (4/7). As unit 3 showed, multiplying the top and bottom of a fraction by the same number does not change its value. Multiply both by 7/4, the reciprocal of the bottom. The bottom becomes 4/7 × 7/4 = 1, and the top becomes 3/5 × 7/4. Anything divided by 1 is itself, so 3/5 ÷ 4/7 = 3/5 × 7/4 = 21/20.

Sharing: dividing by a whole number

Division also means sharing. 3/4 ÷ 3 shares three quarters among 3 people, and each gets 1/4. The rule agrees: 3/4 × 1/3 = 3/12 = 1/4. Dividing by 2 is the same as taking half: 4/5 ÷ 2 = 4/5 × 1/2 = 2/5.

Worked examples

Example 1. 5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 5/4. Check: 5/4 × 2/3 = 10/12 = 5/6.

Example 2. Mixed numbers are converted first. 2 1/2 ÷ 3/4 = 5/2 × 4/3 = 20/6 = 10/3, which is 3 1/3.

Example 3. A 6 m ribbon is cut into pieces 3/4 m long. 6 ÷ 3/4 = 6 × 4/3 = 24/3 = 8 pieces.