Multiplying and Dividing Fractions

Multiplying fractions is simpler than adding them. Multiply the numerators together, multiply the denominators together, then simplify:

2/3 × 4/5 = (2 × 4)/(3 × 5) = 8/15.

Why does this work? Picture a square. Split its width into halves and its height into thirds: it is now cut into 2 × 3 = 6 equal pieces. Shading half of one third shades exactly one of those pieces, so 1/2 × 1/3 = 1/6. Multiplying the denominators counts the pieces; multiplying the numerators counts the shaded ones.

In maths, 'of' between a fraction and an amount means multiply. 3/4 of 20 is 3/4 × 20. Write the whole number over 1 and multiply across: 3/4 × 20/1 = 60/4 = 15. Or do it in your head: divide by the denominator (20 ÷ 4 = 5), then multiply by the numerator (5 × 3 = 15).

Before multiplying, you can divide any numerator and any denominator by a common factor. In 3/8 × 4/9, the 3 and the 9 share a factor of 3, and the 4 and the 8 share a factor of 4. Cancelling gives 1/2 × 1/3 = 1/6. Multiplying first gives 12/72, which also simplifies to 1/6, but with bigger numbers and more chances to slip.

Change mixed numbers into improper fractions before you multiply or divide. 2 1/2 × 1 1/3 becomes 5/2 × 4/3 = 20/6 = 10/3 = 3 1/3. A mixed number is really a sum (2 1/2 = 2 + 1/2), so it cannot be multiplied piece by piece. A whole number becomes a fraction over 1: 6 = 6/1.

To divide by a fraction, multiply by its reciprocal, which is the fraction turned upside down. The reciprocal of 2/5 is 5/2, and a number times its reciprocal always makes 1 (2/5 × 5/2 = 10/10 = 1). Zero has no reciprocal.

Many students remember the method as keep, change, flip: keep the first fraction, change ÷ to ×, flip the second fraction.

3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8.

Only the second fraction, the one you are dividing by, is flipped.

Reason 1: how many fit? Division asks how many times one number fits into another. How many halves fit into 3? Each whole holds 2 halves, so 3 holds 3 × 2 = 6. Dividing by 1/2 is the same as multiplying by 2. In the same way, dividing by 1/4 is multiplying by 4, and dividing by 2/3 is multiplying by 3/2.

Reason 2: make the divisor 1. Multiplying both numbers in a division by the same amount does not change the answer: 12 ÷ 4 = 24 ÷ 8 = 3. So for 2/3 ÷ 4/9, multiply both numbers by 9/4. The second number becomes 4/9 × 9/4 = 1, and dividing by 1 changes nothing. What is left is 2/3 × 9/4 = 18/12 = 3/2. That is exactly keep, change, flip.

A useful check: dividing a positive number by a proper fraction makes it bigger, and multiplying by one makes it smaller. 4 ÷ 1/4 = 16, but 4 × 1/4 = 1.