Multiplying & dividing fractions

To multiply two fractions, multiply the top numbers (numerators) together and the bottom numbers (denominators) together. No common denominator is needed. For example, 2/3 × 1/5 = (2 × 1) / (3 × 5) = 2/15. Always finish by writing the answer in lowest terms.

Sometimes the product can be reduced. 2/3 × 3/4 = 6/12, and both 6 and 12 divide by 6, so 6/12 = 1/2. The word of means multiply too: “3/4 of 8/9” is just 3/4 × 8/9 = 24/36 = 2/3. So taking a fraction of a fraction is one multiplication.

To divide by a fraction, multiply by its reciprocal — the fraction turned upside down. Keep the first fraction, change ÷ to ×, and flip the second. So 3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2. Dividing also answers “how many of these fit?”: 7/8 ÷ 1/8 asks how many eighths are in seven eighths — the answer is 7.

Dividing by 1/2 is the same as asking “how many halves fit?” — and there are two halves in every whole, so you end up with twice as much. Multiplying by the reciprocal (2/1) does exactly that. The reciprocal undoes the original fraction, which is why keep-change-flip always works.

A whole number is just a fraction over 1, so 3 = 3/1. To work out 1/5 × 3, write 3 as 3/1 and multiply across: (1 × 3) / (5 × 1) = 3/5. Only the top is scaled — the bottom stays the same. The classic trap is to multiply the denominator too; don’t. Sometimes the answer comes out whole: 2/3 × 6 = 12/3 = 4.

Dividing by a whole number n is the same as multiplying by 1/n, so the fraction gets smaller: 4/5 ÷ 2 = 4/5 × 1/2 = 4/10 = 2/5. Going the other way, dividing a whole number by a fraction below 1 gives abigger answer, because lots of small pieces fit inside: 6 ÷ 1/3 asks “how many thirds are in 6?” — and the answer is 18.

Turn a mixed number into an improper fraction before you multiply or divide. For 1 1/2, multiply the whole part by the denominator and add the top: 1 × 2 + 1 = 3, so 1 1/2 = 3/2. Then 1 1/2 × 2/3 = 3/2 × 2/3 = 6/6 = 1. Likewise 2 1/4 = 9/4, so 2 1/4 ÷ 3/8 = 9/4 × 8/3 = 6.

The sign rules are the same as for whole numbers: a negative times (or divided by) a positive is negative, and two negatives make a positive. Work out the size first, then attach the sign: -2/3 × 3/4 = 6/12 = 1/2, so the answer is -1/2.

Watch for these: multiplying by a whole number scales only the top, not the bottom (1/5 × 3 = 3/5, not 3/15); dividing by a number below 1 makes the answer bigger, not smaller; you must turn mixed numbers into improper fractions before multiplying; and “keep, change, flip” flips the second fraction only — never the first. Finally, always reduce your answer to lowest terms.

Type fractions with a slash, like 3/8, or as a whole number like 7 when the answer comes out evenly. Any equivalent form is accepted — 6/12 and 1/2 both count — but aim to give your answer in lowest terms.