Teach Me Maths

Multiplying and Dividing Fractions

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How to multiply and divide fractions, whole numbers and mixed numbers without a calculator, how to use reciprocals and cancelling, and how to decide whether to multiply or divide in a problem.

Multiplying and dividing fractions in the exam

In the exam you are usually asked to multiply or divide two fractions or mixed numbers, without a calculator, for 2 or 3 marks. The question often says "Give your answer as a mixed number" or "in its simplest form". Some questions ask you to show that a calculation is correct, or to find the mistake in someone's working.

Fractions also turn up inside problems: the area of a rectangle with fraction sides, an amount shared equally, or how many small pieces can be cut from a length.

  • To multiply, multiply the tops together and the bottoms together.
  • To divide, keep the first fraction and multiply by the second one turned upside down.
  • Change mixed numbers into improper fractions before you do either.

Multiplying fractions

To multiply two fractions, multiply the numerators (the top numbers) together and multiply the denominators (the bottom numbers) together:

\frac{3}{5}\times\frac{1}{2}=\frac{3\times1}{5\times2}=\frac{3}{10}

You don't need a common denominator.

The word "of" means multiply. Half of \frac{3}{5} is \frac{1}{2}\times\frac{3}{5}=\frac{3}{10}. Multiplying by a fraction less than 1 makes the answer smaller.

To multiply by a whole number, write the whole number over 1. Three lots of \frac{2}{5} is \frac{2}{5}\times\frac{3}{1}=\frac{6}{5}, which is 1\frac{1}{5}

Worked example 1: Multiplying two fractions

Work out \frac{5}{8}\times\frac{2}{3}. Give your answer in its simplest form.

  1. Multiply the numerators: 5\times2=10

  2. Your turn. Multiply the denominators. What is 8\times3?

    8\times3=24, so \frac{5}{8}\times\frac{2}{3}=\frac{10}{24}

  3. 10 and 24 are both even, so divide the top and the bottom by 2: \frac{10}{24}=\frac{5}{12}
  4. No number other than 1 divides into both 5 and 12, so this is the simplest form.
  5. Answer: \frac{5}{12}

Cancelling before you multiply

Big fractions make big numbers to simplify. You can make the numbers smaller first, by dividing a numerator and a denominator by the same number. This is called cancelling.

In \frac{4}{9}\times\frac{3}{8}, the 4 on top and the 8 on the bottom can both be divided by 4. The 3 on top and the 9 on the bottom can both be divided by 3:

\frac{4}{9}\times\frac{3}{8}=\frac{1}{3}\times\frac{1}{2}=\frac{1}{6}

Always cancel a top number with a bottom number. Never cancel two top numbers, or two bottom numbers, with each other. If you don't spot a cancel, simplify at the end instead: you get the same answer.

Worked example 2: Cancelling first

Work out \frac{9}{10}\times\frac{4}{15}. Give your answer in its simplest form.

  1. 9 on top and 15 on the bottom are both in the 3 times table. Divide both by 3: the 9 becomes 3 and the 15 becomes 5
  2. 4 on top and 10 on the bottom are both even. Divide both by 2: the 4 becomes 2 and the 10 becomes 5

  3. Your turn. The calculation is now \frac{3}{5}\times\frac{2}{5}. Work it out.

    \frac{3\times2}{5\times5}=\frac{6}{25}

  4. Check without cancelling: \frac{9\times4}{10\times15}=\frac{36}{150}, and dividing the top and the bottom by 6 gives \frac{6}{25} too.
  5. Answer: \frac{6}{25}

Dividing by a fraction

Dividing asks how many times one number fits into another. 2\div\frac{1}{5} asks how many fifths fit into 2. Each whole has 5 fifths, so 2 wholes have 10 fifths: 2\div\frac{1}{5}=10

Dividing by \frac{1}{5} gave the same answer as multiplying by 5. The reciprocal of a number is that number turned upside down. The reciprocal of \frac{3}{8} is \frac{8}{3}. The reciprocal of 6, which is \frac{6}{1}, is \frac{1}{6}. For a mixed number, change it to an improper fraction first: 1\frac{1}{2}=\frac{3}{2}, so its reciprocal is \frac{2}{3}

To divide by a fraction, multiply by its reciprocal:

  • Keep the first fraction as it is.
  • Change the \div to \times.
  • Flip the second fraction, and only the second one.

\frac{3}{8}\div\frac{2}{5}=\frac{3}{8}\times\frac{5}{2}=\frac{15}{16}

Worked example 3: Dividing two fractions

Work out \frac{7}{10}\div\frac{3}{5}. Give your answer as a mixed number.

  1. Keep \frac{7}{10}, change \div to \times and flip \frac{3}{5}: \frac{7}{10}\times\frac{5}{3}

  2. Your turn. Work out \frac{7}{10}\times\frac{5}{3}. Give the fraction before you simplify it.

    \frac{7\times5}{10\times3}=\frac{35}{30}

  3. Divide the top and the bottom by 5: \frac{35}{30}=\frac{7}{6}
  4. 7\div6=1 remainder 1, so \frac{7}{6}=1\frac{1}{6}
  5. Answer: 1\frac{1}{6}

Mixed numbers: change them first

Before you multiply or divide, change every mixed number into an improper fraction. Multiply the whole number by the denominator and add the numerator:

3\frac{1}{4}=\frac{3\times4+1}{4}=\frac{13}{4}

Don't multiply the whole numbers and the fractions separately. 2\frac{1}{2}\times2\frac{1}{2} is not 4\frac{1}{4}. As improper fractions it is \frac{5}{2}\times\frac{5}{2}=\frac{25}{4}, which is 6\frac{1}{4}

If the question asks for a mixed number, change your answer back at the end. Divide the numerator by the denominator: 23\div6=3 remainder 5, so \frac{23}{6}=3\frac{5}{6}

Worked example 4: Multiplying mixed numbers

Work out 2\frac{2}{3}\times1\frac{4}{5}. Give your answer as a mixed number.

  1. Change the first mixed number: 2\frac{2}{3}=\frac{2\times3+2}{3}=\frac{8}{3}

  2. Your turn. 1\frac{4}{5}=\frac{\square}{5}. What number goes in the box?

    1\frac{4}{5}=\frac{1\times5+4}{5}=\frac{9}{5}

  3. Cancel the 3 on the bottom with the 9 on top, dividing both by 3: \frac{8}{3}\times\frac{9}{5}=\frac{8}{1}\times\frac{3}{5}=\frac{24}{5}
  4. 24\div5=4 remainder 4, so \frac{24}{5}=4\frac{4}{5}
  5. Answer: 4\frac{4}{5}

Worked example 5: Dividing mixed numbers

Work out 3\frac{3}{4}\div1\frac{1}{8}. Give your answer as a mixed number.

  1. Change both mixed numbers: 3\frac{3}{4}=\frac{3\times4+3}{4}=\frac{15}{4} and 1\frac{1}{8}=\frac{1\times8+1}{8}=\frac{9}{8}
  2. Keep, change, flip: \frac{15}{4}\div\frac{9}{8}=\frac{15}{4}\times\frac{8}{9}
  3. Cancel 15 and 9 by 3, and 8 and 4 by 4: \frac{15}{4}\times\frac{8}{9}=\frac{5}{1}\times\frac{2}{3}

  4. Your turn. Work out \frac{5}{1}\times\frac{2}{3}

    \frac{5\times2}{1\times3}=\frac{10}{3}

  5. 10\div3=3 remainder 1, so \frac{10}{3}=3\frac{1}{3}
  6. Answer: 3\frac{1}{3}

Multiply or divide? Fractions in problems

  • Multiply for a fraction of an amount, for an amount repeated several times, and for the area of a rectangle (length times width).
  • Divide to find how many pieces fit, to share equally, or to find the amount for one (one hour, one metre).

Sharing \frac{2}{3} of a pizza equally between 4 people is \frac{2}{3}\div4=\frac{2}{3}\times\frac{1}{4}=\frac{2}{12}. Each person gets \frac{1}{6} of the pizza.

For the amount in one hour, divide. A cyclist rides 1\frac{1}{2} km in \frac{1}{4} of an hour. In one hour she rides \frac{3}{2}\div\frac{1}{4}=\frac{3}{2}\times\frac{4}{1}=\frac{12}{2}=6 km.

To find a missing fraction, undo the multiplication by dividing. If \frac{2}{3}\times\square=\frac{8}{15}, then the missing fraction is \frac{8}{15}\div\frac{2}{3}=\frac{8}{15}\times\frac{3}{2}=\frac{24}{30}=\frac{4}{5}. Check: \frac{2}{3}\times\frac{4}{5}=\frac{8}{15}

Worked example 6: How many fit, and what is left

A baker has 5\frac{1}{2} kg of flour. Each loaf needs \frac{3}{4} kg of flour.

(a) Work out the greatest number of loaves the baker can make.

(b) How much flour is left over?

  1. (a) How many loaves fit means divide: 5\frac{1}{2}\div\frac{3}{4}=\frac{11}{2}\times\frac{4}{3}

  2. Your turn. Cancel the 4 and the 2, then work out \frac{11}{2}\times\frac{4}{3}

    \frac{11}{2}\times\frac{4}{3}=\frac{11}{1}\times\frac{2}{3}=\frac{22}{3}=7\frac{1}{3}

  3. Only whole loaves count. There is not enough flour for an 8th loaf, so the baker can make 7 loaves. Don't round 7\frac{1}{3} up.
  4. (b) 7 loaves use 7\times\frac{3}{4}=\frac{21}{4}=5\frac{1}{4} kg of flour.
  5. Flour left over: 5\frac{1}{2}-5\frac{1}{4}=\frac{1}{4} kg
  6. Answer: (a) 7 loaves (b) \frac{1}{4} kg

Common mistakes

  • Flipping a fraction when you multiply. Only flip when you divide; to multiply, multiply top by top and bottom by bottom.
  • Using a common denominator to multiply or divide. You don't need one, and it makes the numbers harder.
  • Multiplying the whole numbers and the fractions separately. Change mixed numbers to improper fractions first.
  • Flipping the wrong fraction when you divide. Keep the first fraction as it is and flip only the second.
  • Giving the answer in the wrong form. Simplify fully, and give a mixed number (not a decimal) when the question asks for one.
  • Rounding up the number of whole items. If 7\frac{1}{3} loaves fit, the answer is 7

Practice questions

More on this topic: Multiplying and Dividing Fractions worksheet with full solutions