Teach Me Maths

Fractions - Four Operations

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How to add, subtract, multiply and divide fractions and mixed numbers without a calculator, change between mixed numbers and improper fractions, and use fractions in word problems.

Fractions in the exam

In the exam you are usually asked to add, subtract, multiply or divide two fractions, without a calculator, for 2 or 3 marks. The fractions are often mixed numbers, such as 3\frac{1}{2}, and the question may say "Give your answer as a mixed number". Some questions ask you to show that a calculation is correct, to find the mistake in someone's working, or to solve a problem in words. In a "Show that" question, write down every step until you reach the answer you were given.

A fraction describes equal parts of a whole. In \frac{3}{5}, the denominator is the bottom number, 5: the whole is split into 5 equal parts. The numerator is the top number, 3: it says how many of those parts you have.

Equivalent fractions have the same value. Multiply the top and the bottom by the same number: \frac{1}{2}=\frac{2}{4} and \frac{1}{3}=\frac{4}{12}. To simplify, divide the top and the bottom by the same number: \frac{6}{8}=\frac{3}{4}. Never change just one of them.

Both bars are the same length. One half and two quarters shade the same amount.

Adding and subtracting: make the pieces the same size

When the denominators are the same, the pieces are the same size. Add the numerators and keep the denominator: \frac{2}{9}+\frac{5}{9}=\frac{7}{9}. You have seven ninths, not seven eighteenths.

When the denominators are different, first write both fractions with a common denominator: a number that both denominators divide into. Use the smallest one you can find. For thirds and quarters, use twelfths.

Subtracting works in the same way. If the second fraction is bigger, the answer is negative: \frac{1}{5}-\frac{3}{5}=-\frac{2}{5}

Worked example 1: Adding thirds and quarters

Work out \frac{1}{3}+\frac{1}{4}

  1. Both 3 and 4 divide into 12, so use twelfths: \frac{1}{3}=\frac{4}{12}

  2. Your turn. \frac{1}{4}=\frac{\square}{12}. What number goes in the box?

    \frac{1}{4}=\frac{3}{12}

  3. Add the numerators and keep the denominator: \frac{4}{12}+\frac{3}{12}=\frac{7}{12}
  4. Answer: \frac{7}{12}

Worked example 2: Subtracting with a common denominator

Work out \frac{7}{8}-\frac{1}{6}

  1. The multiples of 8 are 8, 16, 24, … The first one that 6 also divides into is 24, so use twenty-fourths.
  2. Multiply the top and the bottom of \frac{7}{8} by 3, and of \frac{1}{6} by 4: \frac{7}{8}=\frac{21}{24} and \frac{1}{6}=\frac{4}{24}

  3. Your turn. Work out \frac{21}{24}-\frac{4}{24}

    \frac{21}{24}-\frac{4}{24}=\frac{17}{24}

  4. Answer: \frac{17}{24}. It is already in its simplest form, because no number other than 1 divides into both 17 and 24.

Multiplying fractions

To multiply fractions, multiply the numerators together and multiply the denominators together. You do not need a common denominator.

\frac{2}{5}\times\frac{1}{3}=\frac{2\times1}{5\times3}=\frac{2}{15}

The word "of" means multiply: \frac{2}{3} of \frac{3}{4} is \frac{2}{3}\times\frac{3}{4}. Simplify the answer if you can.

Worked example 3: A fraction of a fraction

Work out \frac{2}{3}\times\frac{3}{4}

  1. The rectangle has 12 equal cells. Three of the four columns show \frac{3}{4}. Two of the three rows in that part show \frac{2}{3} of \frac{3}{4}: 6 cells.
  2. Multiply the numerators and multiply the denominators: \frac{2\times3}{3\times4}=\frac{6}{12}

  3. Your turn. Divide the top and the bottom of \frac{6}{12} by 6. What is the new denominator?

    \frac{6\div6}{12\div6}=\frac{1}{2}

  4. Answer: \frac{1}{2}. The 6 shaded cells are half of the 12 cells.

Dividing by a fraction

\frac{3}{4}\div\frac{1}{8} asks how many eighths fit into three quarters. Three quarters is six eighths, so the answer is 6.

The reciprocal of a fraction is the fraction turned upside down. The reciprocal of \frac{4}{5} is \frac{5}{4}. The reciprocal of 3, which is \frac{3}{1}, is \frac{1}{3}.

To divide by a fraction, keep the first fraction and multiply by the reciprocal of the second:

\frac{3}{4}\div\frac{1}{8}=\frac{3}{4}\times\frac{8}{1}=\frac{24}{4}=6

Only do this for dividing. Never turn a fraction upside down to add, subtract or multiply.

Worked example 4: Dividing two fractions

Work out \frac{5}{6}\div\frac{2}{3}. Give your answer as a mixed number.

  1. Keep \frac{5}{6} and multiply by the reciprocal of \frac{2}{3}: \frac{5}{6}\div\frac{2}{3}=\frac{5}{6}\times\frac{3}{2}

  2. Your turn. Work out \frac{5}{6}\times\frac{3}{2}

    \frac{5\times3}{6\times2}=\frac{15}{12}

  3. Simplify by dividing the top and the bottom by 3: \frac{15}{12}=\frac{5}{4}
  4. 5\div4=1 remainder 1, so \frac{5}{4}=1\frac{1}{4}
  5. Answer: 1\frac{1}{4}. It is more than 1 because \frac{2}{3} fits into \frac{5}{6} more than once.

Mixed numbers and improper fractions

A mixed number, such as 2\frac{3}{4}, is a whole number and a fraction: 2+\frac{3}{4}. An improper fraction, such as \frac{11}{4}, has a numerator bigger than its denominator. They can be the same amount.

To change a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator. Keep the denominator:

2\frac{3}{4}=\frac{2\times4+3}{4}=\frac{11}{4}

Each whole is four quarters, so 2 wholes and 3 quarters make 11 quarters.

To change back, divide the numerator by the denominator. 11\div4=2 remainder 3, so \frac{11}{4}=2\frac{3}{4}. The remainder is the new numerator, and the denominator stays the same.

Before you multiply or divide, always change mixed numbers into improper fractions. To add or subtract, you can work with the whole numbers and the fractions separately, but changing to improper fractions always works. At the end, give your answer in the form the question asks for.

Worked example 5: Subtracting mixed numbers

Work out 4\frac{1}{6}-1\frac{3}{4}. Give your answer as a mixed number.

  1. Change both mixed numbers to improper fractions. One whole is six sixths.

  2. Your turn. 4\frac{1}{6}=\frac{\square}{6}. What number goes in the box?

    4\frac{1}{6}=\frac{4\times6+1}{6}=\frac{25}{6}

  3. In the same way, 1\frac{3}{4}=\frac{1\times4+3}{4}=\frac{7}{4}
  4. Use twelfths: \frac{25}{6}=\frac{50}{12} and \frac{7}{4}=\frac{21}{12}
  5. \frac{50}{12}-\frac{21}{12}=\frac{29}{12}, and 29\div12=2 remainder 5
  6. Answer: 2\frac{5}{12}

Worked example 6: Dividing mixed numbers

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

  1. Change both mixed numbers to improper fractions: 3\frac{1}{3}=\frac{10}{3} and 1\frac{1}{4}=\frac{5}{4}

  2. Your turn. Write down the reciprocal of \frac{5}{4}.

    The reciprocal of \frac{5}{4} is \frac{4}{5}

  3. \frac{10}{3}\div\frac{5}{4}=\frac{10}{3}\times\frac{4}{5}=\frac{40}{15}
  4. Simplify by dividing the top and the bottom by 5: \frac{40}{15}=\frac{8}{3}
  5. 8\div3=2 remainder 2, so \frac{8}{3}=2\frac{2}{3}
  6. Answer: 2\frac{2}{3}

Fractions in word problems

  • A total, or putting amounts together: add.
  • A difference, or the amount left: subtract.
  • A fraction of an amount: multiply.
  • How many equal portions fit: divide.

Read carefully what each fraction is a fraction of. If \frac{1}{4} of a cake is eaten, \frac{3}{4} is left. Half of what is left is \frac{1}{2}\times\frac{3}{4}=\frac{3}{8} of the whole cake, not half of the whole cake.

When you divide to find how many full cups, jars or pieces you can make, the answer may not be a whole number. For example, 5\div\frac{2}{3}=5\times\frac{3}{2}=7\frac{1}{2}, so only 7 cups can be filled completely.

To find a starting number, undo the working. If \frac{3}{4} of a number is 6, the number is 6\div\frac{3}{4}=6\times\frac{4}{3}=8. Check: \frac{3}{4} of 8 is 6.

Common mistakes

  • Adding the denominators. \frac{1}{3}+\frac{1}{4} is not \frac{2}{7}: write both fractions with a common denominator, then add only the numerators.
  • Changing the denominator but not the numerator. Multiply the top and the bottom by the same number.
  • Using a common denominator to multiply. To multiply, just multiply top by top and bottom by bottom.
  • Turning a fraction upside down when multiplying. Use the reciprocal only when you divide.
  • Taking the smaller fraction from the bigger one when subtracting mixed numbers. 4\frac{1}{6}-1\frac{3}{4} is not 3\frac{7}{12}: change to improper fractions first, then use a common denominator.
  • Leaving the answer in the wrong form. If the question asks for the simplest form, divide the top and the bottom by the same number; for a mixed number, divide the numerator by the denominator and use the remainder.

Practice questions

More on this topic: Fractions - Four Operations worksheet with full solutions