Adding and Subtracting Fractions

In the exam you are usually asked to add or subtract two fractions, without a calculator, for 2 or 3 marks. The fractions are often mixed numbers, such as 2\frac{3}{4}, and the question may say "Give your answer as a mixed number". Other questions put fractions into a story: a total, an amount that is left, or the rest of something.

The key idea is that you can only add or subtract pieces of the same size. \frac{1}{2}+\frac{1}{4} is not \frac{2}{6}, because halves and quarters are different sizes. Change the half into quarters first: \frac{2}{4}+\frac{1}{4}=\frac{3}{4}

When the denominators are the same, the pieces are already the same size. Add or subtract the numerators, and keep the denominator:

\frac{5}{12}+\frac{4}{12}=\frac{9}{12}

The answer is nine twelfths, not nine twenty-fourths: the pieces are still twelfths.

Then simplify if you can, by dividing the top and the bottom by the same number. Dividing both by 3 gives \frac{9}{12}=\frac{3}{4}. If the question says "in its simplest form", you must do this.

If the numerator of your answer is bigger than the denominator, the answer is more than 1. An improper fraction such as \frac{7}{5} is fine, unless the question asks for a mixed number. Then write it as 1\frac{2}{5}

When the denominators are different, change one or both fractions into equivalent fractions with the same denominator, a common denominator. Multiply the top and the bottom by the same number, so the value does not change:

\frac{2}{3}=\frac{2\times4}{3\times4}=\frac{8}{12}

Sometimes one denominator divides into the other. For \frac{1}{2}+\frac{3}{10}, 2 divides into 10, so use tenths and change only the half: \frac{1}{2}=\frac{5}{10}. Then \frac{5}{10}+\frac{3}{10}=\frac{8}{10}=\frac{4}{5}

Otherwise, list the multiples of the bigger denominator until you reach one that the smaller denominator also divides into. For quarters and tenths, the multiples of 10 are 10, 20, … and 4 divides into 20. So use twentieths.

Multiplying the two denominators together always gives a common denominator: 4\times10=40 works too. The smallest one keeps the numbers small, so there is less to simplify at the end.

A mixed number is a whole number and a fraction: 2\frac{1}{3} means 2+\frac{1}{3}. To add mixed numbers, add the whole numbers, then add the fractions.

For 1\frac{1}{5}+3\frac{2}{5}, the whole numbers give 1+3=4 and the fractions give \frac{1}{5}+\frac{2}{5}=\frac{3}{5}. So the answer is 4\frac{3}{5}

If the fractions add up to more than 1, change the extra into a whole number and add it on. For 2\frac{2}{3}+1\frac{2}{3}, the whole numbers give 3 and the fractions give \frac{4}{3}=1\frac{1}{3}. So the answer is 3+1\frac{1}{3}=4\frac{1}{3}

You can also change both mixed numbers into improper fractions first. Multiply the whole number by the denominator and add the numerator:

2\frac{2}{3}=\frac{2\times3+2}{3}=\frac{8}{3}

This always works, but the numbers get bigger.

To subtract mixed numbers, you can subtract the whole numbers and the fractions separately. 4\frac{5}{7}-1\frac{2}{7}=3\frac{3}{7}, because 4-1=3 and \frac{5}{7}-\frac{2}{7}=\frac{3}{7}

Take care when the fraction you are taking away is the bigger one. In 3\frac{1}{4}-1\frac{3}{4}, you cannot take \frac{3}{4} from \frac{1}{4}, and you must not swap them round. Instead, change one of the wholes into quarters: 3\frac{1}{4}=2\frac{5}{4}

Now subtract: 2\frac{5}{4}-1\frac{3}{4}=1\frac{2}{4}=1\frac{1}{2}

If this feels tricky, change both mixed numbers into improper fractions instead: \frac{13}{4}-\frac{7}{4}=\frac{6}{4}=1\frac{1}{2}. Both ways give the same answer.

In a problem, "altogether" or "in total" means add. "How much is left", "how much more" or "the difference" means subtract. When fractions describe parts of one thing, all the parts together make 1 whole.

To take a fraction away from a whole number, use one of the wholes. One whole is \frac{3}{3}, so 1-\frac{2}{3}=\frac{1}{3}, and 4-\frac{2}{3}=3\frac{1}{3}

For a missing number, use the opposite operation. If \frac{1}{6}+\square=\frac{3}{4}, the missing number is \frac{3}{4}-\frac{1}{6}=\frac{9}{12}-\frac{2}{12}=\frac{7}{12}

Check that your answer makes sense. An amount that is left must be less than the amount you started with, and you can check a subtraction by adding: \frac{1}{6}+\frac{7}{12}=\frac{2}{12}+\frac{7}{12}=\frac{9}{12}=\frac{3}{4}

• Adding or subtracting the denominators. \frac{2}{5}+\frac{1}{3} is not \frac{3}{8}: write both fractions with a common denominator, then add only the numerators.
• Changing the denominator but not the numerator. Whatever you multiply the bottom by, multiply the top by the same number.
• Swapping the fractions round in a mixed number subtraction. When the fraction you take away is bigger, exchange one whole first, or use improper fractions.
• Turning a fraction upside down. That is only for dividing; never do it when you add or subtract.
• Not simplifying fully. If the question asks for the simplest form, keep dividing the top and the bottom until no number except 1 divides into both.
• Leaving the answer in the wrong form. If the question asks for a mixed number, divide the numerator by the denominator and write the remainder over the denominator.