Adding and subtracting with like denominators

2/7 + 3/7 = 5/7. Sevenths plus sevenths are still sevenths, so only the numerators change.

Counting pieces of the same size

Three apples and two apples make five apples. Nobody adds the word apples to itself and gets apple-apples. Fractions with the same denominator work the same way. The denominator names the size of the piece, like the word apples, and the numerator counts the pieces. So 2/7 + 3/7 means two sevenths plus three sevenths, which is five sevenths: 5/7.

The rule: when the denominators are the same, add or subtract the numerators and keep the denominator. In symbols, a/n + b/n = (a + b)/n and a/n − b/n = (a − b)/n. That is the whole method. The rest of this unit is about using it carefully and tidying up the result.

Why you must not add the denominators

The most common mistake is to add the denominators too: 2/7 + 3/7 = 5/14. A quick check shows the problem. 5/14 is less than 1/2, while 3/7 alone is almost 1/2, and adding 2/7 to it must make it larger. Adding denominators would mean the pieces somehow shrink when you put them together. Adding fractions never changes the size of a piece, only how many pieces you have.

Simplify the answer

The sum of two fractions in simplest form is not always in simplest form. 1/8 + 3/8 = 4/8, and 4/8 simplifies to 1/2. 7/10 − 3/10 = 4/10 = 2/5. 5/12 + 1/12 = 6/12 = 1/2. Always look at the result and divide out any common factor, as in unit 4.

Answers bigger than 1

When the numerators add up to more than the denominator, the answer is more than one whole. 5/6 + 5/6 = 10/6. Simplify by 2 to get 5/3, which is 1 whole and 2/3 more. Unit 8 shows how to switch between 5/3 and the mixed number 1 2/3. Either form is a correct answer; improper fractions are usually easier to keep calculating with.

Subtracting from 1

To subtract a fraction from 1, write 1 as a fraction with the same denominator. 1 − 3/10 = 10/10 − 3/10 = 7/10. If 3/10 of a tank of fuel has been used, 7/10 remains. The same trick works for any whole number: 2 − 3/4 = 8/4 − 3/4 = 5/4.

Walking along the number line

The number line from unit 2 gives another picture of the same rule. Adding a fraction with the same denominator is a walk to the right in steps of that size. For 3/8 + 4/8, start at 3/8 and take 4 more eighth steps: you land on 7/8. Subtracting walks to the left. The picture also shows why a sum can pass 1. Start at 5/6 and take five more sixth steps: the first one brings you to 6/6, which is 1, and the other four carry you on to 10/6.

Subtraction can always be checked by adding back. If 9/10 − 4/10 = 5/10, then 5/10 + 4/10 must give 9/10 again, and it does. A check like this takes a few seconds and catches most slips with signs and numerators.

More than two fractions

The rule extends to any number of fractions with the same denominator. 1/5 + 2/5 + 4/5 = 7/5. Mixed additions and subtractions work left to right: 9/11 − 4/11 + 2/11 = 5/11 + 2/11 = 7/11.

Worked examples

Example 1. You pour 3/4 cup of water into a bowl, then another 3/4 cup. How much water is in the bowl? 3/4 + 3/4 = 6/4 = 3/2 cups, one and a half cups.

Example 2. A trail is 9/10 km long and you have walked 4/10 km. How far is left? 9/10 − 4/10 = 5/10 = 1/2 km.

Example 3. A film runs for 7/4 hours and you have watched 5/4 hours. 7/4 − 5/4 = 2/4 = 1/2 hour, so 30 minutes remain.