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Add fractions with the same denominator

Learn this first

You should already be able to

  • · Read a fraction: a whole split into equal parts, and how many of those parts you mean.
  • · Tell whether two fractions have the same denominator.
  • · Decide whether a fraction is in its simplest form, and simplify it if it is not.

The denominator tells you how many equal parts the whole was cut into. The numerator tells you how many of those parts this fraction is about.

One thing has to be true before any of this works: both fractions must be parts of the same whole, or of wholes the same size. An eighth of a small cake is not an eighth of a large one.

Take this bar as the whole and cut it into 8 equal parts. Each part is one eighth. Shade 3 parts, then 2 more: 5 parts are shaded. No part changed size, so this is 5 lots of one eighth — five eighths.

A bar is one whole, cut into 8 equal parts, so each part is one eighth. Three parts are shaded, then two more, so five of the eight parts are shaded: five eighths.32one whole, cut into 8 equal parts — each part is 1/8

So when the denominators match, add the numerators and keep the denominator — you are counting parts that are all one size. Then check whether the answer can be simplified.

Worked examples

Combining parts of one size. The answer needs no simplifying.

38 + 28 = ?

  1. 1. Both fractions are parts of the same whole, cut into 8 equal parts, so every part is one 8th.
  2. 2. 3 parts and 2 more parts is 5 parts, each still one 8th: 38 + 28 = 5/8.
  3. 3. 5 and 8 have no common factor greater than 1, so 58 is already in its simplest form.

Answer: 58

The same method, then the second step: the answer is simplified, shown division by division.

18 + 38 = ?

  1. 1. Both fractions are parts of the same whole, cut into 8 equal parts, so every part is one 8th.
  2. 2. 1 parts and 3 more parts is 4 parts, each still one 8th: 18 + 38 = 4/8.
  3. 3. 4 and 8 have a common factor greater than 1: 4. Divide both by 4: 48 = (4÷4)/(8÷4) = 1/2.
  4. 4. The way it is written changed; the amount did not. 48 and 12 are the same size.

Answer: 12

Try it, with help

29 + 59 = ?

Hint: Every part is one ninth, so count how many parts there are altogether. Then check whether the answer can be simplified.

Show the answer

79 — 2 + 5 = 7, so the answer is 7/9. 7 and 9 have no common factor greater than 1, so there is nothing to simplify.

210 + 310 = ?

Hint: Add the numerators first. Then look for a common factor greater than 1.

Show the answer

12 — 2 + 3 = 5, so first you get 5/10. 5 and 10 share the factor 5, so divide both by 5: 510 = 1/2.

If it went wrong

The denominators were added as well as the numerators.

Look at the two bars — they are the same length. Shading more parts of the first never cut it into more pieces; it is still cut into 8. Writing 516 means the whole was cut into 16, which makes every part smaller, and nobody cut anything. Ask yourself: after shading 2 more, did any part get smaller?

Two bars of the same length. The first is cut into 8 parts with 5 shaded; the second into 16 parts, also with 5 shaded. The same five parts cover much less of the second bar, because cutting into 16 makes every part smaller.55/8 — the right answer55/16 — the size the wrong answer means

Now try this: 17 + 27 = ?

Show the answer

37 — The whole stays cut into 7. One part plus two parts is three parts, each still one seventh.

The addition was right. The answer is just not in its simplest form yet.

Look for a common factor of the numerator and the denominator that is greater than 1 — every pair shares 1, so 1 tells you nothing. Find one, divide both by it, then look again, until no common factor greater than 1 is left.

The same whole cut into 8 equal parts with 4 shaded. The shaded part is exactly half the bar, so 4/8 and 1/2 are one amount written two ways.44 of 8 parts — exactly half the bar

Now try this: Is 69 in its simplest form? If not, simplify it.

Show the answer

23 — 6 and 9 share the factor 3, so divide both by 3: 69 = 2/3. 2 and 3 share nothing greater than 1, so that is simplest.

Same whole, same denominator: add the numerators, keep the denominator, then simplify if a common factor greater than 1 exists.

Try it now

Question 1

Calculate, giving your answer in its simplest form: 710 + 110

Calculate, giving your answer in its simplest form: 7/10 + 1/10
Question 2

Calculate, giving your answer in its simplest form: 612 + 212

Calculate, giving your answer in its simplest form: 6/12 + 2/12
Question 3

Calculate, giving your answer in its simplest form: 611 + 111

Calculate, giving your answer in its simplest form: 6/11 + 1/11

Topic quiz

Five questions on this topic, marked at the end. No hints and no answers along the way — that is what makes the result mean something.

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