What Is 1/2 + 1/3? A Full Worked Answer

The answer, and the working

1/2 + 1/3 = 5/6

Step 1: the denominators are 2 and 3. The smallest number both divide into is 6.

Step 2: rewrite each fraction over 6. 1/2 = 3/6, because 2 × 3 = 6 so the top is multiplied by 3 too. 1/3 = 2/6, because 3 × 2 = 6 so the top is multiplied by 2.

Step 3: add the numerators. 3 + 2 = 5, so the answer is 5/6.

Step 4: check whether it simplifies. 5 is prime and does not divide 6, so 5/6 is already in lowest terms.

Why it is not 2/5

Adding the tops and the bottoms gives 2/5, and it is by far the most common wrong answer to this question. It is worth understanding why it fails rather than just being told not to do it.

1/2 is exactly half. 1/3 is a bit less than half. Added together, they should come to a little less than one whole.

2/5 is less than a half. So the "answer" is smaller than one of the things you started with, which is impossible when adding two positive amounts.

5/6, on the other hand, is a bit less than 1 — exactly where the answer should be.

Checking with decimals

If you are unsure of a fraction answer, converting to decimals is a quick and reliable check.

1/2 = 0.5. 1/3 = 0.3333… Adding gives 0.8333…

And 5/6 = 0.8333… They match, so the answer is right.

The wrong answer 2/5 is 0.4, which is nowhere near. This check takes fifteen seconds and catches almost every fraction error.

Seeing it as a picture

Picture a chocolate bar divided into six squares.

Half the bar is 3 squares. A third of the bar is 2 squares.

Put them together and you have 5 squares out of 6. There is one square left, which is the 1/6 short of a whole bar.

Six works as the number of squares because it is the smallest number that splits cleanly into both halves and thirds — which is exactly what finding a common denominator is doing.

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