# Adding two fractions with different denominators

### 分数加法 — 不同分母的情形

What are the results of the following two fraction additions?

a) (1 ⁄ 2) + (1 ⁄ 3) = ?

b) (1 ⁄ 2) + (2 ⁄ 3) = ?

The solution is right here!

a) (1 ⁄ 2) + (1 ⁄ 3) = (3 ⁄ 6) + (2 ⁄ 6) = (5 ⁄ 6)

b) (1 ⁄ 2) + (2 ⁄ 3) = (3 ⁄ 6) + (4 ⁄ 6) = (7 ⁄ 6)

All one needs is the concepts and operation of fractions. To elaborate, you need to know how to find the least common multiples.

Once the question a) was given to a student, and he got a surprising result (the teacher was surprised):

a’) (1 ⁄ 2) + (1 ⁄ 3) = 2 ⁄ 5

He (the student) has the following argument: (1 ⁄ 2) is to take one piece from two pieces, and (1 ⁄ 3) is to take one piece from three pieces, so in total there are 5 pieces and out of them we take 3. That’s 3 ⁄ 5!

This is obviously not the rule we are used in doing the arithmetic. And the teacher was persuading the student the result is not correct by turning all fractions into decimals:

1 ⁄ 2 = 0.5, 1 ⁄ 3 = 0.333…., and 2 ⁄ 5 = 0.4,
Note 0.5 + 0.333…. ≠ 0.4, therefore
(1 ⁄ 2) + (1 ⁄ 3) ≠ 2 ⁄ 5.

But this does not explain why the student’s argument is incorrect. If you were the teacher, how would you have explained the reason?
If you can do this, you can explain (b) similarly.

For your convenience, we show you the reference diagram for 1 ⁄ 2 and 1 ⁄ 3, as below.