Question

How do you add rational numbers?

Answer 1

I assume you know that if you multiply both numerator and denominator of a fraction by a same number, you get an equivalent fraction. Thus, for example, if you start from 2/3 and multiply both numerator and denominator by 3, you get 6/9, which is indeed equivalent to 2/3.

Now, if you want to add two fraction, you first of all transform both of them as just shown, obtaining two equivalent fractions with the same denominator. At this point, you have a sum of two fraction of the form `\frac{a}{b}+\frac{c}{b}`, which is easily `\frac{a+c}{b}`.

To do so, you look for the least common multiple of the two denominator. Let's say that we have to calculate `\frac{3}{5} + \frac{5}{8}`. The least common multiple of 5 and 8 is 40, so we have to transform `\frac{3}{5}` into `\frac{24}{40}` (multiplying numerator and denominator by 8), and then we transform `\frac{5}{8}` into `\frac{25}{40}` (multiplying numerator and denominator by 5).

These are equivalent fraction, so we can be sure that `\frac{3}{5} + \frac{5}{8}` equals `\frac{24}{40} + \frac{25}{40}`. The advantage is, of course, that the second one is much easier to compute, since one immediately gets that `\frac{24}{40} + \frac{25}{40}=\frac{49}{40}`

If something isn't clear, don't hesitate to ask:)