How do you add rational numbers?

Jan 21, 2015

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 $\setminus \frac{a}{b} + \setminus \frac{c}{b}$, which is easily $\setminus \frac{a + c}{b}$.

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

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

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