1.5.4 a/b + b/a Trick

Let’s look at when we add the two fractions $\frac{a}{b} + \frac{b}{a}$:

$$\frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab} = \frac{2ab - 2ab + a^2 + b^2}{ab} = 2 + \frac{(a - b)^2}{ab}$$

Here is an example:

$$\frac{5}{7} + \frac{7}{5} = 2 + \frac{(7 - 5)^2}{7 \cdot 5} = 2 \frac{4}{35}$$

There are some variations to this trick. For example:

$$\frac{11}{13} + \frac{2}{11} = \left(\frac{11}{13} + \frac{13}{11}\right) - \frac{11}{11} = 2 + \frac{2^2}{143} - 1 = 1 \frac{4}{143}$$

This is a popular variation that is used especially on the last column of the test because the trick is there but not as obvious.

Problem Set 1.5.4