Common denominator

Statement

Write each expression as a single fraction with a common denominator: (a) $\dfrac{2x}{5} - \dfrac{7}{20x}$; (b) $\dfrac{3 - r^2}{4r} - \dfrac{r}{2}$.

Solution

(a) The least common multiple of the denominators is $20x$: $\frac{2x}{5} - \frac{7}{20x} = \frac{2x \cdot 4x}{20x} - \frac{7}{20x} = \frac{8x^2 - 7}{20x}.$

(b) The least common multiple is $4r$: $\frac{3 - r^2}{4r} - \frac{r}{2} = \frac{3 - r^2}{4r} - \frac{2r^2}{4r} = \frac{3 - 3r^2}{4r} = \frac{3(1 - r^2)}{4r}.$

$\boxed{\dfrac{8x^2 - 7}{20x}\ ;\quad \dfrac{3(1 - r^2)}{4r}}$