JAMB 1989 · UME · Q25

Find the range of values of $r$ which satisfies $\frac ra + \frac rb + \frac rc > 1$ , where $a$ , $b$ and $c$ are positive.

Worked solution (try it first)

Take $r$ out as a common factor: $r\left(\frac1a + \frac1b + \frac1c\right) > 1$ .
Add the fractions over the common denominator $abc$ : $\frac1a + \frac1b + \frac1c = \dfrac{bc + ac + ab}{abc}$ .
This fraction is positive, since $a$ , $b$ and $c$ are, so dividing by it keeps the sign: $r > \dfrac{abc}{bc + ac + ab}$ , option A.