JAMB 1984 · UME · Q36

If $pq + 1 = q^2$ and $t = \dfrac1p - \dfrac1{pq}$ , express $t$ in terms of $q$ .

Worked solution (try it first)

Put $t$ over one denominator: $t = \frac1p - \frac{1}{pq}$
$= \frac{q - 1}{pq}$ .
From $pq + 1 = q^2$ , subtract 1: $pq = q^2 - 1$ .
So $t = \frac{q - 1}{q^2 - 1}$
$= \frac{q - 1}{(q - 1)(q + 1)}$ , and cancelling $q - 1$ leaves $\dfrac{1}{q + 1}$ , option C.