Question

If a + c = mb and ${\displaystyle \frac{1}{b}+\frac{1}{d}=\frac{m}{c}}$, prove that a, b, c and d are in proportion.

Answer 1

a + c = mb and ${\displaystyle \frac{1}{b}+\frac{1}{d}=\frac{m}{c}}$
a + c = mb
${\displaystyle \frac{a}{c}+\frac{c}{d}=m}$    (Dividing by b)...(i)

and ${\displaystyle \frac{1}{b}+\frac{1}{d}=\frac{m}{c}}$

${\displaystyle \frac{c}{b}+\frac{c}{d}=m}$    (Multiplying by c)...(ii)
From (i) and (ii)
${\displaystyle \frac{a}{b}+\frac{c}{b}}$

= ${\displaystyle \frac{c}{b}+\frac{c}{d}}$

⇒ ${\displaystyle \frac{a}{b}=\frac{c}{d}}$
Hence, a, b, c and d are proportional.