Question

If (ma + nb) : b : : (mc + nd) : d, prove that a, b, c, d are in proportion.

Answer 1

Consider, $(ma+nb):b::(mc+nd):d$

$⟹\frac{(ma+nb)}{b}=\frac{(mc+nd)}{d}$

$d(ma+nb)=b(mc+nd)$

$dma+dnb=bmc+bnd$

$dma-bmc=bnd-dnb$

$m(da-bc)=n(bd-db)$

$m(da-bc)=n\left(0\right)$

$m(da-bc)=0$

$da-bc=0$ ---- As $m$ cannot be equal to $0$

$\therefore da=bc$

$\frac{a}{b}=\frac{c}{d}$

Thus, $a,b,c$ and $d$ are in proportion.