Question

If $a:b=c:d,$ prove that $xa+yb:xc+yd=b:d$

Answer 1

Prove that $xa+yb:xc+yd=b:d$
Given: $\frac{a}{b}=\frac{c}{d}$
Multiply both sides with $\frac{x}{y},$ we get
$\Rightarrow \frac{xa}{yb}=\frac{xc}{yd}$
$\Rightarrow \frac{xa+yb}{yb}=\frac{xc+yd}{yd}$ $[\because componendorule]$
By doing cross multiplication,we get,
$\Rightarrow \frac{xa+yb}{xc+yd}=\frac{yb}{yd}=\frac{b}{d}$
$\therefore xa+yb:xc+yd=b:d$
Hence, proved.