Question Number 202315 by MATHEMATICSAM last updated on 24/Dec/23
$$\mathrm{If}\:{x}\::\:{y}\::\:{z}\:=\:{a}\::\:{b}\::\:{c}\:\mathrm{then}\:\mathrm{show}\:\mathrm{that} \\ $$$$\left(\frac{{a}\:+\:{b}\:+\:{c}}{{x}\:+\:{y}\:+\:{z}}\right)^{\mathrm{3}} \:=\:\frac{{abc}}{{xyz}}\:. \\ $$
Answered by AST last updated on 24/Dec/23
$${x}={ka};{y}={kb};{z}={kc} \\ $$$$\left(\frac{{a}+{b}+{c}}{{x}+{y}+{z}}\right)^{\mathrm{3}} =\frac{\mathrm{1}}{{k}^{\mathrm{3}} }=\frac{{abc}}{{k}^{\mathrm{3}} {abc}}=\frac{{abc}}{{xyz}} \\ $$