Question

If a, b, c and d are positive integers with a x b x c = 200, b x c x d = 90, c x d x a = 54 and d x a x b = 48, the value of a x b x c x d is: explain​

Answer 1

$\textbf{Given:}$

$a{\times}b{\times}c=200$

$b{\times}c{\times}d=90$

$c{\times}d{\times}a=54$

$d{\times}a{\times}b=48$

$\textbf{To find:}$

$\text{The value of a{\times}b{\times}c{\times}d }$

$\textbf{Solution:}$

$\text{Consider,}$

$a{\times}b{\times}c=200$

$b{\times}c{\times}d=90$

$c{\times}d{\times}a=54$

$d{\times}a{\times}b=48$

$\text{Multiplying all these equations, we get}$

$a^3{\times}b^3{\times}c^3{\times}d^3=200{\times}90{\times}54{\times}48$

$a^3{\times}b^3{\times}c^3{\times}d^3=1000{\times}2{\times}9{\times}27{\times}2{\times}6{\times}8$

$a^3{\times}b^3{\times}c^3{\times}d^3=10^3{\times}2^3{\times}3^3{\times}3^3{\times}2^3$

$(a{\times}b{\times}c{\times}d)^3=10^3{\times}2^3{\times}3^3{\times}3^3{\times}2^3$

$\text{Taking cube root on bothsides, we get}$

$a{\times}b{\times}c{\times}d=10{\times}2{\times}3{\times}3{\times}2$

$\implies\,a{\times}b{\times}c{\times}d=360$

$\therefore\textbf{The value of \bf\,a{\times}b{\times}c{\times}d is 360}$