Question 23

The product of two positive numbers $$x$$ and $$y$$ is $$4$$ times their sum and the same product is $$8$$ times their difference. If $$x\ge y$$, then $$x$$ is ______.

Correct Answer: 16

Solution

The conditions give $$xy=4(x+y)$$ and $$xy=8(x-y)$$. Equating the right sides gives $$4(x+y)=8(x-y)$$, so $$x=3y$$. Substituting into the first equation gives $$3y^2=16y$$, and positivity gives $$y=\frac{16}{3}$$ and hence $$x=16$$.

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