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jmc

algebra junior

Problem

Let

x

and

y

be real numbers satisfying

\frac{2}{x} = \frac{y}{3} = \frac{x}{y} .

Determine

x^{3} .

Solution

Let

k = \frac{2}{x} = \frac{y}{3} = \frac{x}{y} .

Then

k^{3} = \frac{2}{x} \cdot \frac{y}{3} \cdot \frac{x}{y} = \frac{2}{3} .

Also,

x = \frac{2}{k},

so

x^{3} = \frac{8}{k ^{3}} = \frac{8}{2/3} = 12 .

Final answer

12