Three numbers are in the ratio 1:2:3. The sum of their cubes is 7776. Find the numbers.

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Solution

Let the numbers be x, 2x, and 3x
sum of their cubes is
$x^{3} + 8 x^{3} + 27 x^{3} = 7776 36 x^{3} = 7776 x^{3} = 216 x = \sqrt[3]{6 \times 6 \times 6} x = 6$
2x = 2(6) = 12
3x = 3(6) = 18
The numbers are 6,12,18.

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