Question

Three numbers are in the ratio 1 : 2 : 3 and the sum of their cubes is 4500. Find the numbers.

Answer 1

Let the three numbers be x, 2x and 3x.
According to the question,
$(x{)}^3+(2x{)}^3+(3x{)}^3=4500\Rightarrow x^3+8x^3+27x^3=4500\Rightarrow 36x^3=4500\Rightarrow x^3=\frac{4500}{36}$
$\Rightarrow x^3=125$ [taking cube root on both sides]
$\Rightarrow x=\sqrt[3]{3125}\Rightarrow x=\sqrt[3]{35\times 5\times 5}\therefore x=5$
Hence, the numbers are 5, $2\times 5$ and $3\times 5$ i.e. 5, 10 and 15.