Question

Three numbers are in the ratio $2:3:4$. The sum of their cubes is $0.334125$. Find the numbers.

Answer 1

Step 1: Form an equation based on given information

Let us assume the three number be $2n,3n,4n$

From given data, sum of cube of three numbers is $0.334125$

$\begin{array}{rcl}{\left(2n\right)}^3+{\left(3n\right)}^3+{\left(4n\right)}^3& =& 0.334125\\ 8n^3+27n^3+64n^3& =& 0.334125\\ 99n^3& =& 0.334125\\ n^3& =& \frac{0.334125}{99}\\ n^3& =& 0.003375\end{array}$

Step 2: Find the value of $n$

$\begin{array}{rcl}{n}^3& =& \frac{3375}{1000000}\\ & \Rightarrow & n=\sqrt[3]{\frac{3375}{1000000}}\\ & \Rightarrow & n=\sqrt[3]{\frac{15\times 15\times 15}{\left(10\times 10\times 10\right)\times \left(10\times 10\times 10\right)}}\\ & \Rightarrow & n={\left({\left(\frac{15}{100}\right)}^3\right)}^{\frac{1}{3}}\\ & \Rightarrow & n=\frac{15}{100}\\ & \Rightarrow & n=0.15\end{array}$

Step 3: Find the required numbers

The required numbers are

$2n=2\times 0.15=0.3$

$3n=3\times 0.15=0.45$

$4n=4\times 0.15=0.6$

Hence, the required numbers are $0.3,0.45,0.6$