If the average of s number is r⁴ and the average of r numbers is s⁴, then find the average of all r + s numbers?

Solution (By Examveda Team)

$$\eqalign{ & {\text{Average}} = \frac{{{\text{Sum}}}}{{{\text{Number}}}} \cr & {\text{Sum}} = {r^4} \times s \cr & {\text{Again sum}} = r \times {s^4} \cr & {\text{Average of all }}r{\text{ and }}s{\text{ numbers}} \cr & = \frac{{{r^4} \times s + r \times {s^4}}}{{r + s}} \cr & = \frac{{rs\left( {{r^3} + {s^3}} \right)}}{{r + s}} \cr & = \frac{{rs\left( {r + s} \right)\left( {{r^2} + {s^2} - rs} \right)}}{{r + s}} \cr & = rs\left( {{r^2} + {s^2} - rs} \right) \cr} $$