Question : If the average of $s$ numbers is $r^4$ and the average of $r$ numbers is $s^4$, then find the average of all $r+s$ numbers.
Option 1: $r s\left(r^2+s^2-r s\right)$
Option 2: $rs$
Option 3: ${r}^2+s^2$
Option 4: $rs\left(r^2+s^2\right)$
Correct Answer: $r s\left(r^2+s^2-r s\right)$
Solution :
Given, the average of $s$ numbers is $r^4$ and the average of $r$ numbers is $s^4$
Total sum of $s$ numbers = $s\times r^4=sr^4$
Total sum of $r$ numbers = $r\times s^4= rs^4$
Now, the average of all $r+s$ numbers
= $\frac{(rs^4+sr^4)}{ (r+s)}$
= $\frac{rs(s^3+r^3)}{ (r+s)}$
= $\frac{rs(s+r)(s^2-sr+r^2)}{ (r+s)}$
= $rs(r^2+s^2-rs)$
Hence, the correct answer is $rs(r^2+s^2-rs)$.
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