There are three positive numbers. One third of the average of all the three numbers is 8 less than the value of the highest number. The average of the lowest and the second lowest number is 8. What is the highest number?

There are three positive numbers. One third of the average of all the three numbers is 8 less than the value of the highest number. The average of the lowest and the second lowest number is 8. What is the highest number? Correct Answer 11

Let the three positive numbers in increasing order be a, b and c and the average of these numbers be be A.
Then,
$$\frac{{a + b + c}}{3} = A.....(i)$$
Given,
$$\eqalign{ & c - \frac{A}{3} = 8 \cr & \Rightarrow c - \frac{{a + b + c}}{9} = 8.....(ii) \cr} $$
Also given,
$$\eqalign{ & \frac{{b + a}}{2} = 8 \cr & \Rightarrow a + b = 16.....(iii) \cr} $$
Putting the value of (a + b) in equation (ii), we get
$$\eqalign{ & \Rightarrow c - \left( {\frac{{16 + c}}{9}} \right) = 8 \cr & \Rightarrow 9c - 16 - c = 72 \cr & \Rightarrow 8c = 72 + 16 \cr & \Rightarrow 8c = 88 \cr & \Rightarrow c = 11 \cr} $$
∴ Highest number = 11
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