In 2011, the arithmetic mean of the annual incomes of Ramesh and Suresh was Rs. 3800. The arithmetic mean of the annual incomes of Suresh and Pratap was Rs. 4800, and the arithmetic mean of the annual incomes of Pratap and Ramesh was Rs. 5800. What is the arithmetic mean of the incomes of the three?
Correct Answer: Rs. 4800
Let a, b, and c be the annual incomes of Ramesh, Suresh, and Pratap, respectively.Now, we are given thatThe arithmetic mean of the annual incomes of Ramesh and Suresh was Rs. 3800.Hence,$$\frac{{{\text{a}} + {\text{b}}}}{2}$$ = 3800
⇒ a + b = 2 × 3800 = 7600The arithmetic mean of the annual incomes of Suresh and Pratap was Rs. 4800.Hence,$$\frac{{{\text{b}} + {\text{c}}}}{2}$$ = 4800⇒ b + c = 2 × 4800 = 9600The arithmetic mean of the annual incomes of Pratap and Ramesh was Rs. 5800.Hence,$$\frac{{{\text{c}} + {\text{a}}}}{2}$$ = 5800
⇒ c + a = 2 × 5800 = 11,600Adding these three equations yields:(a + b) + (b + c) + (c + a) = 7600 + 9600 + 11,6002a + 2b + 2c = 28,800a + b + c = 14,400The average of the incomes of the three equals the sum of the incomes divided by 3,$$\eqalign{ & \frac{{{\text{a}} + {\text{b}} + {\text{c}}}}{3} \cr & = \frac{{14,400}}{3} \cr & = {\text{Rs}}{\text{.}}\,4800 \cr} $$
⇒ a + b = 2 × 3800 = 7600The arithmetic mean of the annual incomes of Suresh and Pratap was Rs. 4800.Hence,$$\frac{{{\text{b}} + {\text{c}}}}{2}$$ = 4800⇒ b + c = 2 × 4800 = 9600The arithmetic mean of the annual incomes of Pratap and Ramesh was Rs. 5800.Hence,$$\frac{{{\text{c}} + {\text{a}}}}{2}$$ = 5800
⇒ c + a = 2 × 5800 = 11,600Adding these three equations yields:(a + b) + (b + c) + (c + a) = 7600 + 9600 + 11,6002a + 2b + 2c = 28,800a + b + c = 14,400The average of the incomes of the three equals the sum of the incomes divided by 3,$$\eqalign{ & \frac{{{\text{a}} + {\text{b}} + {\text{c}}}}{3} \cr & = \frac{{14,400}}{3} \cr & = {\text{Rs}}{\text{.}}\,4800 \cr} $$