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?

Solution(By Examveda Team)

Let a, b, and c be the annual incomes of Ramesh, Suresh, and Pratap, respectively.
Now, we are given that
The 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 = 7600
The 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 = 9600
The 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 = 11,600
Adding these three equations yields:
(a + b) + (b + c) + (c + a) = 7600 + 9600 + 11,600
2a + 2b + 2c = 28,800
a + b + c = 14,400
The average of the incomes of the three equals the sum of the incomes divided by 3,
$$\eqalign{ & \frac{{{\text{a}} + {\text{b}} + {\text{c}}}}{3} \cr & = \frac{{14,400}}{3} \cr & = {\text{Rs}}{\text{.}}\,4800 \cr} $$