The average age of Ram and Rahim is 10 years. The average age of Rahim and Ramesh is 15 years. The average age of Ram and Ramesh is 20 years. What is the age of the eldest of the three?

25 years

Average of age of 2 people = $\frac{Sum\; of\; ages}{2}$

⇒ Sum of ages = 2 x average

Using this formula, 

If the average age of Ram and Rahim is 10 years,

Sum of ages of Ram and Rahim = 20      -----(i)

and, if the average age of Rahim and Ramesh is 15 years,

Sum of ages of Rahim and Ramesh = 30          -----(ii)

and, if the average age of Ram and Ramesh is 20 years,

Sum of ages of Ram and Ramesh = 40            -----(iii)

Adding eq. (i), (ii), (iii),

Sum of ages of Ram, Rahim and Ramesh = $\frac{20+30+40}{2}$

                                                            = 45 years

Thus, age of Ram = 15 years, Rahim = 5 years and Ramesh = 25 years

The eldest among the three is Ramesh, whose age is 25 years