Problem Analysis: Average Age Calculation

The problem asks for the average age of a father and his two sons. We are given the average ages for pairs of them.

Let:

• $F$ represent the Father's age
• $E$ represent the Elder son's age
• $Y$ represent the Younger son's age

Formulating Equations

From the given information, we can write the following equations:

• The average age of the father and elder son is 35 years: $ \frac{F + E}{2} = 35 \implies F + E = 70 \quad \text{(Equation 1)} $
• The average age of the father and younger son is 32 years: $ \frac{F + Y}{2} = 32 \implies F + Y = 64 \quad \text{(Equation 2)} $
• The average age of the two sons is 17 years: $ \frac{E + Y}{2} = 17 \implies E + Y = 34 \quad \text{(Equation 3)} $

Calculating Total Age Sum

To find the average age of the father and both sons, we first need their total combined age ($F + E + Y$). We can find this by adding the three equations:

$ (F + E) + (F + Y) + (E + Y) = 70 + 64 + 34 $

Combine the terms:

$ 2F + 2E + 2Y = 168 $

Factor out the 2:

$ 2(F + E + Y) = 168 $

Solve for the sum of the ages:

$ F + E + Y = \frac{168}{2} = 84 $

Calculating the Final Average Age

The average age of the father and his two sons is the total sum of their ages divided by 3 (since there are three people).

Average Age = $ \frac{F + E + Y}{3} $

Substitute the calculated sum:

Average Age = $ \frac{84}{3} $

Average Age = $ 28 $ years