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A family's combined ages

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A father has $2$ children. Today the father's age and the ages of both children add up to $60$ years. In $10$ years the father's age will equal the sum of the children's ages at that time. How old is the father today, in years?

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Hint

Over $10$ years the children's total grows by more than the father's age does.

Solution

Let $f$ be the father's age and $c$ the children's total today, so $f + c = 60$ .
In $10$ years the father gains $10$ years, but the children together gain $2 \times 10 = 20$ .
The condition is $f + 10 = c + 20$ , so $f = c + 10$ .
Substituting gives $2 c + 10 = 60$ , so $c = 25$ and the father is $25 + 10 = 35$ .
So the father is $35$ years old today.

Answer: $35$

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