The present ages of a mother and her son are in the ratio 11 : 5. When the son becomes as old as his mother is now, then the ratio of his father's age to that of his mother is 19 : 17. When the son becomes as old as his father is now, then the sum of his father's age and his age will be 170 years. What is the father's present age?

### Concept & Logic In age-related problems, the absolute difference in ages between any two individuals remains strictly constant over time, regardless of past or future scenarios. ### Step-by-Step Solution 1. Let the present ages of the mother and son be $11x$ and $5x$, respectively. 2. The difference in their ages is $11x - 5x = 6x$. This difference is constant. 3. The first future scenario is "when the son becomes as old as his mother is now". This means the son becomes $11x$. This happens after $6x$ years. 4. In this future scenario ($+6x$ years): - Mother's age = $11x + 6x = 17x$. - The ratio of Father : Mother is given as 19 : 17. - Since the Mother's age is exactly $17x$, the Father's age at this time must be $19x$. 5. Determine the Father's present age: - Father's present age = Father's future age - $6x = 19x - 6x = 13x$. 6. The second future scenario is "when the son becomes as old as his father is now". This means the son becomes $13x$. This happens after $8x$ years (since $13x - 5x = 8x$). 7. In this second future scenario ($+8x$ years): - Son's age = $13x$. - Father's age = $13x + 8x = 21x$. - The sum of their ages is 170: $$ 13x + 21x = 170 $$ $$ 34x = 170 \implies x = 5 $$ 8. Find the Father's present age: - Father's present age = $13x = 13 \times 5 = 65$ years. ### Exam Strategy & Shortcut Track the age increments directly in terms of units ($x$). Son=5, Mother=11. Increment to reach Mother's age = 6 units. So future Mother = 17 units. Father is 19 units then. Present Father = $19 - 6 = 13$ units. To reach Father's present age, Son needs $13 - 5 = 8$ units. Future Father = $13 + 8 = 21$ units. Sum = $13 + 21 = 34$ units. $34 \text{ units} = 170 \implies 1 \text{ unit} = 5$. Father = $13 \times 5 = 65$. No complex algebraic substitution required. ### Common Pitfall A common mistake is adding the future years incorrectly (e.g., adding to the father's future age instead of present age) or losing track of which timeframe you are operating in. ### Final Answer Therefore, the correct answer is **65 years**.