The present ages of three persons are in the proportion 4 : 7 : 9. Eight years ago, the sum of their ages was 56 years. The present age of the eldest person is

### Concept & Logic When given a ratio for present ages and a sum for past ages, we must bring the sum to the present timeline. For every year that passes, the sum of $n$ people's ages increases by $n$. $$ \text{Present Sum} = \text{Past Sum} + (n \times \text{years}) $$ ### Step-by-Step Solution * **Given:** Present ratio = $4:7:9$. Sum 8 years ago = 56. Number of people = 3. * **Calculation:** Find the present sum of their ages. Since 8 years have passed for all 3 people, the total increase is $3 \times 8 = 24$ years. * Present sum = $56 + 24 = 80$ years. * Let the present ages be $4x$, $7x$, and $9x$. * $4x + 7x + 9x = 80 \implies 20x = 80 \implies x = 4$. * The eldest person has the largest ratio share ($9x$). * Present age of the eldest = $9 \times 4 = 36$ years. ### Exam Strategy & Shortcut **Ratio Multiples:** The present age of the eldest is represented by 9 units in the ratio. Therefore, the answer must be a multiple of 9. Among the options, 28 is not, 45 is ($9 \times 5$), and 36 is ($9 \times 4$). You only have to test two options, or just quickly find the sum multiplier. ### Common Pitfall Forgetting to multiply the 8 years by the number of people. Many students simply add 8 to 56, getting 64, which leads to incorrect fractional ages and confusion. ### Final Answer Therefore, the correct answer is **36 years**.