The ratio of the present ages of X and Y is 4:5. Which of the following options cannot represent the ratio of their ages 20 years ago?

Introduction / Context:
This is a conceptual problem on ages and ratios. We are given the ratio of the present ages of two people, X and Y, and we must determine which ratio among the options is impossible for their ages 20 years ago. The question checks understanding of how age differences remain constant over time while ratios change.

Given Data / Assumptions:
The present age ratio of X to Y is 4:5. We consider their ages 20 years ago and check which of the given ratios could or could not represent that past ratio. Ages are in years and assumed to be positive.

Concept / Approach:
Let the present ages be 4k and 5k for some positive number k. Twenty years ago, their ages were 4k - 20 and 5k - 20. For any proposed past ratio a:b to be possible, there must exist a positive k such that (4k - 20) : (5k - 20) = a : b. We can test each option by forming an equation and checking whether a valid positive k exists.

Step-by-Step Solution:
Step 1: Let present ages be 4k and 5k years.Step 2: Twenty years ago, X was 4k - 20 years and Y was 5k - 20 years.Step 3: Test option 2 : 5. Set (4k - 20) / (5k - 20) = 2 / 5. Cross multiply: 5 * (4k - 20) = 2 * (5k - 20). This gives 20k - 100 = 10k - 40, so 10k = 60 and k = 6 (valid). So 2 : 5 is possible.Step 4: Test option 8 : 15. Set (4k - 20) / (5k - 20) = 8 / 15. Cross multiply: 15 * (4k - 20) = 8 * (5k - 20), so 60k - 300 = 40k - 160, giving 20k = 140 and k = 7 (valid). So 8 : 15 is possible.Step 5: Test option 3 : 5. Set (4k - 20) / (5k - 20) = 3 / 5. Cross multiply: 5 * (4k - 20) = 3 * (5k - 20), so 20k - 100 = 15k - 60, giving 5k = 40 and k = 8 (valid). So 3 : 5 is possible.Step 6: Test option 9 : 10. Set (4k - 20) / (5k - 20) = 9 / 10. Cross multiply: 10 * (4k - 20) = 9 * (5k - 20), so 40k - 200 = 45k - 180, giving -5k = 20 and k = -4, which is not valid because k must be positive. Thus, 9 : 10 is impossible.

Verification / Alternative check:
For the valid k values, we can compute actual ages and see that they are reasonable. For example, with k = 6, present ages are 24 and 30, and 20 years ago they were 4 and 10, which form the ratio 2 : 5. Similar checks work for k = 7 and k = 8. However, for 9 : 10, the required k is negative, which would make the ages negative and impossible in a real context. So option 9 : 10 cannot represent the ratio 20 years ago.

Why Other Options Are Wrong:
The options 2 : 5, 8 : 15, and 3 : 5 all correspond to valid positive k values and yield meaningful positive ages 20 years ago, so they can be correct ratios in the past. Only 9 : 10 leads to a negative k, which makes it impossible for real ages. Therefore, 9 : 10 is the only option that cannot be the ratio 20 years ago.

Common Pitfalls:
Some learners may try to guess by inspection without considering the algebraic relationship between present and past ratios. Others may incorrectly assume that any ratio close to 4:5 must be valid in the past. The key is to remember that difference in ages is fixed, so not every arbitrary ratio is possible for past or future ages.

Final Answer:
The ratio that cannot represent the ages 20 years ago is 9 : 10.