The present ages of Deepa and Hyma are in the ratio 5 : 6. After 4 years, the ratio of their ages will become 6 : 7. What is the difference between their present ages?

Introduction / Context:
This age problem uses two different ratios of the same two people taken at two different times. By expressing each person's age in terms of a common variable, we can form an equation from the future ratio and solve for their current ages. Then we find the difference between those ages.

Given Data / Assumptions:

• The ratio of the present ages of Deepa and Hyma is 5 : 6.
• After 4 years, the ratio of their ages will become 6 : 7.
• We need to determine the difference between their present ages.
• Ages increase linearly and there are no other conditions.

Concept / Approach:
Let the present ages be 5k and 6k. After 4 years, their ages will be 5k + 4 and 6k + 4. The ratio 6 : 7 in the future gives an equation that we solve for k. Once k is known, actual ages and their difference are straightforward to obtain.

Step-by-Step Solution:
Let Deepa's present age be 5k years and Hyma's present age be 6k years. After 4 years, Deepa's age will be 5k + 4 years. After 4 years, Hyma's age will be 6k + 4 years. Given that (5k + 4) / (6k + 4) = 6 / 7. Cross multiply: 7(5k + 4) = 6(6k + 4). So 35k + 28 = 36k + 24. Rearrange to get 28 - 24 = 36k - 35k, so 4 = k. Deepa's present age = 5k = 5 * 4 = 20 years. Hyma's present age = 6k = 6 * 4 = 24 years. Difference between present ages = 24 - 20 = 4 years.

Verification / Alternative check:
Check the ratio after 4 years. Deepa will be 24 and Hyma will be 28. The ratio 24 : 28 simplifies to 6 : 7 when both terms are divided by 4. This matches the condition given in the question. The present ratio 20 : 24 also simplifies to 5 : 6, confirming the setup and the solution.

Why Other Options Are Wrong:
A difference of 6 or 8 years would not preserve the ratio changes as given. A difference of 2 years is too small and results in ratios that do not match 5 : 6 and 6 : 7 at the required times. The option 5 years also fails the ratio check. Only a 4 year difference satisfies all conditions.

Common Pitfalls:
Some students try to handle the ratios by guessing rather than setting up an equation, which often leads to errors. Another common mistake is to forget to add 4 to both ages when forming the future ratio. Writing a clear equation based on the ratio definition avoids such mistakes.

Final Answer:
Therefore, the difference between their present ages is 4 years.