The present ages of Hema and Chinnu are in the ratio 13 : 17. Four years ago, their ages were in the ratio 11 : 15. What will be the ratio of their ages six years from now?

Introduction / Context:
This question involves present and past ratios of the ages of two people, Hema and Chinnu. From these ratios, we must determine their current ages and then compute the ratio of their ages six years in the future. It is a good test of working with ratios and linear equations in age-related problems.

Given Data / Assumptions:

- Present age ratio of Hema : Chinnu = 13 : 17.
- Four years ago, the age ratio of Hema : Chinnu was 11 : 15.
- Both grow older by the same number of years over time.
- We must find the ratio of their ages six years from now.

Concept / Approach:
We first represent the present ages as multiples of a common variable and then use the ratio from four years ago to form an equation. Solving that equation gives the value of the variable and hence the present ages of Hema and Chinnu. After finding their present ages, we add six years to each and form the required future ratio.

Step-by-Step Solution:
Step 1: Let the present ages of Hema and Chinnu be 13x and 17x years respectively. Step 2: Four years ago, Hema's age was 13x − 4 and Chinnu's age was 17x − 4. Step 3: The ratio four years ago is given as (13x − 4) : (17x − 4) = 11 : 15. Step 4: Form the equation (13x − 4) / (17x − 4) = 11 / 15. Step 5: Cross-multiply to get 15(13x − 4) = 11(17x − 4). Step 6: Expand both sides: 195x − 60 = 187x − 44. Step 7: Rearrange to obtain 8x = 16, so x = 2. Step 8: Present ages: Hema = 13 × 2 = 26 years, Chinnu = 17 × 2 = 34 years. Step 9: Six years from now, Hema will be 26 + 6 = 32 years and Chinnu will be 34 + 6 = 40 years. Step 10: The future ratio is 32 : 40, which simplifies to 4 : 5.

Verification / Alternative check:
We can recheck the ratio four years ago using the found values. Four years ago, Hema was 22 years and Chinnu was 30 years old. The ratio 22 : 30 simplifies to 11 : 15 when divided by 2, which matches the given past ratio. This consistency confirms that the present ages and the future ratio 4 : 5 are correct.

Why Other Options Are Wrong:
Ratios such as 3 : 7, 5 : 6, 7 : 9, and 1 : 3 do not match the ages 32 and 40 that we obtain after 6 years. When we simplify 32 : 40, we only get 4 : 5, so all other options must be rejected.

Common Pitfalls:
Common errors include using the wrong time frame when applying ratios (for example, mixing the present and four-years-ago ratios incorrectly) or not simplifying the final ratio properly. It is important to carefully translate each statement into equations and to verify the ratios at each stage.

Final Answer:
The ratio of the ages of Hema and Chinnu six years hence will be 4 : 5.