The present ages of Manisha and Sudeshna are in the ratio 5 : 6. After 8 years, the ratio of their ages will be 7 : 8. What is the difference between their present ages?

Introduction / Context:
This question involves present and future age ratios of two people. By using the ratio form at two different times, we can set up an equation to determine their actual ages and then find the difference between them. Such problems test understanding of ratios and linear equations in age-related situations.

Given Data / Assumptions:

- Present age ratio of Manisha : Sudeshna = 5 : 6.
- After 8 years, the ratio of their ages will be 7 : 8.
- Both individuals grow older by the same 8 years over the period.
- We are required to find the difference between their present ages.

Concept / Approach:
When present ages are given in a ratio, we can represent them as multiples of a common variable. Then we apply the future ratio condition after adding the given years to both ages. This leads to a solvable linear equation in one variable. Once the variable value is found, we compute the actual ages and take their difference.

Step-by-Step Solution:
Step 1: Let the present ages of Manisha and Sudeshna be 5x and 6x years respectively. Step 2: After 8 years, their ages will be 5x + 8 and 6x + 8 years. Step 3: The ratio after 8 years is given as (5x + 8) : (6x + 8) = 7 : 8. Step 4: Form the equation (5x + 8) / (6x + 8) = 7 / 8. Step 5: Cross-multiply to get 8(5x + 8) = 7(6x + 8). Step 6: Expand both sides: 40x + 64 = 42x + 56. Step 7: Rearrange to find 2x = 8, so x = 4. Step 8: Present ages are Manisha = 5x = 20 years and Sudeshna = 6x = 24 years. Step 9: Difference between their present ages = 24 − 20 = 4 years.

Verification / Alternative check:
Check the ratio after 8 years: Manisha will be 20 + 8 = 28 years and Sudeshna will be 24 + 8 = 32 years. Their ratio is 28 : 32, which simplifies to 7 : 8 when both numbers are divided by 4. This matches the given future ratio, confirming that the ages and the difference of 4 years are correct.

Why Other Options Are Wrong:
Differences of 2 years, 6 years, 8 years, or 10 years do not satisfy both the current ratio 5 : 6 and the future ratio 7 : 8 when actual ages are computed. Only a difference of 4 years leads to consistent ages that match both ratio conditions.

Common Pitfalls:
A frequent mistake is to treat the ratio numbers as direct ages, ignoring that they must be multiplied by a common factor x. Another mistake is to forget to add the 8 years to both ages when using the future ratio. Carefully setting up the equation with variables and the correct time adjustment is essential.

Final Answer:
The difference between the present ages of Manisha and Sudeshna is 4 years.