The ages of X and Y are in the ratio 6 : 5 and the sum of their present ages is 44 years. What will be the ratio of their ages exactly 8 years from now?

Introduction / Context:
This problem on ages uses a simple ratio along with the total of ages to form linear equations. Once the present ages are known, we project them into the future and compute the new ratio. Questions like this are very common in aptitude tests and are a good example of how proportional reasoning and basic algebra work together.

Given Data / Assumptions:

• The present ages of X and Y are in the ratio 6 : 5.
• The sum of their present ages is 44 years.
• We are asked to find the ratio of their ages after 8 years.
• Ages increase linearly with time and there is no other change in the relationship.

Concept / Approach:
We represent ages in terms of a common multiplier using the ratio. Then we use the given total age to find the value of this multiplier. Once we know the actual current ages of X and Y, we add 8 years to each and simplify the resulting ratio. The key algebra ideas are ratios, linear equations and simplification of fractions.

Step-by-Step Solution:
Let the present ages of X and Y be 6k and 5k years respectively. Given that 6k + 5k = 44, so 11k = 44. Therefore k = 44 / 11 = 4. So the present age of X = 6 * 4 = 24 years. The present age of Y = 5 * 4 = 20 years. After 8 years, the age of X will be 24 + 8 = 32 years. After 8 years, the age of Y will be 20 + 8 = 28 years. Required ratio after 8 years = 32 : 28. Divide both terms by 4 to simplify: 32 : 28 = 8 : 7.

Verification / Alternative check:
We can quickly check the total present age: 24 + 20 = 44, which matches the given sum. The ratio 24 : 20 simplifies to 6 : 5, which matches the given ratio. After 8 years, the ages 32 and 28 differ by 4 years, the same difference as now, which is consistent. Simplifying 32 : 28 by 4 correctly gives 8 : 7, confirming the result.

Why Other Options Are Wrong:
The ratio 6 : 5 corresponds to the present ages, not the future ages. The ratio 7 : 6 does not come from any correct computation. The ratio 9 : 5 would require one person to age faster than the other, which does not happen. The ratio 5 : 4 is unrelated to the given data and does not match the calculations.

Common Pitfalls:
A common error is to take 6 and 5 directly as ages instead of 6k and 5k. Another mistake is to forget to add 8 years to both ages before taking the ratio or to add 8 only to one of them. Some students also simplify the ratio incorrectly or mix up the order of X and Y. Carefully writing each algebraic step helps avoid these issues.

Final Answer:
Thus, the ratio of the ages of X and Y after 8 years will be 8 : 7.