The ratio of the present ages of Mahesh and Nilesh is 5 : x. Mahesh is 18 years younger than Ramesh. After 9 years, Ramesh will be 47 years old. If the absolute difference between the present ages of Mahesh and Nilesh is equal to the present age of Ramesh, what is the value of x?

Introduction / Context:
This problem combines age ratios, fixed age differences, and a condition involving the difference between the ages of two people being equal to the age of a third person. It is a multi-step algebraic question typical of higher level problems on ages in aptitude tests.

Given Data / Assumptions:

• The present age ratio of Mahesh to Nilesh is 5 : x.
• Mahesh is 18 years younger than Ramesh.
• After 9 years, Ramesh will be 47 years old.
• The absolute difference between the present ages of Mahesh and Nilesh is equal to the present age of Ramesh.
• We need to find the value of x.

Concept / Approach:
The first step is to determine Ramesh present age using the future information. Then, using the relationship between Mahesh and Ramesh, we compute Mahesh age. Using the given ratio 5 : x, we express Nilesh age in terms of x. The condition that the absolute difference between Mahesh and Nilesh ages equals Ramesh age gives an equation in x, which we solve. Finally, we match the numeric value of x to the options.

Step-by-Step Solution:
After 9 years, Ramesh will be 47 years old.Therefore, Ramesh present age is 47 - 9 = 38 years.Mahesh is 18 years younger than Ramesh.So Mahesh present age = 38 - 18 = 20 years.The ratio of the ages of Mahesh and Nilesh is 5 : x.Let Mahesh age be 5 * k and Nilesh age be x * k.Since Mahesh age is 20, 5 * k = 20, so k = 4.Then Nilesh present age = x * k = 4 * x years.We are told that the absolute difference between Mahesh and Nilesh ages equals Ramesh present age.So |20 - 4 * x| = 38.This gives two possible equations: 20 - 4 * x = 38 or 4 * x - 20 = 38.The first equation 20 - 4 * x = 38 leads to -4 * x = 18, so x = -4.5, which is not a valid ratio value.The second equation is 4 * x - 20 = 38, so 4 * x = 58 and x = 58 / 4 = 14.5.

Verification / Alternative check:
If x = 14.5, then Nilesh present age is 4 * 14.5 = 58 years.Mahesh present age is 20 years and Ramesh present age is 38 years.The absolute difference between Mahesh and Nilesh ages is |20 - 58| = 38 years.This exactly matches Ramesh present age, confirming that x = 14.5 satisfies all conditions.

Why Other Options Are Wrong:
11.8, 12.9, 13.7: Each of these values for x produces an age for Nilesh that does not yield a difference of 38 years when compared with Mahesh age of 20 years.Furthermore, negative or illogical ratios would contradict the idea of meaningful age comparisons.

Common Pitfalls:
Some students forget that the difference is an absolute value and only consider one algebraic case.Others make sign errors while solving 20 - 4 * x = 38 or 4 * x - 20 = 38.It is also easy to mis-handle the ratio and incorrectly relate Mahesh age to x rather than to 5.

Final Answer:
The required value of x in the ratio of the ages of Mahesh and Nilesh is 14.5.