The present age of Rajan is three times the age of his daughter and equal to nine thirteenths of the present age of his mother. The sum of the present ages of all three of them is 125 years. What is the difference between the present age of Rajan's daughter and the present age of Rajan's mother?

Introduction / Context:
This question involves three related people: Rajan, his daughter, and his mother. The relations between their ages are given using multiplication and a fractional comparison, and the total of their present ages is known. From this, we must find the difference between the ages of the daughter and the mother. Problems like this are common in aptitude tests on ages and require solving a small system of equations.

Given Data / Assumptions:
Rajan's present age is three times the age of his daughter. Rajan's present age is also nine thirteenths of his mother's present age. The sum of their present ages (Rajan, daughter, and mother) is 125 years. We must find the difference between the present age of Rajan's daughter and that of his mother.

Concept / Approach:
We introduce variables for the ages of the daughter and Rajan and express all other ages in terms of those variables. The relation that Rajan's age is three times his daughter's age gives one equation. The relation involving nine thirteenths gives a connection between Rajan's age and his mother's age. Adding these and using the total sum of ages gives a solvable system. Once the ages are found, the required difference can be calculated.

Step-by-Step Solution:
Step 1: Let the daughter's present age be D years.Step 2: Rajan's present age is three times his daughter's age, so his age is 3D years.Step 3: Rajan's age is nine thirteenths of his mother's age. So 3D = (9 / 13) * M, where M is the mother's age.Step 4: Rearrange to express M in terms of D: multiply both sides by 13 and divide by 9, giving M = (13 / 9) * 3D = (39D) / 9 = (13D) / 3.Step 5: The sum of their present ages is D + 3D + M = 125.Step 6: Substitute M = 13D / 3 into the sum: D + 3D + 13D / 3 = 125.Step 7: Combine terms: D + 3D = 4D. So 4D + 13D / 3 = (12D + 13D) / 3 = 25D / 3.Step 8: Set 25D / 3 = 125. Multiply both sides by 3: 25D = 375, so D = 375 / 25 = 15 years.Step 9: Then Rajan's age is 3D = 45 years, and M = 13D / 3 = 13 * 15 / 3 = 65 years.Step 10: The required difference between the mother's age and the daughter's age is 65 - 15 = 50 years.

Verification / Alternative check:
Check the sum: 15 (daughter) + 45 (Rajan) + 65 (mother) = 125 years, which matches the problem statement. Check the fractional relation: Rajan's age as nine thirteenths of his mother's age gives (9 / 13) * 65 = 585 / 13 = 45 years, which matches Rajan's age. This confirms that our calculations are correct.

Why Other Options Are Wrong:
A difference of 32, 48, or 62 years would correspond to different combinations of ages that do not satisfy all the constraints simultaneously. For example, they would either violate the three times relation between Rajan and his daughter or the nine thirteenths relation between Rajan and his mother, or fail to sum to 125 years. Only a difference of 50 years preserves all given relationships.

Common Pitfalls:
Common mistakes include mishandling the fraction nine thirteenths, incorrectly cross multiplying, or forgetting to express all three ages in terms of a single variable before using the sum. Another pitfall is to treat 9/13 as 13/9 by mistake. Writing each relationship slowly and checking each step prevents such errors.

Final Answer:
The difference between the present ages of Rajan's daughter and his mother is 50 years.