Eight years ago Ajay's age was four third of Vijay's age, and eight years from now Ajay's age will be six fifth of Vijay's age. What is Ajay's present age in years?

Introduction / Context:
This is a classic two person age problem where ratios of ages are given at two different times, one in the past and one in the future. By translating the statements into equations, we can solve for the present ages. Such questions train you to handle linear equations and time shifts in aptitude exams.

Given Data / Assumptions:

• Eight years ago, Ajay's age was 4/3 times Vijay's age at that time.
• Eight years from now, Ajay's age will be 6/5 times Vijay's age at that time.
• We need to find Ajay's present age.
• Ages increase uniformly and there are no other hidden conditions.

Concept / Approach:
Let the present ages of Ajay and Vijay be A and V years. Use the condition from eight years ago to express A and V in one equation. Then use the condition from eight years in the future for a second equation. Solving these two linear equations simultaneously gives us both ages. We finally report Ajay's present age.

Step-by-Step Solution:
Let Ajay's present age be A years and Vijay's present age be V years. Eight years ago: A - 8 = (4/3) * (V - 8). Eight years hence: A + 8 = (6/5) * (V + 8). From the second equation, cross multiply: 5(A + 8) = 6(V + 8). This gives 5A + 40 = 6V + 48 so 5A = 6V + 8. From the first equation: 3(A - 8) = 4(V - 8). So 3A - 24 = 4V - 32 which simplifies to 3A = 4V - 8. Now solve 5A = 6V + 8 and 3A = 4V - 8 simultaneously. From 3A = 4V - 8, we get A = (4V - 8) / 3. Substitute into 5A = 6V + 8. 5 * (4V - 8) / 3 = 6V + 8 gives 20V - 40 = 18V + 24 so 2V = 64 and V = 32. Then A = (4 * 32 - 8) / 3 = (128 - 8) / 3 = 120 / 3 = 40.

Verification / Alternative check:
Check eight years ago: Ajay was 40 - 8 = 32 and Vijay was 32 - 8 = 24. Ratio 32 : 24 simplifies to 4 : 3, which matches. Eight years ahead: Ajay will be 48 and Vijay 40, so 48 / 40 = 6 / 5, which again matches. Hence the solution is correct.

Why Other Options Are Wrong:
If Ajay were 41, the ratios would not match exactly when checked in past and future. Ages 37 or 33 also fail either the 4/3 or the 6/5 relation. Only 40 years satisfies both time based ratios simultaneously, so the other choices are not consistent with the given conditions.

Common Pitfalls:
Typical errors include mixing up the direction of time shifts, forgetting to subtract or add 8 correctly, or mismanaging fractions when forming equations. Another common issue is solving only one of the equations and guessing the ages instead of solving the system. Careful algebra and checking both conditions prevent these mistakes.

Final Answer:
Therefore, Ajay's present age is 40 years.