Six years ago, Parvez’s age was equal to the present age of Manish. Parvez’s present age is one-fourth more than Manish's present age. In how many years from now will Parvez’s age become exactly double Manish’s present age?

Introduction / Context:
This age problem links Parvez and Manish with both a past equality and a current percentage relationship. We are told that Parvez's age six years ago was the same as Manish's present age, and Parvez is now one-fourth more than Manish. We must determine after how many years Parvez's age will be exactly double Manish's present age. This tests your ability to translate word statements into algebraic equations and use both past and future time references correctly.

Given Data / Assumptions:

- Six years ago, Parvez's age was equal to Manish's present age.
- Parvez's present age is one-fourth more than Manish's present age (i.e., Parvez = 1.25 × Manish).
- We must find the number of years from now when Parvez's age will be double Manish's present age.
- Ages are in whole years.

Concept / Approach:
We introduce variables for the present ages of Parvez and Manish and use the two given statements to set up simultaneous equations. Solving them gives their current ages. Then, we find the time t such that Parvez's age after t years equals twice Manish's present age. The key is careful handling of time shifts and percentage relations (one-fourth more means 5/4 times, not plus 1/4 year).

Step-by-Step Solution:
Step 1: Let Manish's present age be M years and Parvez's present age be P years. Step 2: Six years ago, Parvez's age was P − 6 years, and it was equal to Manish's present age, so P − 6 = M. Step 3: Parvez's present age is one-fourth more than Manish's, meaning P = M + (1 / 4)M = (5 / 4)M. Step 4: Substitute P from Step 3 into P − 6 = M, giving (5 / 4)M − 6 = M. Step 5: Rearrange: (5 / 4)M − M = 6 ⇒ (1 / 4)M = 6 ⇒ M = 24 years. Step 6: Parvez's present age P = (5 / 4) × 24 = 30 years. Step 7: We want Parvez's age to become double Manish's present age. Double Manish's present age = 2M = 2 × 24 = 48 years. Step 8: Let t years from now be the required time. Then, Parvez's age after t years will be P + t, and we must have P + t = 48. Step 9: Substitute P = 30: 30 + t = 48 ⇒ t = 48 − 30 = 18 years.

Verification / Alternative check:
Check the given conditions: Six years ago, Parvez's age was 30 − 6 = 24 years, which equals Manish's present age, 24 years, confirming the first condition. Also, Parvez's present age is 30, which is one-fourth more than 24 because 24 + (1 / 4) × 24 = 24 + 6 = 30. After 18 years, Parvez will be 30 + 18 = 48 years old, exactly double Manish's present age of 24 years. All conditions are satisfied.

Why Other Options Are Wrong:
Values such as 6 years, 12 years, 15 years, or 20 years would not make Parvez's future age equal to twice Manish's present age when we use the correct present ages of 30 and 24 years. Only 18 years gives 48 years for Parvez, which is exactly double 24.

Common Pitfalls:
Common mistakes include misinterpreting "one-fourth more" as P = M + 1/4 instead of P = (5 / 4)M, or using Manish's future age rather than his present age when forming the "double" condition. Another typical error is to mis-handle the timeline and forget to subtract or add years consistently. Setting up the equations clearly and checking each condition carefully helps avoid these issues.

Final Answer:
Parvez's age will become double Manish's present age after 18 years.