Manish is 12 years old and currently three times as old as his brother Rajeev. How old will Manish be when he becomes exactly twice as old as Rajeev?

Introduction / Context:
This is a straightforward problem on ages involving two brothers. We know Manish's present age and his current relationship to his younger brother Rajeev. We must find Manish's age at the time when he will be exactly twice as old as Rajeev. Such questions are commonly used to test basic algebraic reasoning in aptitude exams.

Given Data / Assumptions:
Manish is 12 years old at present. He is currently three times as old as his brother Rajeev. We are asked to find Manish's age when he becomes twice as old as Rajeev. Ages are assumed to be in whole years.

Concept / Approach:
We first compute Rajeev's present age using the three times relation. Then we introduce a variable t for the number of years after which Manish will be twice as old as Rajeev. We then form a simple linear equation using their future ages and solve for t. Finally, we add t to Manish's current age to get the required age.

Step-by-Step Solution:
Step 1: Let Manish's present age be M = 12 years.Step 2: Since Manish is three times as old as his brother, Rajeev's present age R satisfies M = 3R.Step 3: Substitute M = 12 into 12 = 3R to get R = 12 / 3 = 4 years.Step 4: Let t be the number of years from now when Manish will be twice as old as Rajeev.Step 5: After t years, Manish's age will be 12 + t and Rajeev's age will be 4 + t.Step 6: At that time, 12 + t must equal 2 * (4 + t).Step 7: Expand: 12 + t = 8 + 2t. Rearranging gives 12 - 8 = 2t - t, so 4 = t.Step 8: Therefore, after 4 years Manish will be 12 + 4 = 16 years old.

Verification / Alternative check:
At present, Manish is 12 years old and Rajeev is 4 years old. After 4 years, Manish will be 16 and Rajeev will be 8. At that time, 16 is exactly twice 8, satisfying the required condition. So the solution is valid.

Why Other Options Are Wrong:
If Manish's age were 12 or 14 years at the moment he is twice as old as Rajeev, simple backward calculations show that the age relation 2:1 would not hold. For example, at 12 years, Rajeev would have to be 6 years old, which contradicts the given information that he is currently 4. If Manish were 18 at that moment, Rajeev would be 9, which would not arise from the original three times relation at age 12. Therefore, only 16 years satisfies all conditions.

Common Pitfalls:
A common mistake is to assume that Manish will be twice as old as Rajeev at some fixed age like 24 without actually setting up the equation. Another error is to forget that both ages increase by the same amount t when moving into the future. Treating the problem systematically with variables prevents such errors.

Final Answer:
Manish will be 16 years old when he is twice as old as Rajeev.