Eight years ago Rama mother was five times as old as Rama. After 8 years from now, the mother will be twice as old as Rama. What is the present age of Rama?

Introduction / Context:
This problem involves the ages of Rama and her mother at two different times, 8 years in the past and 8 years in the future. The relationships given are multiplicative, making it a classic two-variable linear equation question under the topic of problems on ages.

Given Data / Assumptions:

• Eight years ago, the mother was five times as old as Rama.
• After 8 years from now, the mother will be twice as old as Rama.
• We are asked to find the present age of Rama.
• Ages are measured in years and may be fractional.

Concept / Approach:
The strategy is to define variables for their present ages and then translate both time-based statements into equations. Each equation links the same two unknown ages but at different times, which allows us to solve the system. Even if the result is fractional, that is mathematically acceptable, and the answer choices include fractional ages.

Step-by-Step Solution:
Let the present age of Rama be R years.Let the present age of the mother be M years.Eight years ago, Rama age was R - 8 and mother age was M - 8.At that time, the mother was five times as old as Rama: M - 8 = 5 * (R - 8).After 8 years, Rama age will be R + 8 and mother age will be M + 8.At that future time, the mother will be twice as old as Rama: M + 8 = 2 * (R + 8).Now expand both equations.From the first equation: M - 8 = 5 * R - 40, so M = 5 * R - 32.From the second equation: M + 8 = 2 * R + 16, so M = 2 * R + 8.Set the two expressions for M equal: 5 * R - 32 = 2 * R + 8.Rearrange: 5 * R - 2 * R = 8 + 32.This gives 3 * R = 40, so R = 40 / 3 years.

Verification / Alternative check:
Rama present age = 40 / 3 years, which is approximately 13.33 years.Mother present age M = 2 * R + 8 = 2 * (40 / 3) + 8 = 80 / 3 + 24 / 3 = 104 / 3 years.Eight years ago: Rama was 40 / 3 - 8 = 40 / 3 - 24 / 3 = 16 / 3 years, mother was 104 / 3 - 8 = 104 / 3 - 24 / 3 = 80 / 3 years.Check first condition: 5 times Rama age then is 5 * 16 / 3 = 80 / 3, equal to the mother age then.After 8 years: Rama will be 40 / 3 + 8 = 64 / 3 years, mother will be 104 / 3 + 8 = 128 / 3 years.Check second condition: Twice Rama age then is 2 * 64 / 3 = 128 / 3, equal to mother age then.

Why Other Options Are Wrong:
10 years: Substitution gives inconsistent values for the mother age that do not satisfy the future condition.20/3 years: This smaller age does not simultaneously satisfy both relationships.50/3 years: This larger age for Rama makes the mother younger than expected based on the given ratios.

Common Pitfalls:
Errors often occur when shifting ages by 8 years in the wrong direction for past or future conditions.Some learners incorrectly assume ages must be integers and reject correct fractional results.Algebra mistakes when equating 5 * R - 32 and 2 * R + 8 can easily lead to the wrong value of R.

Final Answer:
The present age of Rama is 40/3 years.