One year ago, Vijaya was four times as old as her daughter Simran. Six years from now, Vijaya age will exceed Simran age by 9 years. What is the ratio of their present ages?

Introduction / Context:
This question presents a mother daughter age situation at two different times: one year in the past and six years in the future. The relationships describe a multiple of ages at one time and a fixed difference at another time. The task is to find the ratio of their present ages, which is a typical type of problem in the ages topic.

Given Data / Assumptions:

• One year ago, Vijaya was four times as old as her daughter Simran.
• Six years from now, Vijaya age will exceed Simran age by 9 years.
• We need the ratio of their present ages.
• All ages are in years and positive.

Concept / Approach:
We assign variables to represent the current ages of Vijaya and Simran. Then we use each time-based statement to write an equation involving those variables. Solving the pair of equations gives the exact ages, from which we derive the required ratio. The key is correctly shifting ages backward and forward in time and translating the relationships accurately.

Step-by-Step Solution:
Let Vijaya present age be V years.Let Simran present age be S years.One year ago, their ages were V - 1 and S - 1 respectively.Given that one year ago Vijaya was four times as old as Simran: V - 1 = 4 * (S - 1).Simplify: V - 1 = 4 * S - 4, so V = 4 * S - 3.Six years from now, their ages will be V + 6 and S + 6.At that time, Vijaya age will exceed Simran age by 9 years, so V + 6 = S + 6 + 9.This simplifies to V + 6 = S + 15, hence V = S + 9.Now we have two expressions for V: V = 4 * S - 3 and V = S + 9.Set them equal: 4 * S - 3 = S + 9.Rearrange: 4 * S - S = 9 + 3, so 3 * S = 12.Thus S = 4 years, and V = S + 9 = 4 + 9 = 13 years.Therefore the ratio of their present ages is V : S = 13 : 4.

Verification / Alternative check:
One year ago, Vijaya was 13 - 1 = 12 years old, Simran was 4 - 1 = 3 years old.Check first condition: 12 is four times 3, so it holds.Six years from now, Vijaya will be 13 + 6 = 19, Simran will be 4 + 6 = 10.Check second condition: 19 - 10 = 9, so the future difference is 9 years, as given.

Why Other Options Are Wrong:
8 : 1: This would imply a much larger age gap and fails both given conditions.10 : 2: This simplifies to 5 : 1, which is inconsistent with being four times at one time and a difference of 9 at another time.11 : 4: Close in appearance but does not satisfy either the past four times condition or the future difference condition.

Common Pitfalls:
Some students misinterpret four times as V - 1 = 4 * S instead of 4 * (S - 1).Forgetting to adjust both ages when moving backward or forward in time is another frequent source of error.Errors in forming or solving the simultaneous equations can easily lead to a wrong ratio.

Final Answer:
The ratio of the present ages of Vijaya and Simran is 13 : 4.