Directions: Each of these questions is followed by three statements. You have to study the question and all the three statements given to decide whether any information provided in the statement(s) is redundant and can be dispensed with while answering the given question. What is the ratio of the present ages of Anna and her mother ? I. The sum of the ages of Anna, her mother and her father is 62. II. Five years ago, Anna's age was one-fifth of her father's age. III. Two years ago, the sum of the ages of Anna and her father was 36.

### Concept & Logic This is a data redundancy problem requiring us to find if any statements can be dropped. To find the ratio of Anna to her mother, we must determine both of their individual ages. If we have three unknown variables (Anna, Mother, Father), we generally need three independent equations to solve the system. ### Step-by-Step Solution Let the present ages of Anna, Mother, and Father be $A$, $M$, and $F$ respectively. * **Analyze Statement I:** $A + M + F = 62$ This is the only statement that includes the Mother ($M$). Therefore, Statement I is absolutely necessary and cannot be redundant. * **Analyze Statement II:** $A - 5 = \frac{1}{5}(F - 5)$ $5A - 25 = F - 5$ $5A - F = 20$ (Equation with two variables: $A$ and $F$) * **Analyze Statement III:** $(A - 2) + (F - 2) = 36$ $A + F = 40$ (Equation with two variables: $A$ and $F$) * **Deduction:** We need $A$ and $M$ to answer the question. Statement I gives us a way to find $M$, but only if we know $A$ and $F$. To find $A$ and $F$, we need two independent equations involving them. Statements II and III provide exactly these two equations. Thus, all three statements are required to solve the problem. ### Exam Strategy & Shortcut **Variable Isolation Rule:** In data sufficiency/redundancy, scan for unique variables. The Mother's age only appears in Statement I, making it instantly non-redundant. The remaining variables (Anna and Father) require two distinct equations to be solved, meaning both II and III are mandatory. No calculations are actually needed to realize you need all three! ### Common Pitfall Students often solve the equations fully to find the exact ratio, wasting precious exam time. Remember, the question only asks *what is redundant*, not the final answer. Stop evaluating as soon as you confirm the necessity of the equations. ### Final Answer **Therefore, the correct answer is All I, II and III are required.**