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 will be the ratio between the ages of Sam and Albert after 5 years ? I. Sam's present age is more than Albert's present age by 4 years. II. Albert's present age is 20 years. III. The ratio of Albert's present age to Sam's present age is 5 : 6.

### Concept & Logic To find the ratio of their ages after 5 years, we must first find their current ages. If any two statements provide enough information to find both current ages, then the third statement is essentially giving us duplicate information (redundant). ### Step-by-Step Solution Let Sam's age be $S$ and Albert's age be $A$. Let's translate the statements: * **I:** $S - A = 4$ * **II:** $A = 20$ * **III:** $\frac{A}{S} = \frac{5}{6} \Rightarrow 6A = 5S$ Let's test pairs to see if they can solve the system: * **Using I and II:** If $A = 20$, then from I, $S = 24$. We found both ages. (III is redundant) * **Using II and III:** If $A = 20$, then from III, $6(20) = 5S \Rightarrow S = 24$. We found both ages. (I is redundant) * **Using I and III:** Substitute $S = A + 4$ into III: $6A = 5(A + 4) \Rightarrow 6A = 5A + 20 \Rightarrow A = 20$. Then $S = 24$. We found both ages. (II is redundant) Since any pair of statements is sufficient to solve for both ages, any single statement can be discarded without losing the ability to answer the question. ### Exam Strategy & Shortcut **The Golden Triangle of Values:** Statements I, II, and III give you the Difference, an Absolute Value, and a Ratio, respectively. Mathematically, having any two of these three elements allows you to find the exact numerical values for both variables. Recognizing this pattern means you can instantly conclude that any one statement is dispensable. ### Common Pitfall A common mistake is selecting an option like "III only" because statements I and II seem like the "easiest" or most direct way to solve the problem. Redundancy means checking if *any* other combination works, not just finding the most convenient one. ### Final Answer **Therefore, the correct answer is I or II or III only.**