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 difference between the present ages of Ayush and Deepak ? I. The ratio between Ayush's present age and his age after 8 years is 4 : 5. II. The ratio between the present ages of Ayush and Deepak is 4 : 3. III. The ratio between Deepak's present age and his age four years ago is 6 : 5.

### Concept & Logic To find the difference between two variables (Ayush and Deepak), we need sufficient information to calculate both of their values. We must analyze whether the statements independently solve for a variable or link the variables together. ### Step-by-Step Solution Let Ayush's present age be $A$ and Deepak's present age be $D$. * **Statement I:** $\frac{A}{A + 8} = \frac{4}{5} \Rightarrow 5A = 4A + 32 \Rightarrow A = 32$. (This statement alone gives Ayush's exact age). * **Statement II:** $\frac{A}{D} = \frac{4}{3} \Rightarrow 3A = 4D$. (This is a linking relationship). * **Statement III:** $\frac{D}{D - 4} = \frac{6}{5} \Rightarrow 5D = 6D - 24 \Rightarrow D = 24$. (This statement alone gives Deepak's exact age). Let's evaluate combinations: * **Using I and III:** We know $A = 32$ and $D = 24$. Difference = 8. (II is redundant). * **Using I and II:** We know $A = 32$. Substituting into II: $3(32) = 4D \Rightarrow D = 24$. Difference = 8. (III is redundant). * **Using II and III:** We know $D = 24$. Substituting into II: $3A = 4(24) \Rightarrow A = 32$. Difference = 8. (I is redundant). Because we only ever need two statements to fully solve for both ages, any one of the three statements can be dropped. ### Exam Strategy & Shortcut **Single Variable Recognition:** Notice that Statement I only talks about Ayush and Statement III only talks about Deepak. Equations with only one variable solve themselves instantly. Statement II connects them. Thus, picking the two independent statements works (I & III), or picking one independent statement plus the connector works (I & II, or II & III). Therefore, any single one is dispensable. ### Common Pitfall Students often solve using the most obvious path (Statement I and Statement III) and incorrectly assume Statement II is the *only* redundant one. You must test all pairs to see if they yield a valid solution. ### Final Answer **Therefore, the correct answer is Any one of the three.**