Introduction / Context:
This is a data sufficiency question about ages. The main question is: 'What is Guna's age?' Two separate statements give relationships among the ages of Guna, Vinay, Keshav and Arjun. We have to decide whether each statement alone is enough, whether both together are needed, or whether even together they are not sufficient. The focus is on algebraic reasoning and understanding when information is sufficient to find a unique value.
Given Data / Assumptions:
- Main Question: What is Guna's age?
- Statement 1: Guna, Vinay and Keshav are all of the same age.
- Statement 2: The sum of ages of Vinay, Keshav and Arjun is 32, and Arjun is as old as Vinay and Keshav together.
- We assume all ages are in years and are positive integers, as is typical in such questions.
Concept / Approach:
The technique is to express each statement in algebraic form, see how many unknowns there are and whether we can solve for Guna's age uniquely. If one statement alone gives multiple possibilities, it is not sufficient. If combining the two statements allows us to find a single numerical value for Guna's age, then both are needed.
Step-by-Step Solution:
Step 1: Assign variables. Let G be Guna's age, V be Vinay's age, K be Keshav's age and A be Arjun's age.
Step 2: Translate Statement 1: 'Guna, Vinay and Keshav are all of the same age.' This means G = V = K.
Step 3: Check sufficiency of Statement 1 alone. If G = V = K, we still do not know any actual numerical value. Many values are possible (e.g., all three could be 8, or 10, or 15). So Statement 1 alone is not sufficient to find Guna's age.
Step 4: Translate Statement 2: 'Sum of ages of Vinay, Keshav and Arjun is 32 and Arjun is as old as Vinay and Keshav together.' In symbols, V + K + A = 32 and A = V + K.
Step 5: Use the condition A = V + K in V + K + A = 32. Substituting, we get V + K + (V + K) = 32, that is 2(V + K) = 32, so V + K = 16. Then A = V + K = 16.
Step 6: Check sufficiency of Statement 2 alone. From Statement 2, we know V + K = 16, but we do not know the individual values of V and K, and we have no information relating G to them. Guna's age G could be anything; we cannot determine it. So Statement 2 alone is also not sufficient.
Step 7: Now combine Statement 1 and Statement 2. From Statement 1, we have G = V = K. From Statement 2, we already found V + K = 16. Since V = K, we can write V + K = G + G = 2G = 16, which gives G = 8.
Step 8: Thus, when both statements are used together, Guna's age is uniquely determined as 8 years.
Verification / Alternative check:
We can verify with concrete numbers: If G = V = K = 8, then V + K = 8 + 8 = 16 and A = 16. The sum V + K + A is 8 + 8 + 16 = 32, satisfying Statement 2. So both statements are consistent and produce a unique age for Guna.
Why Other Options Are Wrong:
Option A (Neither 1 nor 2 needed) is wrong because we do need information from the statements to determine Guna's age.
Option B (Only statement 2 is needed) is wrong because Statement 2 does not involve G directly and gives no way to find his age alone.
Option D (Only statement 1 is needed) is wrong because Statement 1 only tells us that G shares an age with Vinay and Keshav but not what that age is.
Common Pitfalls:
A typical mistake is to think that since Statement 2 contains actual numbers (like 32), it must be sufficient. But without a link to G, it is not.
Another pitfall is not combining the equations correctly or forgetting that G, V and K are equal, which is crucial to solving the problem.
Final Answer:
We can find Guna's age only when we use the equality of ages from Statement 1 together with the numerical relations from Statement 2, giving G = 8. Hence, both statements 1 and 2 are needed.
Discussion & Comments