What is the age of Gagan? Statement I: Gagan, Vimal and Kunal are all of the same age. Statement II: The total age of Vimal, Kunal and Anil is 32 years and the age of Anil is equal to the sum of the ages of Vimal and Kunal.

Introduction / Context:
This data sufficiency question combines age relationships with simple algebra. We are asked to find the age of Gagan. The statements relate the ages of four individuals: Gagan, Vimal, Kunal and Anil. The reasoning involves setting up equations, solving for unknowns and then checking if the information is sufficient to obtain a unique value for Gagan's age.

Given Data / Assumptions:

• We need the numerical age of Gagan.
• Statement I: Gagan, Vimal and Kunal are all of the same age.
• Statement II: The total age of Vimal, Kunal and Anil is 32 years, and Anil's age is equal to the sum of the ages of Vimal and Kunal.
• All ages are positive real numbers.
• We must find whether the age of Gagan can be uniquely determined.

Concept / Approach:
The main concepts here are:

• Expressing relationships using algebraic equations.
• Using data sufficiency rules: test each statement alone, then both together.
• Checking whether a unique value is obtained for Gagan's age.

Statement I gives equality among three ages but no numerical value. Statement II gives a total and a relation among three other ages. When combined, these statements should allow us to solve for the common age.

Step-by-Step Solution:
Step 1: Let the common age of Gagan, Vimal and Kunal be x years. Thus Gagan = x, Vimal = x and Kunal = x. Step 2: From statement I alone, x is unknown. We have no equation that gives a numerical value, so statement I alone is not sufficient. Step 3: From statement II, let the ages of Vimal, Kunal and Anil be V, K and A respectively. Step 4: According to statement II, the total age is V + K + A = 32 and A = V + K. Step 5: Substitute A = V + K into the total: V + K + (V + K) = 32. This simplifies to 2(V + K) = 32. Step 6: Divide both sides by 2 to get V + K = 16. Since A = V + K, Anil is 16 years old. Step 7: Statement II alone still does not give the individual ages of Vimal or Kunal, and it says nothing directly about Gagan. Step 8: Now combine both statements. From statement I, Vimal and Kunal each have age x, so V = x and K = x. Step 9: From step 5, V + K = 16. Substituting V = x and K = x, we get x + x = 16, so 2x = 16. Step 10: Solve for x: x = 16 / 2 = 8. Therefore, Gagan is 8 years old. Step 11: A unique value is obtained for Gagan's age when both statements are used together.

Verification / Alternative check:
We can verify by checking all ages. If Gagan, Vimal and Kunal are each 8 years old, then Vimal plus Kunal have a combined age of 16 years. Anil's age, according to statement II, is equal to the sum of the ages of Vimal and Kunal, which is 16 years. So Anil is 16 years old. Now the total age of Vimal, Kunal and Anil is 8 + 8 + 16 = 32, which exactly matches the data in statement II. Thus our solution is consistent.

Why Other Options Are Wrong:
Option a is wrong because statement I without numerical information leaves x undetermined. Option b is wrong because statement II alone does not refer to Gagan and does not provide individual ages of Vimal and Kunal. Option c is wrong because neither statement alone is sufficient; both are needed to create a solvable system of equations. Option e is wrong because once we combine both statements, we obtain a unique age for Gagan, namely 8 years.

Common Pitfalls:
A common error is to stop after finding that V + K = 16 and conclude that each of Vimal and Kunal is 8 years old, without properly justifying that they are equal. Statement I is essential to ensure they have the same age. Without that, any pair of positive numbers that sum to 16 would work. Another trap is to think that since the total is 32, one can simply divide by the number of people mentioned, ignoring the specific relation that Anil is as old as the sum of Vimal and Kunal. Data sufficiency questions test not only algebraic manipulation but also disciplined interpretation of wording.

Final Answer:
Using both statements together, Gagan's age is 8 years. Correct option: The data in both statements I and II together are sufficient to answer the question.