Three states X, Y, and Z had students appearing for an exam in 1998 in the ratio 3 : 5 : 6. Next year, the numbers rose by 20%, 10%, and 20% respectively. If in the next year the ratio of students in states X and Z is 1 : 2, determine whether the 1998 numbers can be uniquely found.

Introduction / Context:This is a data sufficiency style question disguised as a percentage/ratio problem. We are asked whether the original numbers (1998) can be uniquely determined using the given ratios and percentage changes.

Given Data / Assumptions:

• 1998 ratio X : Y : Z = 3 : 5 : 6 ⇒ numbers = 3k, 5k, 6k.
• Next year increases: X by 20%, Y by 10%, Z by 20%.
• Next year X : Z is 1 : 2.

Concept / Approach:Compute next-year values as factors of k and check the X : Z ratio. If the ratio condition holds identically (for all k), then k is not determined and original totals cannot be uniquely fixed.

Step-by-Step Solution:

Verification / Alternative check:Even if Y’s growth is included, it doesn’t affect the X : Z ratio. Without an absolute count or another independent equation, 1998 totals are undetermined.

Why Other Options Are Wrong:Numerical totals (5000, 6000, 75000, 7000) cannot be justified uniquely from the data.

Common Pitfalls:Assuming ratios alone can produce exact totals; missing that a free scale factor k remains.

Final Answer:Data is insufficient