Divide ₹ 990 into three parts such that half of the first part, one-third of the second part, and one-fifth of the third part are all equal. Find the three parts.

Introduction / Context:
The condition equates particular fractions of the three parts. Converting those equalities to a common parameter allows us to express each part in terms of that parameter and then use the total to find exact values.

Given Data / Assumptions:
Let parts be x, y, z such that (1/2)x = (1/3)y = (1/5)z.

Concept / Approach:
Set the equal value to k. Then solve x, y, z in terms of k and sum to 990 to find k. Finally compute each part.

Step-by-Step Solution:
(1/2)x = k ⇒ x = 2k. (1/3)y = k ⇒ y = 3k. (1/5)z = k ⇒ z = 5k. Sum: x + y + z = 2k + 3k + 5k = 10k = 990 ⇒ k = 99. Thus x = 198, y = 297, z = 495.

Verification / Alternative check:
Check the condition: half of 198 is 99, one-third of 297 is 99, one-fifth of 495 is 99; all equal.

Why Other Options Are Wrong:
Other triplets do not satisfy both the total 990 and the fractional equalities simultaneously.

Common Pitfalls:
Equating x : y : z directly to 1/2 : 1/3 : 1/5 rather than converting via a common k.

Final Answer:
198, 297, 495