Rs 7,750 is divided among three people X, Y and Z such that three times the share of X equals five times the share of Y, which also equals two times the share of Z. Under this condition, what is the share in rupees received by Z?

Introduction / Context:
This question tests ratio and proportion concepts in the context of dividing a total amount of money among three people according to a given relationship. The relations between the shares of X, Y and Z are given in terms of multiples, and you must convert these into a consistent ratio to find the share of Z.

Given Data / Assumptions:

• Total sum to be divided = Rs 7,750.
• Let the shares be X, Y and Z respectively.
• 3X = 5Y = 2Z.
• All values are non-negative and represent real monetary amounts.

Concept / Approach:
If 3X, 5Y and 2Z are all equal to the same constant value, say k, then we can express X, Y and Z in terms of k. Specifically, X = k / 3, Y = k / 5 and Z = k / 2. The total sum X + Y + Z must equal Rs 7,750. This gives an equation in k that we can solve, then use to obtain Z.

Step-by-Step Solution:
Step 1: Let 3X = 5Y = 2Z = k. Step 2: Then X = k / 3, Y = k / 5 and Z = k / 2. Step 3: The total sum is X + Y + Z = Rs 7,750. Step 4: Substitute the expressions in terms of k: k / 3 + k / 5 + k / 2 = 7,750. Step 5: Find the common denominator of 3, 5 and 2, which is 30. Step 6: Rewrite each term: k / 3 = 10k / 30, k / 5 = 6k / 30, k / 2 = 15k / 30. Step 7: Add them: (10k + 6k + 15k) / 30 = 31k / 30. Step 8: Set this equal to 7,750: 31k / 30 = 7,750. Step 9: Solve for k: k = 7,750 * 30 / 31 = 7,500. Step 10: Now find Z: Z = k / 2 = 7,500 / 2 = 3,750.

Verification / Alternative check:
Check the individual shares. X = k / 3 = 7,500 / 3 = 2,500. Y = k / 5 = 7,500 / 5 = 1,500. Z = 3,750. Sum: 2,500 + 1,500 + 3,750 = 7,750, which matches the given total. Also verify the ratio condition: 3X = 7,500, 5Y = 7,500, 2Z = 7,500, so the condition 3X = 5Y = 2Z is satisfied.

Why Other Options Are Wrong:
Other values such as 4,250, 3,875, 4,475 or 3,500 do not produce a set of X, Y and Z values that both sum to 7,750 and satisfy 3X = 5Y = 2Z. They fail either the total-sum check or the ratio relationship when back-substituted.

Common Pitfalls:
A frequent error is to misinterpret 3X = 5Y = 2Z as simple ratios like 3 : 5 : 2, which is incorrect in this context. You must treat each expression as equal to the same constant and then derive the shares accordingly. Another pitfall is arithmetic mistakes when finding the common denominator or solving for k.

Final Answer:
The share received by Z is Rs 3,750.