In a partnership, the capitals of three partners P, Q and R are such that 6 times P's capital equals 8 times Q's capital and equals 12 times R's capital. Out of a total profit of Rs. 4,650, how much profit does R receive?

Introduction / Context:
This question examines understanding of proportional relationships in partnership capital. Instead of providing the capitals directly, it relates them through multiple equations: 6 times P's capital equals 8 times Q's capital and 12 times R's capital. From this, we must determine the ratio of P, Q and R's investments and then the share of R in the total profit of Rs. 4,650.

Given Data / Assumptions:

• 6 times the capital of P equals 8 times the capital of Q.
• 6 times the capital of P also equals 12 times the capital of R.
• Total profit to be divided among P, Q and R is Rs. 4,650.
• Profit sharing is directly proportional to the capital invested.

Concept / Approach:
When several products of capitals with constants are equal, we set them equal to a common variable and then derive each capital in terms of that variable. Once we have expressions for P, Q and R, we can obtain the ratio of their capitals. The share of R in the total profit is then equal to R's ratio share divided by the sum of all ratio shares, multiplied by the total profit.

Step-by-Step Solution:
Step 1: Let 6P = 8Q = 12R = k, where k is a common constant.Step 2: From 6P = k, we get P = k / 6.Step 3: From 8Q = k, we get Q = k / 8.Step 4: From 12R = k, we get R = k / 12.Step 5: The ratio P : Q : R is (k / 6) : (k / 8) : (k / 12).Step 6: Cancel k to obtain 1/6 : 1/8 : 1/12.Step 7: Express them with common denominator 24: P = 4/24, Q = 3/24, R = 2/24, giving P : Q : R = 4 : 3 : 2.Step 8: Total ratio parts = 4 + 3 + 2 = 9.Step 9: Share of R = (2 / 9) of total profit = (2 / 9) * 4,650.Step 10: Calculate (2 / 9) * 4,650 = 2 * 516.666... = approximately Rs. 1,033.33.

Verification / Alternative check:
We can compute all shares. One share (1 part) is 4,650 / 9 ≈ Rs. 516.67. Then P's share is 4 parts ≈ 4 * 516.67 = 2,066.67. Q's share is 3 parts ≈ 1,550.00 and R's share is 2 parts ≈ 1,033.33. Adding these, 2,066.67 + 1,550.00 + 1,033.33 ≈ 4,650, confirming that the ratio and calculations are consistent to two decimal places.

Why Other Options Are Wrong:
The other numerical options (Rs. 1,024, Rs. 1,148.14 and Rs. 1,245.32) do not correspond to 2/9 of 4,650. When plugged back into the ratio, they would not maintain the 4 : 3 : 2 balance among the three partners. 'None of these' is not correct because approximately Rs. 1,033.33 exactly matches the computed share of R.

Common Pitfalls:
A frequent mistake is to misinterpret 6P = 8Q = 12R as giving a ratio of 6 : 8 : 12 directly, which is actually the opposite of what is required. Another common error is to forget to convert fractions like 1/6, 1/8 and 1/12 into a clean integer ratio before applying the total profit. Always derive the actual capital ratio before dividing the profit.

Final Answer:
The share of R from the total profit is approximately Rs. 1,033.33.