U and V start a business by investing Rs 184000 and Rs 224000 respectively. If the share of U in the profit at the end of the year is Rs 20700, what is the total profit earned by them together?

Introduction / Context:
This is a direct partnership proportion question where two partners invest for the same duration. When the time period is equal for all partners, profit is shared in the ratio of their capitals. We are given the capital investments of U and V and the profit share of U, and we need to find the total profit for the year.

Given Data / Assumptions:
U invests Rs 184000. V invests Rs 224000. Both investments are for the same time period of one year. U share in the profit is Rs 20700. We must find total profit earned by U and V together.

Concept / Approach:
When partners invest for equal times, profit share is directly proportional to capital. So the ratio of U and V profits is the same as the ratio of their capitals, 184000 : 224000. We simplify this ratio, treat U share as corresponding to the first term, and then scale the ratio to find the full profit corresponding to the sum of both terms. This standard ratio scaling approach makes the calculation simple and transparent.

Step-by-Step Solution:
Step 1: Capital ratio U : V = 184000 : 224000. Step 2: Divide both terms by 8000 to simplify: 184000 / 8000 = 23 and 224000 / 8000 = 28. Step 3: So profit ratio U : V = 23 : 28. Step 4: Let common profit factor be k, so profit of U = 23k and profit of V = 28k. Step 5: Given profit of U is 20700, so 23k = 20700. Step 6: Therefore k = 20700 / 23 = 900. Step 7: Total profit = 23k + 28k = 51k. Step 8: Substitute k = 900, total profit = 51 * 900 = 45900.

Verification / Alternative check:
We can compute V share as 28k = 28 * 900 = 25200. Adding U and V shares gives 20700 + 25200 = 45900, which matches the total profit obtained from 51k. The ratio 20700 : 25200 simplifies to 23 : 28 when divided by 900, confirming that the profit shares are correctly proportional to the capitals 184000 and 224000. This double check validates that the total profit is Rs 45900.

Why Other Options Are Wrong:
Total profit of Rs 43200, Rs 52300 or Rs 56400 would lead to different shares for U and V that are not in the ratio 23 : 28 and would not give U exactly Rs 20700. Since the profit ratio must match the capital ratio and U share is fixed, only one total profit value can satisfy both conditions. That value is Rs 45900.

Common Pitfalls:
A common mistake is to subtract capitals or to compute some percentage difference instead of working directly with the ratio. Another error is solving 23k = 20700 incorrectly or approximating without proper division. Using a clear ratio approach, simplifying 184000 : 224000 correctly, and then solving for k ensures that such questions are handled accurately and quickly in exams.

Final Answer:
The total profit earned by U and V together at the end of the year is Rs 45900.