Rs. 1200 is divided among P, Q and R. P gets half of the total amount received by Q and R together, and Q gets one third of the total amount received by P and R together. What is the amount received by R?

Introduction / Context:
This partnership style question uses relational statements between the shares of three people instead of simple ratios. The aim is to convert these relationships into equations, use the total amount, and solve systematically for the share of R.

Given Data / Assumptions:

• Total amount shared among P, Q and R is Rs. 1200.
• P gets half of the combined amount received by Q and R.
• Q gets one third of the combined amount received by P and R.
• We must find R's share.

Concept / Approach:
When we have conditions like one person getting a fraction of the sum of the other two, we can represent each person's share with variables and set up equations. Solving a system of three equations gives exact values for P, Q and R. The key idea is to keep the expressions simple and combine them carefully, using substitution or simultaneous solving.

Step-by-Step Solution:
Step 1: Let the shares of P, Q and R be P, Q and R respectively in rupees. Step 2: Total amount condition: P + Q + R = 1200. Step 3: P gets half of the total received by Q and R, so P = 1/2 * (Q + R). Step 4: Q gets one third of the total received by P and R, so Q = 1/3 * (P + R). Step 5: Substitute P from Step 3 into the equation in Step 4 to express Q in terms of Q and R. However, it is more efficient to solve these three equations together. Step 6: From P = 1/2 * (Q + R), we can write 2P = Q + R. Step 7: From Q = 1/3 * (P + R), we can write 3Q = P + R. Step 8: Now use P + Q + R = 1200 along with 2P = Q + R and 3Q = P + R to solve. Solving this system gives P = 400, Q = 300 and R = 500. Step 9: Therefore, the amount received by R is Rs. 500.

Verification / Alternative check:
Check P's condition: Q + R = 300 + 500 = 800. Half of this is 800 / 2 = 400, which matches P's share. Check Q's condition: P + R = 400 + 500 = 900. One third of this is 900 / 3 = 300, which equals Q's share. The total P + Q + R = 400 + 300 + 500 = 1200, which matches the given total. All conditions are satisfied, so the values are correct.

Why Other Options Are Wrong:
Rs. 1100 and Rs. 1200 cannot be R's share because they are larger than or equal to the total or would leave negative or zero amounts for the others.
Rs. 700 does not satisfy both fractional conditions when we back compute P and Q. Only Rs. 500 fits all three equations simultaneously.

Common Pitfalls:
Learners sometimes misinterpret the phrases and treat P as half of Q or R individually instead of half of the sum Q + R. Another mistake is to plug in approximate values without solving the equations exactly, which can quickly lead to inconsistent totals. The safest method is to write clear equations for all three statements and solve them systematically.

Final Answer:
The amount received by R is Rs. 500.