The average monthly income of $P$ and $Q$ is ₹ 5050. The average monthly income of $Q$ and $R$ is ₹ 6250 and the average monthly income of $P$ and $R$ is ₹ 5200. The monthly income of $P$ is

### Concept & Formula When you are given the averages of overlapping pairs, you can find the total sum of all entities by adding the sums of the pairs and dividing by 2. $$Sum = Average \times Number of Entities$$ ### Step-by-Step Solution * **Sum of P and Q:** Since their average is ₹ 5050, $$P + Q = 5050 \times 2 = 10100$$ * **Sum of Q and R:** Since their average is ₹ 6250, $$Q + R = 6250 \times 2 = 12500$$ * **Sum of P and R:** Since their average is ₹ 5200, $$P + R = 5200 \times 2 = 10400$$ * **Find the Total Sum (P + Q + R):** Add all three equations together: $$(P + Q) + (Q + R) + (P + R) = 10100 + 12500 + 10400$$ $$2(P + Q + R) = 33000$$ $$P + Q + R = 16500$$ * **Calculate P's Income:** Subtract the sum of (Q + R) from the total sum: $$P = (P + Q + R) - (Q + R)$$ $$P = 16500 - 12500 = 4000$$ ### Exam Strategy & Shortcut Instead of writing out full variables, simply add the three given averages and subtract the average of the pair that *does not* include the person you are looking for. P's income = Sum of averages containing P - Average not containing P Wait, the formula for this specific shortcut is: $$P = (Average of P+Q) + (Average of P+R) - (Average of Q+R)$$ $$P = 5050 + 5200 - 6250 = 10250 - 6250 = 4000$$. This is incredibly fast! ### Common Pitfall Students often add the totals together ($33000$) and forget to divide by $2$, erroneously thinking $33000$ is the sum of $P, Q,$ and $R$. Always remember that each person is counted twice when adding overlapping pairs. ### Final Answer **Therefore, the correct answer is ₹ 4000.**