Introduction / Context:
This is an average and linear-equation problem involving three people P, Q and R. We are given the average of all three together and the averages of two pairs: (P, Q) and (Q, R). Using this information, we can form a system of equations and solve for the individual weight of Q.
Given Data / Assumptions:
- Average weight of P, Q, R = 47 kg.
- Average weight of P and Q = 32.5 kg.
- Average weight of Q and R = 48.5 kg.
- Let weights be P, Q, R in kg.
- We must find Q.
Concept / Approach:
We convert the averages into total-weight equations by multiplying by the number of people in each group:
- P + Q + R = 3 * 47.
- P + Q = 2 * 32.5.
- Q + R = 2 * 48.5.
Then we use these equations to eliminate P and R and isolate Q. It is straightforward algebra with careful arithmetic.
Step-by-Step Solution:
Step 1: Translate averages into sums.
P + Q + R = 3 * 47 = 141.
P + Q = 2 * 32.5 = 65.
Q + R = 2 * 48.5 = 97.
Step 2: Add the equations for P + Q and Q + R.
(P + Q) + (Q + R) = 65 + 97.
Left side is P + 2Q + R; right side is 162.
So P + 2Q + R = 162.
Step 3: Subtract P + Q + R = 141 from this equation.
(P + 2Q + R) − (P + Q + R) = 162 − 141.
Left side simplifies to Q; right side is 21.
Hence Q = 21 kg.
Verification / Alternative check:
From P + Q = 65, we get P = 65 − Q = 65 − 21 = 44.
From Q + R = 97, we get R = 97 − Q = 97 − 21 = 76.
Check the total: P + Q + R = 44 + 21 + 76 = 141.
Average of all three = 141 / 3 = 47 kg, matching the given average.
Why Other Options Are Wrong:
If Q were 25, 29, 33 or 31, the corresponding values of P and R computed from P + Q = 65 and Q + R = 97 would not give the correct total 141.
Thus, those alternatives violate at least one of the original average conditions.
Common Pitfalls:
A common mistake is to incorrectly double the averages (e.g., miscomputing 2 * 32.5 as 64 or 66 instead of 65).
Another error is adding or subtracting the equations incorrectly, which can yield wrong values for Q.
Final Answer:
The weight of Q is 21 kilograms.
Discussion & Comments