The average weight of P, Q and R is 45 kg. If the average weight of P and Q is 40 kg and that of Q and R is 43 kg, what is the weight of Q?

Options

A. 31
B. 67
C. 65
D. 32

Answer

To find the weight of Q, we can follow these steps:

### Step 1: Calculate the total weight of P, Q, and R
Given that the average weight of P, Q, and R is 45 kg, we can calculate their total weight as follows:

\[ \text{Total weight of P, Q, R} = \text{Average weight} \times \text{Number of persons} = 45 \, \text{kg} \times 3 = 135 \, \text{kg} \]

### Step 2: Calculate the total weight of P and Q
We are given that the average weight of P and Q is 40 kg. Therefore, we can calculate their total weight:

\[ \text{Total weight of P and Q} = \text{Average weight} \times \text{Number of persons} = 40 \, \text{kg} \times 2 = 80 \, \text{kg} \]

### Step 3: Calculate the total weight of Q and R
We are also given that the average weight of Q and R is 43 kg. Hence, we can calculate their total weight:

\[ \text{Total weight of Q and R} = \text{Average weight} \times \text{Number of persons} = 43 \, \text{kg} \times 2 = 86 \, \text{kg} \]

### Step 4: Set up the equations
From the above calculations, we can set up the following equations:
1. \( P + Q + R = 135 \) (Equation 1)
2. \( P + Q = 80 \) (Equation 2)
3. \( Q + R = 86 \) (Equation 3)

### Step 5: Combine equations to find Q
Now, we can add Equation 2 and Equation 3:

\[ (P + Q) + (Q + R) = 80 + 86 \]
This simplifies to:

\[ P + 2Q + R = 166 \quad \text{(Equation 4)} \]

### Step 6: Substitute Equation 1 into Equation 4
Now, we can substitute Equation 1 into Equation 4 to eliminate P and R:

From Equation 1, we know that \( P + Q + R = 135 \). Therefore, we can express \( P + R \) as:

\[ P + R = 135 - Q \]

Substituting this into Equation 4 gives us:

\[ (135 - Q) + 2Q = 166 \]

### Step 7: Solve for Q
Now, simplify the equation:

\[ 135 - Q + 2Q = 166 \]
\[ 135 + Q = 166 \]
\[ Q = 166 - 135 \]
\[ Q = 31 \, \text{kg} \]

### Final Answer
The weight of Q is **31 kg**.

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