The average ages of (P,Q),(Q,R),(R,P) are 13,14 and 12 years respectively. What is the age of Q ?

Options

A. A)11 years
B. B)15 years
C. C)13 years
D. D)12 years

Answer

To find the age of Q given the average ages of (P, Q), (Q, R), and (R, P), we can follow these steps:

### Step 1: Set up the equations based on the averages.
- The average age of P and Q is 13 years:
\[ \frac{P + Q}{2} = 13 \implies P + Q = 26 \quad \text{(Equation 1)} \]

- The average age of Q and R is 14 years:
\[ \frac{Q + R}{2} = 14 \implies Q + R = 28 \quad \text{(Equation 2)} \]

- The average age of R and P is 12 years:
\[ \frac{R + P}{2} = 12 \implies R + P = 24 \quad \text{(Equation 3)} \]

### Step 2: Add all three equations.
Now we will add Equation 1, Equation 2, and Equation 3:
\[ (P + Q) + (Q + R) + (R + P) = 26 + 28 + 24 \]
This simplifies to:
\[ 2P + 2Q + 2R = 78 \]

### Step 3: Divide by 2 to find the total age of P, Q, and R.
\[ P + Q + R = \frac{78}{2} = 39 \]

### Step 4: Use the total age to find Q.
Now we know:
\[ P + Q + R = 39 \]
We can express Q in terms of P and R using the equations we derived earlier.

From Equation 1:
\[ P + Q = 26 \implies Q = 26 - P \quad \text{(Equation 4)} \]

From Equation 3:
\[ R + P = 24 \implies R = 24 - P \quad \text{(Equation 5)} \]

### Step 5: Substitute Equation 4 and Equation 5 into the total age equation.
Substituting Equation 4 and Equation 5 into \(P + Q + R = 39\):
\[ P + (26 - P) + (24 - P) = 39 \]
This simplifies to:
\[ 26 + 24 - P = 39 \]
\[ 50 - P = 39 \]
\[ P = 11 \]

### Step 6: Find Q using the value of P.
Now substitute \(P = 11\) back into Equation 4:
\[ Q = 26 - P = 26 - 11 = 15 \]

### Final Answer:
The age of Q is \(15\) years.

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