If the average of the ages of Rakesh and Mohan is 15, average of the ages of Mohan and Ramesh is 12 and the average of the ages of Rakesh and Ramesh is 13, then the age of Mohan, is

Options

A. 16
B. 13
C. 14
D. 12

Answer

To find the age of Mohan based on the given averages, we can follow these steps:

### Step 1: Understand the given averages
1. The average age of Rakesh and Mohan is 15.
2. The average age of Mohan and Ramesh is 12.
3. The average age of Rakesh and Ramesh is 13.

### Step 2: Set up the equations
From the averages, we can set up the following equations:
1. For Rakesh and Mohan:
\[ \frac{R + M}{2} = 15 \implies R + M = 30 \quad \text{(Equation 1)} \]
2. For Mohan and Ramesh:
\[ \frac{M + R'}{2} = 12 \implies M + R' = 24 \quad \text{(Equation 2)} \]
3. For Rakesh and Ramesh:
\[ \frac{R + R'}{2} = 13 \implies R + R' = 26 \quad \text{(Equation 3)} \]

### Step 3: Solve the equations
Now we have three equations:
1. \( R + M = 30 \) (Equation 1)
2. \( M + R' = 24 \) (Equation 2)
3. \( R + R' = 26 \) (Equation 3)

We can express \( R' \) in terms of \( R \) from Equation 3:
\[ R' = 26 - R \quad \text{(Equation 4)} \]

### Step 4: Substitute Equation 4 into Equation 2
Now substitute Equation 4 into Equation 2:
\[ M + (26 - R) = 24 \]
This simplifies to:
\[ M + 26 - R = 24 \implies M - R = -2 \implies M = R - 2 \quad \text{(Equation 5)} \]

### Step 5: Substitute Equation 5 into Equation 1
Now substitute Equation 5 into Equation 1:
\[ R + (R - 2) = 30 \]
This simplifies to:
\[ 2R - 2 = 30 \implies 2R = 32 \implies R = 16 \]

### Step 6: Find Mohan's age using Equation 5
Now that we have \( R = 16 \), we can find \( M \):
\[ M = R - 2 = 16 - 2 = 14 \]

### Conclusion
The age of Mohan is **14 years**.

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