The average age of Ram and Rahim is 18 years. The average age of Rahim and Ramesh is 25 years. The average age of Ram and Ramesh is 29 years. What is the age (in years) of the oldest of the three?

Options

A. 14
B. 22
C. 28
D. 36

Answer

To solve the problem step by step, we will denote the ages of Ram, Rahim, and Ramesh as \(X\), \(Y\), and \(Z\) respectively.

### Step 1: Set Up the Equations
From the problem, we know the following averages:
1. The average age of Ram and Rahim is 18 years:
\[ \frac{X + Y}{2} = 18 \]
Multiplying both sides by 2 gives:
\[ X + Y = 36 \quad \text{(Equation 1)} \]

2. The average age of Rahim and Ramesh is 25 years:
\[ \frac{Y + Z}{2} = 25 \]
Multiplying both sides by 2 gives:
\[ Y + Z = 50 \quad \text{(Equation 2)} \]

3. The average age of Ram and Ramesh is 29 years:
\[ \frac{X + Z}{2} = 29 \]
Multiplying both sides by 2 gives:
\[ X + Z = 58 \quad \text{(Equation 3)} \]

### Step 2: Add the Equations
Now, we will add all three equations:
\[ (X + Y) + (Y + Z) + (X + Z) = 36 + 50 + 58 \]
This simplifies to:
\[ 2X + 2Y + 2Z = 144 \]
Dividing the entire equation by 2 gives:
\[ X + Y + Z = 72 \quad \text{(Equation 4)} \]

### Step 3: Find Individual Ages
Now we can find the individual ages by substituting values from Equations 1, 2, and 3 into Equation 4.

1. **Finding \(Z\)**:
From Equation 1:
\[ Z = 72 - (X + Y) = 72 - 36 = 36 \]

2. **Finding \(X\)**:
From Equation 2:
\[ X = 72 - (Y + Z) = 72 - 50 = 22 \]

3. **Finding \(Y\)**:
From Equation 3:
\[ Y = 72 - (X + Z) = 72 - 58 = 14 \]

### Step 4: Determine the Oldest Age
Now we have the ages:
- \(X = 22\) (Ram)
- \(Y = 14\) (Rahim)
- \(Z = 36\) (Ramesh)

The oldest of the three is:
\[ \text{Oldest Age} = \max(X, Y, Z) = \max(22, 14, 36) = 36 \]

### Final Answer
The age of the oldest of the three is **36 years**.

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