The average age of Rustam and Preetam Is 39 years. The average age of Preetam and Geetam is 28 years. The average age of Rustam and Geetam is 34 years. What will be the age (in years) of the youngest of the three after 10 years?

Options

A. 13
B. 23
C. 33
D. 43

Answer

To solve the problem step by step, we can denote the ages of Rustam, Preetam, and Geetam as R, P, and G respectively.

### Step 1: Set Up the Equations
From the information given in the question, we can set up the following equations based on the averages provided:

1. The average age of Rustam and Preetam is 39 years:
\[ \frac{R + P}{2} = 39 \]
Multiplying both sides by 2 gives:
\[ R + P = 78 \quad \text{(Equation 1)} \]

2. The average age of Preetam and Geetam is 28 years:
\[ \frac{P + G}{2} = 28 \]
Multiplying both sides by 2 gives:
\[ P + G = 56 \quad \text{(Equation 2)} \]

3. The average age of Rustam and Geetam is 34 years:
\[ \frac{R + G}{2} = 34 \]
Multiplying both sides by 2 gives:
\[ R + G = 68 \quad \text{(Equation 3)} \]

### Step 2: Solve the System of Equations
Now we have a system of three equations:
1. \( R + P = 78 \) (Equation 1)
2. \( P + G = 56 \) (Equation 2)
3. \( R + G = 68 \) (Equation 3)

We can solve these equations step by step.

#### Step 2.1: Express P in terms of R
From Equation 1:
\[ P = 78 - R \quad \text{(Equation 4)} \]

#### Step 2.2: Substitute P in Equation 2
Substituting Equation 4 into Equation 2:
\[ (78 - R) + G = 56 \]
Simplifying this gives:
\[ G = 56 - 78 + R \]
\[ G = R - 22 \quad \text{(Equation 5)} \]

#### Step 2.3: Substitute G in Equation 3
Now substitute Equation 5 into Equation 3:
\[ R + (R - 22) = 68 \]
This simplifies to:
\[ 2R - 22 = 68 \]
Adding 22 to both sides:
\[ 2R = 90 \]
Dividing by 2:
\[ R = 45 \quad \text{(Rustam's age)} \]

### Step 3: Find P and G
#### Step 3.1: Find Preetam's Age
Using Equation 4:
\[ P = 78 - R = 78 - 45 = 33 \quad \text{(Preetam's age)} \]

#### Step 3.2: Find Geetam's Age
Using Equation 5:
\[ G = R - 22 = 45 - 22 = 23 \quad \text{(Geetam's age)} \]

### Step 4: Determine the Youngest Age After 10 Years
Now we have the ages:
- Rustam (R) = 45 years
- Preetam (P) = 33 years
- Geetam (G) = 23 years

The youngest among them is Geetam, who is currently 23 years old. After 10 years, Geetam's age will be:
\[ G + 10 = 23 + 10 = 33 \text{ years} \]

### Final Answer
The age of the youngest of the three after 10 years will be **33 years**.