The average age of Sachin and Gangili is 35 years if Kaif replaces Sachin the average age become 32 years and if Kaif replaces Ganguli then the average age becomes 38 years if the average age of Dhoni and Irfan is be half of the average age of Sachin Ganguli and Kaif then the average age of the five people is:

Options

A. 28 years
B. 32 years
C. 25 years
D. none of these

Answer

To solve the problem step by step, we will first extract the necessary information and then calculate the required average ages.

### Step 1: Calculate the total age of Sachin and Ganguly
Given that the average age of Sachin and Ganguly is 35 years, we can calculate their total age.

\[ \text{Average} = \frac{\text{Total Age}}{\text{Number of People}} \]
\[ 35 = \frac{\text{Total Age of Sachin and Ganguly}}{2} \]
\[ \text{Total Age of Sachin and Ganguly} = 35 \times 2 = 70 \text{ years} \]

### Step 2: Determine the age of Kaif
When Kaif replaces Sachin, the average age becomes 32 years. Thus, we can set up the equation:

\[ \text{Average} = \frac{\text{Total Age of Kaif and Ganguly}}{2} \]
\[ 32 = \frac{\text{Age of Kaif} + \text{Age of Ganguly}}{2} \]
Let \( G \) be the age of Ganguly. Then we have:
\[ \text{Age of Kaif} + G = 32 \times 2 = 64 \text{ years} \quad (1) \]

### Step 3: Determine the age of Sachin
When Kaif replaces Ganguly, the average age becomes 38 years. Thus, we can set up another equation:

\[ \text{Average} = \frac{\text{Total Age of Sachin and Kaif}}{2} \]
\[ 38 = \frac{\text{Age of Sachin} + \text{Age of Kaif}}{2} \]
Let \( S \) be the age of Sachin. Then we have:
\[ S + \text{Age of Kaif} = 38 \times 2 = 76 \text{ years} \quad (2) \]

### Step 4: Solve equations (1) and (2)
From equation (1):
\[ \text{Age of Kaif} = 64 - G \quad (3) \]

Substituting equation (3) into equation (2):
\[ S + (64 - G) = 76 \]
\[ S - G = 76 - 64 \]
\[ S - G = 12 \quad (4) \]

### Step 5: Substitute the total age of Sachin and Ganguly
We know from Step 1 that \( S + G = 70 \). Now we can solve for \( S \) and \( G \) using equations (4) and the total age:

From equation (4):
\[ S = G + 12 \]

Substituting into \( S + G = 70 \):
\[ (G + 12) + G = 70 \]
\[ 2G + 12 = 70 \]
\[ 2G = 70 - 12 \]
\[ 2G = 58 \]
\[ G = 29 \text{ years} \]

Now substituting \( G \) back to find \( S \):
\[ S = 29 + 12 = 41 \text{ years} \]

### Step 6: Find the age of Kaif
Using equation (3):
\[ \text{Age of Kaif} = 64 - G = 64 - 29 = 35 \text{ years} \]

### Step 7: Calculate the average age of Dhoni and Irfan
The average age of Sachin, Ganguly, and Kaif is:
\[ \text{Average} = \frac{S + G + \text{Age of Kaif}}{3} = \frac{41 + 29 + 35}{3} = \frac{105}{3} = 35 \text{ years} \]

The average age of Dhoni and Irfan is half of this:
\[ \text{Average of Dhoni and Irfan} = \frac{35}{2} = 17.5 \text{ years} \]

### Step 8: Calculate the average age of all five people
Now, we can find the total age of all five people:
\[ \text{Total Age} = S + G + \text{Age of Kaif} + \text{Age of Dhoni} + \text{Age of Irfan} \]
\[ = 41 + 29 + 35 + 17.5 + 17.5 = 141 \text{ years} \]

The average age of the five people is:
\[ \text{Average} = \frac{\text{Total Age}}{5} = \frac{141}{5} = 28.2 \text{ years} \]

### Final Answer
The average age of the five people is **28.2 years**.

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