The captain of a cricket team of 11 members is 28 years old and the wicket keeper is 4 years older than him. If the ages of these two are excluded, the average age of the remaining players is two years less than the average age of the whole team. What is the average age of the team?

Options

A. 15 years
B. 22 years
C. 21 years
D. 29 years

Answer

To solve the problem step by step, we will break down the information given and use algebra to find the average age of the cricket team.

### Step 1: Define Variables
Let:
- \( A \) = average age of the whole team
- \( T \) = sum of the ages of all players in the team

Since there are 11 members in the team, we can express the total sum of ages as:
\[ T = 11A \]
This is our **Equation 1**.

### Step 2: Determine the Ages of the Captain and Wicketkeeper
The captain's age is given as 28 years. The wicketkeeper is 4 years older than the captain, so:
- Captain's age = 28 years
- Wicketkeeper's age = \( 28 + 4 = 32 \) years

### Step 3: Calculate the Total Age of Captain and Wicketkeeper
The total age of the captain and wicketkeeper combined is:
\[ 28 + 32 = 60 \text{ years} \]

### Step 4: Find the Average Age of the Remaining Players
When the captain and wicketkeeper are excluded, there are 9 players left. The problem states that the average age of these remaining players is 2 years less than the average age of the whole team. Therefore, the average age of the remaining players is:
\[ A - 2 \]

### Step 5: Set Up the Equation for Remaining Players
The total age of the remaining players can be expressed as:
\[ T - 60 \]
The average age of the remaining players can also be expressed as:
\[ \frac{T - 60}{9} \]

Setting these two expressions equal gives us:
\[ A - 2 = \frac{T - 60}{9} \]

### Step 6: Substitute \( T \) from Equation 1
Substituting \( T = 11A \) into the equation:
\[ A - 2 = \frac{11A - 60}{9} \]

### Step 7: Clear the Fraction
Multiply both sides by 9 to eliminate the fraction:
\[ 9(A - 2) = 11A - 60 \]
Expanding the left side:
\[ 9A - 18 = 11A - 60 \]

### Step 8: Rearrange the Equation
Rearranging gives:
\[ 9A - 11A = -60 + 18 \]
\[ -2A = -42 \]

### Step 9: Solve for \( A \)
Dividing both sides by -2:
\[ A = \frac{42}{2} = 21 \]

### Conclusion
The average age of the team is:
\[ \boxed{21} \]