The average age of 11 players of a cricket team is decreased by 2 months when two of them aged 17 years and 20 years are replaced by two new players. The average age of the new players is

Options

A. (a) 17 years 7 months
B. (b) 19 years 3 months
C. (c) 21 years 9 months
D. (d) 23 years 11 months

Answer

To solve the problem step by step, we will first analyze the information given and then calculate the average age of the new players.

### Step 1: Understand the initial average age of the players
Let the initial average age of the 11 players be \( x \) years. Therefore, the total age of the 11 players can be expressed as:
\[ \text{Total age of 11 players} = 11x \]

### Step 2: Calculate the total age after replacement
When two players aged 17 years and 20 years are replaced, the total age of the team decreases. The total age of the two players being replaced is:
\[ \text{Total age of replaced players} = 17 + 20 = 37 \text{ years} \]

### Step 3: Find the new total age of the players
The new average age of the team decreases by 2 months. Since 2 months is \(\frac{2}{12} = \frac{1}{6}\) years, the new average age becomes:
\[ \text{New average age} = x - \frac{1}{6} \]
The total age of the 11 players after the replacement is:
\[ \text{Total age after replacement} = 11 \left( x - \frac{1}{6} \right) = 11x - \frac{11}{6} \]

### Step 4: Set up the equation for the new total age
After replacing the two players, the new total age can also be expressed as:
\[ \text{Total age after replacement} = \text{Total age of 11 players} - \text{Total age of replaced players} + \text{Total age of new players} \]
Let the total age of the new players be \( y \). Then we have:
\[ 11x - 37 + y = 11x - \frac{11}{6} \]

### Step 5: Solve for the total age of the new players
Simplifying the equation:
\[ 11x - 37 + y = 11x - \frac{11}{6} \]
Subtract \( 11x \) from both sides:
\[ -37 + y = -\frac{11}{6} \]
Now, isolate \( y \):
\[ y = -\frac{11}{6} + 37 \]
Convert 37 into a fraction with a denominator of 6:
\[ 37 = \frac{222}{6} \]
Now substitute:
\[ y = \frac{222}{6} - \frac{11}{6} = \frac{211}{6} \]

### Step 6: Calculate the average age of the new players
The average age of the two new players can be calculated as follows:
\[ \text{Average age of new players} = \frac{y}{2} = \frac{\frac{211}{6}}{2} = \frac{211}{12} \]

### Final Answer
The average age of the new players is:
\[ \frac{211}{12} \text{ years} \approx 17.58 \text{ years} \]