The average age of 11 players of a cricket team is increased by 2 months when two of them aged 18 years and 20 years are replaced by two new players. The average age of the new players is

Answer

To solve the problem, we need to find the average age of the two new players who replaced the two players aged 18 years and 20 years. Here’s a step-by-step breakdown of the solution:

### Step 1: Calculate the total age of the original 11 players.
Let the average age of the original 11 players be \( A \). The total age of these players can be calculated as:
\[ \text{Total age of original players} = 11A \]

### Step 2: Determine the new average age after replacement.
The average age of the team increases by 2 months when the two players are replaced. Since there are 12 months in a year, an increase of 2 months is equivalent to:
\[ \text{New average age} = A + \frac{2}{12} = A + \frac{1}{6} \]

### Step 3: Calculate the total age of the team after the replacement.
The total age of the team after replacing the two players is:
\[ \text{Total age after replacement} = 11 \left( A + \frac{1}{6} \right) = 11A + \frac{11}{6} \]

### Step 4: Calculate the total age of the two players being replaced.
The total age of the two players being replaced (aged 18 and 20) is:
\[ \text{Total age of replaced players} = 18 + 20 = 38 \]

### Step 5: Set up the equation for the total age after replacement.
Let the total age of the two new players be \( x \). The total age after replacement can also be expressed as:
\[ \text{Total age after replacement} = (11A - 38) + x \]

### Step 6: Equate the two expressions for total age after replacement.
We can set the two expressions for total age after replacement equal to each other:
\[ 11A + \frac{11}{6} = (11A - 38) + x \]

### Step 7: Simplify the equation.
Subtract \( 11A \) from both sides:
\[ \frac{11}{6} = -38 + x \]
Now, add 38 to both sides:
\[ x = 38 + \frac{11}{6} \]

### Step 8: Convert 38 into a fraction with a common denominator.
Convert 38 into sixths:
\[ 38 = \frac{228}{6} \]
Thus,
\[ x = \frac{228}{6} + \frac{11}{6} = \frac{239}{6} \]

### Step 9: Find the average age of the new players.
The average age of the two new players is:
\[ \text{Average age of new players} = \frac{x}{2} = \frac{\frac{239}{6}}{2} = \frac{239}{12} \]

### Step 10: Convert the average age into years and months.
To convert \( \frac{239}{12} \) into years and months:
\[ \frac{239}{12} = 19 \text{ years and } 11 \text{ months} \]

### Final Answer:
The average age of the new players is \( 19 \) years and \( 11 \) months.

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