`{:("Column A","The average ages of the players on team A and team B , are 20 and 30 years, respectively. The Average age of the players of the teams togeter is 26","Column B"),("The number of players on team A",,"The number of players on team B"):}`

Options

A. If column A is larger
B. If column B is larger
C. If the columns are equal
D. If there is not enough information to decide

Answer

To solve the problem, we need to compare the number of players on team A and team B based on the average ages provided. Here’s a step-by-step breakdown of the solution:

### Step 1: Define Variables
Let:
- \( a \) = number of players in team A
- \( b \) = number of players in team B

### Step 2: Write Total Ages
The average age of players in team A is 20 years, so the total age of players in team A is:
\[ \text{Total Age of Team A} = 20a \]
The average age of players in team B is 30 years, so the total age of players in team B is:
\[ \text{Total Age of Team B} = 30b \]

### Step 3: Write Combined Average Age
The average age of players from both teams combined is given as 26 years. Therefore, we can express the combined average age as:
\[ \text{Combined Average Age} = \frac{20a + 30b}{a + b} \]
Setting this equal to 26 gives us the equation:
\[ \frac{20a + 30b}{a + b} = 26 \]

### Step 4: Clear the Fraction
To eliminate the fraction, we multiply both sides by \( a + b \):
\[ 20a + 30b = 26(a + b) \]

### Step 5: Expand and Rearrange
Expanding the right side:
\[ 20a + 30b = 26a + 26b \]
Now, rearranging the equation to group terms involving \( a \) and \( b \):
\[ 20a + 30b - 26a - 26b = 0 \]
This simplifies to:
\[ -6a + 4b = 0 \]

### Step 6: Solve for the Ratio
Rearranging gives:
\[ 4b = 6a \]
Dividing both sides by 2:
\[ 2b = 3a \]
Thus, we find the ratio of \( b \) to \( a \):
\[ \frac{b}{a} = \frac{3}{2} = 1.5 \]

### Step 7: Compare the Values
Since \( \frac{b}{a} = 1.5 \), it indicates that \( b > a \). Therefore, the number of players in team B is greater than the number of players in team A.

### Conclusion
Since column A represents the number of players in team A and column B represents the number of players in team B, we conclude that:
\[ \text{Column B is larger than Column A.} \]
Thus, the correct option is option 2.

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