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{:("Column A",a^(**)b = a//b - b//a","m ...

`{:("Column A",a^(**)b = a//b - b//a","m > n > 0,"ColumnB"),(1/m **1/n,,1/n ** 1/m):}`

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

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to evaluate the expressions in Column A and Column B using the defined operation \( A \star B = \frac{A}{B} - \frac{B}{A} \). ### Step-by-Step Solution: 1. **Define Column A**: \[ \text{Column A} = \frac{1}{m} \star \frac{1}{n} \] Using the operation defined: \[ \frac{1}{m} \star \frac{1}{n} = \frac{\frac{1}{m}}{\frac{1}{n}} - \frac{\frac{1}{n}}{\frac{1}{m}} = \frac{n}{m} - \frac{m}{n} \] 2. **Simplify Column A**: \[ \text{Column A} = \frac{n}{m} - \frac{m}{n} \] To combine these fractions, we can find a common denominator: \[ \text{Column A} = \frac{n^2 - m^2}{mn} = \frac{(n - m)(n + m)}{mn} \] 3. **Define Column B**: \[ \text{Column B} = \frac{1}{n} \star \frac{1}{m} \] Using the operation defined: \[ \frac{1}{n} \star \frac{1}{m} = \frac{\frac{1}{n}}{\frac{1}{m}} - \frac{\frac{1}{m}}{\frac{1}{n}} = \frac{m}{n} - \frac{n}{m} \] 4. **Simplify Column B**: \[ \text{Column B} = \frac{m}{n} - \frac{n}{m} \] Similarly, we can combine these fractions: \[ \text{Column B} = \frac{m^2 - n^2}{mn} = \frac{(m - n)(m + n)}{mn} \] 5. **Compare Column A and Column B**: We have: - Column A: \( \frac{(n - m)(n + m)}{mn} \) - Column B: \( \frac{(m - n)(m + n)}{mn} \) Since \( m > n \), we know: - \( n - m < 0 \) (negative) - \( m - n > 0 \) (positive) Therefore: - Column A is negative (because it has a negative numerator). - Column B is positive (because it has a positive numerator). 6. **Conclusion**: Since Column A is negative and Column B is positive, we conclude that: \[ \text{Column B} > \text{Column A} \] ### Final Answer: **Column B is larger than Column A.**
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