Question

If \(\frac{1}{x}\) is a positive fraction and \(\frac{1}{y}\) is a negative fraction, then which of the following statements are true? 
(A) \(\frac{1}{x} + \frac{1}{y}\) is positive 
(B) \(\frac{1}{x} - \frac{1}{y}\) is positive 
(C) \(\frac{x - y}{xy}\) is negative 
(D) \(\frac{1}{xy}\) is positive 
(E) \(\frac{1}{x} - \frac{1}{y}\) is positive

• A. Only (A) & (B) are true
• B. Only (B) & (C) are true
• C. Only (D) & (E) are true
• D. Only (A) & (E) are true

Answer 1

The Correct Option is B

To determine which statements are true, let’s analyze each option individually.

1. Statement (A): \( \frac{1}{x} + \frac{1}{y} \) is positive.
   • \( \frac{1}{x} \) is a positive fraction.
   • \( \frac{1}{y} \) is a negative fraction.
   • Adding a positive number and a negative number results in a number whose sign depends on their magnitudes.
   • Without specific values, we cannot conclude that \( \frac{1}{x} + \frac{1}{y} \) is positive.
2. Statement (B): \( \frac{1}{x} - \frac{1}{y} \) is positive.
   • Subtracting a negative fraction \( \frac{1}{y} \) from a positive fraction \( \frac{1}{x} \) can be written as: \[ \frac{1}{x} + \left(-\frac{1}{y}\right) = \frac{1}{x} + \frac{1}{y} \], where the second fraction is actually positive.
   • Therefore, \( \frac{1}{x} - \frac{1}{y} \) is positive.
3. Statement (C): \( \frac{x - y}{xy} \) is negative.
   • Since \( \frac{1}{x} \) is positive, \( x \) must be positive.
   • Since \( \frac{1}{y} \) is negative, \( y \) must be negative.
   • As a result, \( xy \) (product of a positive and a negative) is negative.
   • Because \( xy \) is negative, \(\frac{x - y}{xy} \) has a negative denominator.
   • Thus, \( \frac{x - y}{xy} \) is negative.
4. Statement (D): \( \frac{1}{xy} \) is positive.
   • We already established \( xy \) is negative.
   • Therefore, \( \frac{1}{xy} \), which has a positive numerator and a negative denominator, is negative, not positive.
5. Statement (E): \( \frac{1}{x} - \frac{1}{y} \) is positive.
   • This is the same concept as Statement (B).
   • This statement is indeed positive, as already explained.

Analyzing each statement demonstrates that only Statement (B) and Statement (C) hold true. Thus, the correct answer is: Only (B) & (C) are true.