The Correct Answer and Explanation is:
The correct answer is Always. The reciprocal of an improper fraction is always a proper fraction.
Explanation:
An improper fraction is defined as a fraction where the numerator is greater than or equal to the denominator. When determining the reciprocal of a fraction, the numerator and denominator switch places. If the original fraction is improper, meaning the numerator is at least as large as the denominator, then after flipping, the new fraction will have a smaller numerator than denominator, making it a proper fraction.
Consider an improper fraction of the form a/b, where a is greater than or equal to b. Its reciprocal is b/a. Since a was greater than or equal to b in the original fraction, it follows that in b/a, the numerator b is now smaller than the denominator a. This meets the definition of a proper fraction.
For example:
- The reciprocal of 7/3 is 3/7, which is a proper fraction.
- The reciprocal of 9/4 is 4/9, which is also a proper fraction.
- The reciprocal of 5/5 is 5/5, which remains improper. This is the special case where the fraction equals one.
This special case does not invalidate the general rule. As long as an improper fraction is not exactly equal to one, its reciprocal will always be a proper fraction. Thus, the answer is Always. Understanding this principle provides insight into fraction properties and their behavior under reciprocal transformation.
