## Basic College Mathematics (10th Edition)

Proper fractions have a numerator that is smaller than the denominator, while improper fractions have a numerator that is greater than or equal to the denominator. Since the numerator is smaller than the denominator in a proper fraction, the overall fraction must be less than $1$. This means that when one divides by a proper fraction, one is dividing by a number less than 1. Dividing by a number less than 1 means that the quotient must be greater than the original fraction (but not necessarily greater than the proper fraction itself). For instance, assume that the original fraction is $\frac{a}{b}$ and that the improper fraction is $\frac{c}{d}$. Then the division results in: $\displaystyle \frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a}{b}*\frac{d}{c}\gt \frac{a}{b}$ (because $d\gt c$ in a proper fraction) Since the numerator in the proper fraction is smaller than the denominator ($c\lt d$), the resulting fraction must be larger than the original fraction. However, the result is not necessarily larger than the proper fraction itself. For example: Original fraction: $\frac{1}{2}$, proper fraction: $\frac{3}{4}$: $\displaystyle \frac{\frac{1}{2}}{\frac{3}{4}}=\frac{1}{2}*\frac{4}{3}=\frac{4}{6}=\frac{2}{3}$ $\frac{2}{3}\gt \frac{1}{2}$, but $\frac{2}{3}\lt \frac{3}{4}$