Sorting fractions

Given two integers we can determine easily which is bigger. This relation of order can be defined also between fractions.

Let's consider the fractions $\displaystyle \frac{a}{b}$ and $\displaystyle \frac{c}{d}$ with $b$ and $d$ positives. The fraction $\displaystyle \frac{a}{b}$ is bigger than the fraction $\displaystyle \frac{c}{d}$ if $$a\cdot d >c \cdot b$$.

This relation is natural because $\displaystyle \frac{a}{b}=\frac{a\cdot d}{b \cdot d}$ and $\displaystyle \frac{c}{d}=\frac{c\cdot b}{d\cdot b}$, and, as they have the same denominator, we can just focus on the numerator.

Let's see some example where we are going to sort the numbers $\displaystyle \frac{1}{3}, \frac{2}{5}$ and $\displaystyle \frac{1}{4}$.

We write them with a common denominator, $$\displaystyle \frac{1}{3}=\frac{1\cdot5\cdot4}{3\cdot5\cdot4}=\frac{20}{60}$$ $$\displaystyle \frac{2}{5}=\frac{2\cdot3\cdot4}{5\cdot3\cdot4}=\frac{24}{60}$$ $$\displaystyle \frac{1}{4}=\frac{1\cdot3\cdot5}{4\cdot3\cdot5}=\frac{15}{60}$$

We have $15 < 20 < 24$ and therefore $\dfrac{1}{4} < \dfrac{1}{3} < \dfrac{2}{5}$