# Fraction Comparison: An Interactive Illustration

The applet below displays two fractions and their graphical representation with rows of rectangular areas. In both fractions, the digits of both the numerator and the denominator can be modified by clicking a little of their vertical midline. Clicking to the right of a digit will cause it to increase by 1; clicking to the left will decrease the digit by $1.$

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As with other numbers, the simplest way to compare fractions is to take their difference: $\displaystyle\frac{p}{q} \gt \frac{r}{s}\,$ if and only if $\displaystyle\frac{p}{q} - \frac{r}{s} \gt 0.\,$ The way to find the difference is to rewrite them with a common denominator, e.g.,

$\displaystyle\frac{p}{q}-\frac{r}{s}=\frac{ps}{qs}-\frac{rq}{sq}=\frac{ps-rq}{qs}.$

Now, provided $qs\,$ is positive, $\displaystyle\frac{p}{q} \gt \frac{r}{s},\,$ iff $\displaystyle ps - qr \gt 0,\,$ i.e., when $\displaystyle ps \gt qr.\,$ This is the cross-miltiplication criteria:

$\displaystyle\frac{p}{q} \gt\frac{r}{s}\,$ if and only if $\displaystyle ps \gt qr.$

The applet can find the least common denominator of two fractions or reduce the fractions to lowest terms (just press the suitable button). For technical reasons (there is simply not enough room) to present graphically (at least, in the manner chosen) fractions with the denominator in excess of 100. You'll see why when the common denominator of two fractions you wish to compare is greater than 100. When this happens, you'll need to first simplify the fractions back before proceeding further.

You may want to practice fraction comparison on special page.

Here are additional pages related to the definitions, properties of and operations on, fractions: