One more post before I return to core algebra. We need to look a bit more at division and fractions.

Now, as you’ve seen, something like 8 ÷ 2 indicates division, but another way to show exactly the same thing is \[\frac{8}{2}\]. In other words, fractions are just another way to show division. Now before I expand on this, let’s review how fractions multiply together.

When two fractions are to be multiplied, the process is very simple. You just multiply the numerators (the numbers above the line) together and the denominators (the numbers below the line) together.

\[\frac{1}{2}\hspace{0.33em}\times\hspace{0.33em}\frac{3}{4}\hspace{0.33em}{=}\hspace{0.33em}\frac{{1}\hspace{0.33em}\times\hspace{0.33em}{3}}{{2}\hspace{0.33em}\times\hspace{0.33em}{4}}\hspace{0.33em}{=}\hspace{0.33em}\frac{3}{8}\]

Now this can be used to our advantage to simplify fractions. Each of the numbers in the above example are called “factors”. Factors are things that are multiplied together. So if we can show the factors of the parts of a fraction, we can effectively cancel factors that are common between the numerator and the denominator because we can split off the common factors as \[\frac{\mathrm{number}}{\mathrm{number}}\] and any number divided by itself is 1 and anything multiplied by 1 is the same anything.

\[\frac{8}{2}\hspace{0.33em}{=}\hspace{0.33em}\frac{{2}\hspace{0.33em}\times\hspace{0.33em}{4}}{{2}\hspace{0.33em}\times\hspace{0.33em}{1}}\hspace{0.33em}{=}\hspace{0.33em}\frac{2}{2}\hspace{0.33em}\times\hspace{0.33em}\frac{4}{1}\hspace{0.33em}{=}\hspace{0.33em}{1}\hspace{0.33em}\times\hspace{0.33em}{4}\hspace{0.33em}{=}\hspace{0.33em}{4}\]

A shortcut version of this is

\[\frac{8}{2}\hspace{0.33em}{=}\hspace{0.33em}\frac{\rlap{/}{2}\hspace{0.33em}\times\hspace{0.33em}{4}}{\rlap{/}{2}\hspace{0.33em}\times\hspace{0.33em}{1}}\hspace{0.33em}{=}\hspace{0.33em}\frac{4}{1}\hspace{0.33em}{=}\hspace{0.33em}{4}\]

So note that when you cross out the only factor in the numerator or denominator, a “1” is left, and this “1” can be left out of the result since it does not change the value of the remaining numbers. Also note that this works for known and unknown numbers such as *x* which we will see in my next post.

A couple more examples:

\[\begin{array}{c}{\frac{16}{4}\hspace{0.33em}{=}\hspace{0.33em}\frac{\rlap{/}{4}\hspace{0.33em}\times\hspace{0.33em}{4}}{\rlap{/}{4}\hspace{0.33em}\times\hspace{0.33em}{1}}\hspace{0.33em}{=}\hspace{0.33em}{4}}\\

{\frac{6}{9}\hspace{0.33em}{=}\hspace{0.33em}\frac{\rlap{/}{3}\hspace{0.33em}\times\hspace{0.33em}{2}}{\rlap{/}{3}\hspace{0.33em}\times\hspace{0.33em}{3}}\hspace{0.33em}{=}\hspace{0.33em}\frac{2}{3}}

\end{array}\]