# Fractions and Decimals

I want to do more interesting algebra problems but there is a bit more  to talk about regarding fractions.

Now remember that multiplying fractions is relatively simple:

$\frac{2}{3}\hspace{0.33em}\times\hspace{0.33em}\frac{4}{5}\hspace{0.33em}{=}\hspace{0.33em}\frac{{2}\hspace{0.33em}\times\hspace{0.33em}{4}}{{3}\hspace{0.33em}\times\hspace{0.33em}{5}}\hspace{0.33em}{=}\hspace{0.33em}\frac{8}{15}$

This and the fact that a “1” can be assumed anywhere as long as it is multiplying or dividing a number can make the same fraction look different. For example, dividing a number say x by 2, is the same as multiplying x by 1/2:

${x}\hspace{0.33em}\div\hspace{0.33em}{2}\hspace{0.33em}{=}\hspace{0.33em}\frac{x}{2}\hspace{0.33em}{=}\hspace{0.33em}\frac{1}{2}\hspace{0.33em}\times\hspace{0.33em}\frac{x}{1}\hspace{0.33em}{=}\hspace{0.33em}\frac{1}{2}\hspace{0.33em}\times\hspace{0.33em}{x}\hspace{0.33em}{=}\hspace{0.33em}\frac{1}{2}{x}$

If some of this is confusing, please review my previous post on fractions.

Now decimals are really fractions in disguise where the denominators are multiples of 10. So,

${0}{.}{5}\hspace{0.33em}{=}\hspace{0.33em}\frac{5}{10}{,}\hspace{0.33em}{0}{.}{67}\hspace{0.33em}{=}\hspace{0.33em}\frac{67}{100}{,}\hspace{0.33em}{0}{.}{108}\hspace{0.33em}{=}\hspace{0.33em}\frac{108}{1000}$

Notice a couple of things here. The number of decimal places to the right of the “.” is the same as the number of “0”s in the denominator. Also, for pure decimals where there is no number to the left of “.”, it is customary to put a “0” because the “.” by itself is easy to miss.

Now, if you were to divide 1 by 2, long hand or on your calculator, you would get 0.5 as the answer.  Now 1 divided by 2 is indicated by the fraction $\frac{1}{2}$. All fractions have a decimal equivalent which you can find by doing the indicated division. Another way to show that $\frac{1}{2}$ = 0.5 is

$\frac{1}{2}\hspace{0.33em}{=}\hspace{0.33em}\frac{1}{2}\hspace{0.33em}\times\hspace{0.33em}\frac{5}{5}\hspace{0.33em}{=}\hspace{0.33em}\frac{{1}\hspace{0.33em}\times\hspace{0.33em}{5}}{{2}\hspace{0.33em}\times\hspace{0.33em}{5}}{=}\hspace{0.33em}\frac{5}{10}\hspace{0.33em}{=}\hspace{0.33em}{0}{.}{5}$

So another way to show x ÷ 2 is

$\frac{x}{2}\hspace{0.33em}{=}\hspace{0.33em}\frac{1}{2}{x}\hspace{0.33em}{=}\hspace{0.33em}{0}{.}{5}{x}$

So in the future, when we solve equations where fractions or divisions are present, we can change their form to suit us as we need to in order to solve the equation or express the answer in a desired form. Let’s see this tomorrow.