Learning Lab

Simplifying Fractions

You have probably seen numerical fractions like \(\frac{18}{24}\) before. Generally we like to reduce fractions to their simplest form. This is done by dividing the numerator and denominator by the same number.11 The numerator is the number on top and the denominator is the number at the bottom of the fraction.

So \[\begin{alignat*}{1} \frac{18}{24} & =\frac{9}{12}\quad\mathrm{\mathrm{dividing}\ top\ and\ bottom\ numbers\ by\ two}\\ & =\frac{3}{4}\quad\mathrm{\mathrm{dividing}\ top\ and\ bottom\ numbers\ by\ three} \end{alignat*}\] this is the simplest form as there is no number that divides into \(3\) and \(4\). Note that this can be done in one step if you realise that \(6\) divides into \(18\) and \(24\). Then you have \[ \frac{18}{24}=\frac{3}{4}\quad\mathrm{\mathrm{dividing}\ top\ and\ bottom\ numbers\ by\ six.} \] Simplifying fractions by dividing the same number into both the numerator and denominator is called cancelling.

Simplifying Algebraic Fractions

We can use the same technique to simplify algebraic fractions as we did in simplifying numerical fractions. For example:

1. Simplify \(\frac{4x^{2}y}{6y^{2}}\).
   \[\begin{align*} \frac{4x^{2}y}{6y^{2}} & =\frac{2x^{2}y}{3y^{2}}\quad\mathrm{\mathrm{dividing}\ top\ and\ bottom\ by\ two}\\ & =\frac{2x^{2}}{3y}\quad\mathrm{\mathrm{dividing}\ top\ and\ bottom\ by}\:y. \end{align*}\]

2. Simplify \(\frac{a(b+2c)}{2ab}.\)
   This example shows it is important to divide all terms on the top by the same number.22 Note that you cannot divide the \(2c\) on top and the \(2b\) on the bottom by \(2\). This is because all the terms on the top and bottom have to be divided by \(2\). So it would be wrong to write \[ \frac{b+2c}{2b}=\frac{b+c}{b} \] because the \(b\) on the top line was not divided by \(2\). \[\begin{alignat*}{1} \frac{a(b+2c)}{2ab} & =\frac{(b+2c)}{2b}\quad\mathrm{\mathrm{dividing}\ top\ and\ bottom\ by}\:a\\ & =\frac{b+2c}{2b}. \end{alignat*}\]

3. Simplify \(\frac{m-n}{\left(m-n\right)^{2}}\). 33 This example uses the fact that \(\left(m-n\right)^{2}=\left(m-n\right)\left(m-n\right)\). Since \(m\) and \(n\) are just numbers, \(m-n\) is also a number and so we can divide by \(m-n\) provided that \(m\neq n\). If \(m=n\), \(m-n=0\) and we mustn’t divide anything by zero. \[\begin{alignat*}{1} \frac{m-n}{\left(m-n\right)^{2}} & =\frac{m-n}{\left(m-n\right)\left(m-n\right)}\quad\left(m\neq n\right)\\ & =\frac{1}{m-n}\quad\mathrm{\mathrm{dividing}\ top\ and\ bottom\ by}\:m-n. \end{alignat*}\]

4. Simplify \(\frac{3x^{2}y}{6x+9y}.\) 44 This example factorizes the denominator first. We then divide the top and bottom of the fraction by the common factor \(3\). \[\begin{alignat*}{1} \frac{3x^{2}y}{6x+9y} & =\frac{3x^{2}y}{3\left(2x+3y\right)}\quad\mathrm{\mathrm{taking\:out}\ the\;factor\;of\:3\ in\ the\:denominator}\\ & =\frac{x^{2}y}{2x+3y}\quad\mathrm{\mathrm{dividing}\ top\ and\ bottom\ by}\:3. \end{alignat*}\]

5. Simplify \(\frac{p-2}{\left(6p-3p^{2}\right)}.\) 55 This example uses the fact that \(p-2=-\left(2-p\right).\) \[\begin{alignat*}{1} \frac{p-2}{\left(6p-3p^{2}\right)} & =\frac{p-2}{3\left(2p-p^{2}\right)}\quad\mathrm{\mathrm{taking\:out}\ the\;factor\;of\:3\ in\ the\:denominator}\\ & =\frac{p-2}{3p\left(2-p\right)}\quad\mathrm{\mathrm{taking\:out}\ the\;factor\;of\:\mathit{p}\ in\ the\:denominator}\\ & =-\frac{p-2}{3p\left(p-2\right)}\quad\mathrm{\mathrm{using\;\mathit{p-\mathrm{2}=-\left(\mathrm{2}-p\right)}}}\\ & =-\frac{1}{3p}\quad\mathrm{\mathrm{dividing}\ top\ and\ bottom\ by}\:p-2. \end{alignat*}\]

Note that you can only divide the denominator and numerator if ALL the terms have a common factor. For example, it would be WRONG to write \[\begin{alignat*}{1} \frac{x+2}{2y} & =\frac{x+1}{y} \end{alignat*}\]

because the \(x\) in the top line has not been divided by 2.

To practice these techniques, please look at Exercise 1.

Multiplying Fractions

When we multiply two fractions, we multiply the numerators and the denominators of each fraction. For example, \[ \frac{3}{4}\times\frac{7}{5}=\frac{21}{20}. \]

It is best to simplify each fraction, if possible, before you do the multiplication. For example, \[\begin{alignat*}{1} \frac{15}{8}\times\frac{24}{35} & =\frac{15}{1}\times\frac{3}{35}\quad\mathrm{dividing\:the\;24\:in\:the\:top\:line\:and\:the\:8\:in\:the\:bottom\:line\:by\:8}\\ & =\frac{3}{1}\times\frac{3}{7}\quad\mathrm{dividing\:the\;15\:in\:the\:top\:line\:and\:the\:35\:in\:the\:bottom\:line\:by\:5}\\ & =\frac{9}{7}. \end{alignat*}\]

Multiplying Algebraic Fractions

You can also use these ideas with algebraic fractions. For example, \[\begin{alignat*}{1} \frac{5a}{7}\times\frac{14}{a} & =\frac{5}{7}\times\frac{14}{1}\quad\mathrm{dividing\:the\;5\mathit{a}\:in\:the\:top\:line\:and\:the\:\mathit{a}\:in\:the\:bottom\:line\:by\:\mathit{a}}\\ & =\frac{5}{1}\times\frac{2}{1}\quad\:\mathrm{dividing\:the\;14\:in\:the\:top\:line\:and\:the\:\mathit{7}\:in\:the\:bottom\:line\:by\:\mathit{7}}\\ & =\frac{10}{1}\\ & =10. \end{alignat*}\]

Here are some more examples:66 Note that we are showing every step in the examples below and so the solutions may appear long and complicated. You don’t have to do this. You can take as many steps as you like. As you get more familiar with algebra you will naturally use fewer steps to get a result.

1. Simplify \(\frac{x}{6\left(x-2\right)}\times\frac{3\left(x-2\right)}{x^{2}}\). \[\begin{alignat*}{1} \frac{x}{6\left(x-2\right)}\times\frac{3\left(x-2\right)}{x^{2}} & =\frac{x}{6}\times\frac{3}{x^{2}}\quad\mathrm{dividing\:the\:top\:and\:bottom\:lines\:by\:\mathit{x-2}}\\ & =\frac{x}{2}\times\frac{1}{x^{2}}\quad\mathrm{dividing\:the\:top\:and\:bottom\:lines\:by\:\mathit{3}}\\ & =\frac{1}{2}\times\frac{1}{x}\quad\mathrm{dividing\:the\:top\:and\:bottom\:lines\:by\:\mathit{x}}\\ & =\frac{1}{2x}. \end{alignat*}\]

2. Simplify \(\frac{3m+12}{10}\times\frac{5}{m^{2}+4m}.\) \[\begin{alignat*}{1} \frac{3m+12}{10}\times\frac{5}{m^{2}+4m} & =\frac{3\left(m+4\right)}{10}\times\frac{5}{m^{2}+4m}\quad\mathrm{taking\:out\:a\:factor\:of\:\mathit{3}}\\ & =\frac{3\left(m+4\right)}{10}\times\frac{5}{m\left(m+4\right)}\quad\mathrm{taking\:out\:a\:factor\:of\:\mathit{m}}\\ & =\frac{3}{10}\times\frac{5}{m}\quad\mathrm{dividing\:the\:top\:and\:bottom\:by\:\mathit{m+4}}\\ & =\frac{3}{2}\times\frac{1}{m}\quad\mathrm{dividing\:the\:top\:and\:bottom\:lines\:by\:\mathit{5}}\\ & =\frac{3}{2m}. \end{alignat*}\]

Dividing Fractions

The reciprocal of a fraction is just the fraction turned upside down. So the reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\) and the reciprocal of \(\frac{2}{7}\) is \(\frac{7}{2}\). Dividing by a fraction is the same as multiplying by the reciprocal.77 When dividing fractions we change the divide sign to times and turn the last fraction upside down.

For example: \[\begin{alignat*}{1} \frac{5}{4}\div\frac{19}{8} & =\frac{5}{4}\times\frac{8}{19}\\ & =\frac{5}{1}\times\frac{2}{19}\\ & =\frac{10}{19}. \end{alignat*}\]

Dividing Algebraic Fractions

We use the same technique for dividing algebraic fractions as we use for dividing numerical fractions.

Exercise 1

Simplify the following fractions:

1. \(\frac{12ab^{2}}{8bc}\)

2. \(\frac{5x-20}{5}\)

3. \(\frac{9u-18}{2u-4}\)

4. \(\frac{6t-9}{12-8t}\)

5. \(\frac{b}{b^{2}+7b}\)

6. \(\frac{\left(j+4\right)\left(j-4\right)}{3j+12}\)

7. \(\frac{2\left(5-v\right)}{3v-15}\)

8. \(\frac{9r^{2}-3r}{16r-48r^{2}}\)

Exercise 2

A. Simplify

1. \(\frac{4}{5}\times\frac{15}{16}\)

2. \(\frac{4a}{3}\times\frac{9}{a}\)

3. \(\frac{32h^{2}}{9j}\times\frac{27j}{48h}\)

4. \(\frac{3d-2}{3}\times\frac{4}{3d-2}\)

5. \(\frac{2r+4}{3r-9}\times\frac{5r-15}{7r+14}\)

6. \(\frac{10p-5}{3}\times\frac{3q+3}{2p-1}\)

7. \(\frac{4g^{2}-6g}{8}\times\frac{3}{6g-9}\)

8. \(\frac{3-2y}{33y-11}\times\frac{18y^{2}-6y}{7-2y}\)

B. Simplify

1. \(\frac{4m-16}{m}\div\frac{8m-32}{8m}\)

2. \(\frac{6xy-5y^{2}}{4x+10y}\div\frac{12x^{2}-10xy}{12x+30y}\)