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# Fractions

## Fractions and simplification of fractions

A fraction represents a rational number as the ratio of two integers, the numerator and the denominator. The denominator cannot be zero.

$\textrm{Fraction} = \frac{\Tblue{\textrm{Numerator}}}{\Tred{\textrm{Denominator}}}$

The fraction $\Tblue{17}/\Tred{26}$ has numerator $\Tblue{17}$ and denominator $\Tred{26}$.

A fraction is in its simplest form, or irreducible, if the numerator and denominator have no common factor. A fraction has a unique simplest form. To simplify a fraction, find the highest common factor of the numerator and the denominator and divide both by it.

$\Tblue{7}/\Tred{9}$ is irreducible. $\Tblue{18}/\Tred{27}$ is not. The HCF of $\Tblue{18}$ and $\Tred{27}$ is $9$. The fraction can be simplified to $\Tblue{2}/\Tred{3}$. $\frac{\Tblue{18}}{\Tred{27}} = \frac{9\times \Tblue{2}}{9\times \Tred{3}} = \frac{\Tblue{2}}{\Tred{3}}.$

Two fractions are equivalent if they have the same value. They have the same simplest form.

$\Tblue{3}/\Tred{9}$ and $\Tblue{2}/\Tred{6}$ are equivalent; their simplest form is $\Tblue{1}/\Tred{3}$.

Every fraction has infinitely many equivalent forms.

## Proper, improper fractions and mixed fractions

A fraction is proper when its numerator is smaller than its denominator in absolute value. The fraction is then a rational number between $-1$ and $1$

When a fraction is not proper, it is improper or top-heavy.

$1/2$ and $-18/19$ are proper; $19/18$ and $-3/2$ are improper.

A mixed fraction is a whole number combined with a proper fraction. It is easier to compare two mixed fractions than two improper fractions.

Negative fractions are treated just the same way as positive fractions.

$\Torange{1} \frac{1}{2}$, $\Torange{8} \frac{5}{17}$ and $\Torange{-3} \frac{9}{20}$ are mixed fractions.

$\Torange{2} \frac{1}{4}$ and $\Torange{1} \frac{83}{91}$ are easier to compare than $\frac{9}{4}$ and $\frac{174}{91}$ although the values are the same.

It is easy to convert mixed and improper fractions.

• Convert to mixed: $\displaystyle \frac{19}{\Tred{5}} = \frac{\Torange{3}\times \Tred{5} + \Tblue{4}}{\Tred{5}} = \frac{\Torange{3}\times \Tred{5}}{\Tred{5}} + \frac{\Tblue{4}}{\Tred{5}} = \Torange{3} \frac{\Tblue{4}}{\Tred{5}}$
• To improper: $\displaystyle -\Torange{2}\frac{\Tblue{1}}{\Tred{2}} = -\left( \frac{\Torange{2}\times\Tred{2}}{\Tred{2}} + \frac{\Tblue{1}}{\Tred{2}}\right) = -\left( \frac{5}{\Tred{2}}\right) = -\frac{5}{\Tred{2}}$
Equivalent improper and mixed fractions

## Reciprocal

The reciprocal of a number is one divided by that number. It is the same as raising it to the power $-1$

The reciprocal of $3$ is a third. It can be written as $1/3$ or $3^{-1}$.

The reciprocal of a fraction is found by swapping the numerator and the denominator.

$\left(\frac{\Tblue{2}}{\Tgreen{7}}\right)^{-1} = \frac{\Tgreen{7}}{\Tblue{2}} = 3.5$

These are the main properties of the reciprocal:

• Every real number except for zero has a reciprocal.
• The reciprocal of a number has the same sign as the number.
• The reciprocal of the reciprocal is the original number.
• The reciprocal multiplied by the original number is $1$.
• The reciprocal of a positive number between $0$ and $1$ is larger than $1$, and vice versa.
• The reciprocal of a negative number between $0$ and $-1$ is less than $-1$, and vice versa.
$(-3)^{-1} = -\frac{1}{3},\; (2^{-1})^{-1} = 2,\; (0.25)^{-1} = 4,\;(-0.25)^{-1}=-4$
A graph showing $y$ and the reciprocal of $x$

## Multiplication and division of fractions

To multiply fractions, the numerators are multiplied to give the new numerator, and the denominators are multiplied to give the new denominator. The fraction can then be simplified.

For instance: $\frac{\Tred{4}}{\Tblue{5}} \times \frac{\Tred{7}}{\Tblue{6}} = \frac{\Tred{4}\times \Tred{7}}{\Tblue{5}\times \Tblue{6}} = \frac{\Tred{28}}{\Tblue{30}} = \frac{14}{15}$

To divide fractions, we take the reciprocal of the fraction we are dividing by and multiply it by the other fraction.

For instance, $\displaystyle\frac{4}{5} \div \frac{\Tred{2}}{\Tblue{3}} = \frac{4}{5} \times \frac{\Tblue{3}}{\Tred{2}} = \frac{4\times \Tblue{3}}{5\times \Tred{2}} = \frac{12}{10} = \frac{6}{5}$

$1/3 \times 1/2$ is the same as finding half of a third, or a third of one half.

## Addition and subtraction of fractions

If you want to add or substract fractions such as $\displaystyle \frac{1}{\Tred{8}} + \frac{5}{\Tblue{6}},$ you need to go through three steps.

• Re-write the fractions so that they have the same denominator. This can be done by cross multiplying by the denominators. $\frac{1}{\Tred{8}} + \frac{5}{\Tblue{6}} = \frac{1\times\Tblue{6}}{\Tred{8}\times\Tblue{6}} + \frac{5\times\Tred{8}}{\Tblue{6}\times \Tred{8}} = \frac{6}{\Tviolet{48}} + \frac{40}{\Tviolet{48}}$
• Add or subtract the numerators $\displaystyle \frac{\Tgreen{6}}{\Tviolet{48}} + \frac{\Tgreen{40}}{\Tviolet{48}} = \frac{\Tgreen{6} + \Tgreen{40}}{\Tviolet{48}}= \frac{\Tgreen{46}}{\Tviolet{48}}$
• Simplify the resulting fraction $\displaystyle \frac{46}{48} = \frac{23\times\Torange{2}}{24\times\Torange{2}} = \frac{23}{24}$

Here are some other examples

\begin{align*} &\frac{5}{6} + \frac{2}{3} = \frac{15 + 12}{18} = \frac{27}{18} = \frac{3}{2},\; &\frac{3}{5} + \frac{1}{3} = \frac{9 + 5}{15} = \frac{14}{15},\\ &\frac{5}{6} - \frac{2}{3} = \frac{15 - 12}{18} = \frac{3}{18} = \frac{1}{6}, \; &\frac{3}{5} - \frac{1}{3} = \frac{9 - 5}{15} = \frac{4}{15} \end{align*}
Fractions can be added when the denominators are the same.

## Addition and subtraction of fractions (using LCM)

To add fractions, we need to re-write them with the same denominators. One way is to multiply the denominators.

$\frac{1}{8} + \frac{5}{6} = \frac{(1\times 6)+ (8\times 5)}{8\times 6} = \frac{46}{48} = \frac{23}{24}$

There is a smarter way which uses smaller numbers.

• Find the lowest common multiple of the denominators $\LCM(8,6) = \Tviolet{24}.$
• Re-write the fractions so that their denominators are equal to the lowest common multiple. $\frac{1}{\Tred{8}} + \frac{5}{\Tred{6}} = \frac{1\times \Tblue{3}}{\Tred{8} \times \Tblue{3}} + \frac{5\times \Tblue{4}}{\Tred{6} \times \Tblue{4}} = \frac{3}{\Tviolet{24}} + \frac{20}{\Tviolet{24}}$
• Add the numerators and simplify if necessary. $\displaystyle \frac{3}{\Tviolet{24}} + \frac{20}{\Tviolet{24}} = \frac{23}{\Tviolet{24}}.$

Here are some other examples:

\begin{align*} \frac{5}{6} + \frac{2}{3} = \frac{5 + 4}{6} = \frac{9}{6} = \frac{3}{2}, &\quad \frac{2}{15} + \frac{1}{6} = \frac{4 + 5}{30} = \frac{9}{30} = \frac{3}{10},\\ \frac{5}{6} - \frac{2}{3} = \frac{5 - 4}{6} = \frac{1}{6}, &\quad \frac{2}{15} - \frac{1}{6} = \frac{4 - 5}{30} = \frac{-1}{30} \end{align*}

## Summary of the operations on fractions

Here is a summary of the operations on fractions.

Operation Formula Example
Addition $\displaystyle \frac{\Tblue{a}}{\Tviolet{b}} + \frac{\Tgreen{c}}{\Torange{d}} = \frac{\Tblue{a}\Torange{d}+\Tviolet {b}\Tgreen{c}}{\Tviolet{b}\Torange{d}}$ $\displaystyle\frac{1}{5} + \frac{2}{3} = \frac{13}{15}$
Subtraction $\displaystyle \frac{\Tblue{a}}{\Tviolet {b}} - \frac{\Tgreen{c}}{\Torange{d}} = \frac{\Tblue{a}\Torange{d}-\Tviolet {b}\Tgreen{c}}{\Tviolet{b}\Torange{d}}$ $\displaystyle\frac{2}{5} - \frac{1}{6} = \frac{7}{30}$
Multiplication $\displaystyle \frac{\Tblue{a}}{\Tviolet {b}}\times \frac{\Tgreen{c}}{\Torange{d}} = \frac{\Tblue{a}\Tgreen{c}}{\Tviolet {b}\Torange{d}}$ $\displaystyle\frac{2}{7} \times \frac{3}{5} = \frac{6}{35}$
Division $\displaystyle \frac{\Tblue{a}}{\Tviolet {b}} \div \frac{\Tgreen{c}}{\Torange{d}}= \frac{\Tblue{a}\Torange{d}}{\Tviolet {b} \Tgreen{c}}$ $\displaystyle\frac{2}{7} \div \frac{3}{5} = \frac{10}{21}$