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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.
Every fraction has infinitely many equivalent forms.

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
Equivalent improper and mixed fractions

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 $$x$$
A graph showing $$y$$ and the reciprocal of $$x$$

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

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.
Fractions can be added when the denominators are the same.

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*}

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 $$\displaystyle \frac{\Tblue{a}}{\Tviolet {b}} \div \frac{\Tgreen{c}}{\Torange{d}}= \frac{\Tblue{a}\Torange{d}}{\Tviolet {b} \Tgreen{c}}$$ $$\displaystyle\frac{2}{7} \div \frac{3}{5} = \frac{10}{21} $$