Fractions

Skill 1: Simplifying Fractions

To simplify a fraction, we divide the numerator (the top of the fraction) and the denominator (the bottom of the fraction) by the same amount, until we can’t simplify anymore.

Note: Simplifying fractions doesn’t change the value of the fraction

Example: Write \dfrac{12}{30} in its simplest form.

\textcolor{red}{12} and \textcolor{red}{30} both contain 6 as a factor, meaning we can divide both by 6.

\dfrac{12}{30}= \dfrac{12\div6}{30\div6} = \textcolor{black}{\dfrac{2}{5}}

2 and 5 are both prime numbers so cannot simplified anymore.

Mixed fractions are numbers with an integer part and a fraction part, such as 6 \dfrac{1}{4}.

Improper fractions are fractions where the numerator (top) is bigger than the denominator (bottom), such as \dfrac{9}{2}. We also call these top-heavy fractions for this reason.

We need to be able to convert between these two types.

Example: Write \textcolor{limegreen}{3} \textcolor{blue}{\dfrac{3}{4}} as an improper fraction.

We can write

\textcolor{limegreen}{3} \textcolor{blue}{\dfrac{3}{4}} = \textcolor{limegreen}{3} + \textcolor{blue}{\dfrac{3}{4}}

Then

3 + \dfrac{3}{4} = \dfrac{12}{4} + \dfrac{3}{4} = \textcolor{orange}{\dfrac{15}{4}}

Skill 4: Subtracting Fractions

When subtracting fractions you need to make sure you have a common denominator. Then we subtract one fraction from the other by subtracting one numerator from the other.

Example: Solve \dfrac{4}{5} - \dfrac{1}{2}

\dfrac{4}{5}=\dfrac{4\times2}{5\times 2}=\dfrac{8}{10}

\dfrac{1}{2}=\dfrac{1\times 5}{2\times 5}=\dfrac{5}{10}

Now, to subtract two fractions with the same denominator, simply subtract the second numerator from the first. Doing so, we get

\dfrac{4}{5} - \dfrac{1}{2}=\dfrac{8}{10} - \dfrac{5}{10}= \textcolor{black}{\dfrac{3}{10}}

Skill 7: Calculations involving Mixed Fractions

When doing calculations involving mixed fractions, it is easier to convert them to improper fractions first. We then perform the calculation using the above skills.

Example: Calculate \textcolor{red}{2 \dfrac{1}{4}} \times \textcolor{blue}{3 \dfrac{1}{2}}.

\textcolor{red}{2 \dfrac{1}{4}} = \dfrac{9}{4} \, and \, \textcolor{blue}{3 \dfrac{1}{2}} = \dfrac{7}{2}

Then

2 \dfrac{1}{4} \times 3 \dfrac{1}{2} = \dfrac{9}{4} \times \dfrac{7}{2} = \dfrac{9 \times 7}{4 \times 2} = \textcolor{black}{\dfrac{63}{8}}

We cannot simplify it any further, so our answer is in its simplest form.

Skill 8: Fractions of Amounts

To work out a fraction of an amount, for example money, we divide the amount by the denominator (bottom) and then multiply it by the numerator (top) or vice versa

Example: Calculate \dfrac{\textcolor{limegreen}{3}}{\textcolor{red}{5}} of \textcolor{blue}{\$ 170}.

Divide \textcolor{blue}{\$ 170} by \textcolor{red}{5} first and then multiply by \textcolor{limegreen}{3}:

 \begin{aligned} \dfrac{3}{5} \text{ of } \textcolor{blue}{\$ 170} &= (\textcolor{blue}{\$ 170} \div \textcolor{red}{5}) \times \textcolor{limegreen}{3} \\ &= \$ 34 \times 3 \\ & = \textcolor{black}{\$ 102} \end{aligned}