2.1 Fractions
We will learn to solve basic operations involving fractions.
  • An improper fraction is a fraction with the numerator equal to or greater than the denominator.

As example,

\(\dfrac{24}{5}\) is an improper fraction because the value of the numerator (\(\color{red}{24}\)) is greater than the value of the denominator (\(\color{red}{5}\)).

  • mixed number consists of a whole number and a proper fraction.

As example,\(\dfrac{24}{5}\) can be converted to mixed numbers as follows:


Based on the above picture, \(\dfrac{24}{5}=\color{magenta}{4\dfrac{4}{5}}\).


Or, the simple steps to convert improper fractions to mixed numbers are as follows:

\(\begin{aligned}\space \space 4 \\5\space\overline{)2\space4} \\\underline{-\space2\space0}\\4 \end{aligned}\)\(=\color{magenta}{4\dfrac{4}{5}}\)


The simple steps to convert mixed numbers to improper fractions are as follows:

\(\begin {aligned} \color{black}{4\dfrac{4}{5}} &= \dfrac {4 \times 5 +4}{5} \\ &= \color{magenta}{\dfrac {24}{5}} \end {aligned}\)

  • To add fractions with the same denominators, add the numerators without changing the denominators.

As example,

\(\begin{aligned} \dfrac{1}{5}+\dfrac{2}{5}&=\dfrac{1+2}{5} \\&=\color{red}{\dfrac{3}{5}} \end{aligned}\)

  • To add fractions with different denominators, find the common multiple value of the denominator.

As example,

\(\begin{aligned} \dfrac{1}{6}+\dfrac{3}{4}&=\dfrac{1{\color{green}{\times2}}}{6{\color{green}{\times2}}}+\dfrac{3{\color{green}{\times3}}}{4{\color{green}{\times3}} }\\&=\dfrac{2}{12}+\dfrac{9}{12} \\&={\color{magenta}{\dfrac{11}{12}}} \end{aligned}\)

  • To subtract fractions with the same denominator, the subtraction operation is performed as usual.
The denominator is retained.

As example,


Thus, \(1-\dfrac{2}{5}=\color{magenta}{\dfrac{3}{5}}\).


To subtract the fractions with different denominators, change to equivalent fractions with the same denominators before subtracting the numerators.

Retain the denominator.

As example,

\(\begin{aligned} \dfrac{3}{5}-\dfrac{1}{2}&=\dfrac{3\color{green}{\times2}}{5\color{green}{\times2}}-\dfrac{1\color{green}{\times5}}{2\color{green}{\times5}} \\&=\dfrac{6}{10}-\dfrac{5}{10} \\&={\color{magenta}{\dfrac{1}{10}}} \end{aligned}\)

  • For mixed operations involving the addition and subtraction of fractions, the calculation is done from the left to the right.

As example,

\(\begin{aligned} \dfrac{1}{7}+\dfrac{5}{7}-\dfrac{2}{7}&=\dfrac{1+5-2}{7} \\&={\color{red}{\dfrac{4}{7}}} \end{aligned}\)

  • 'from' means multiply

As example,

find the value of \(\dfrac{5}{7}\) from \(56\).

\(\begin{aligned} \dfrac{5}{7}\text{ from }56&=\dfrac{5}{7}\times56 \\&=\dfrac{5\times56}{7} \\&=\dfrac{280}{7} \\&={\color{red}{40}} \end{aligned}\)