A proper fraction is a fraction whose numerator is smaller than its denominator.
For example: \( \dfrac{1}{2} , \dfrac{4}{5} , \dfrac{3}{7} \)
Consider the proper fraction \( \dfrac{4}{5} \).
It can be visualised as:
An improper fraction is a fraction whose numerator is greater than its denominator.
For example: \( \dfrac{3}{2} , \dfrac{7}{4} , \dfrac{9}{5} \)
Consider the improper fraction \( \dfrac{7}{4} \).
It can be visualised as:
A mixed fraction or mixed number is a combination of a whole number and a proper fraction.
For example: \( 3\dfrac{1}{2} , 5\dfrac{2}{5} , 1\dfrac{3}{4} \)
The mixed fraction \(1\dfrac{3}{4}\) means there is one whole part plus \(\dfrac{3}{4}\) of a whole part.
In fact, \(1\dfrac{3}{4} = 1 + \dfrac{3}{4}\) and can be visualised as:
Notice that the shaded region in the circles representing \(\dfrac{7}{4}\) and \(1\dfrac{3}{4}\) are the same!
In fact, \(1\dfrac{3}{4} = \dfrac{7}{4}\).
They are different ways of representing the same thing!
To switch from mixed to improper:
Examples
To switch from improper to mixed:
The improper fraction in mixed form is: \(\text{Quotient}\dfrac{\text{Remainder}}{\text{Denominator}}\)
Example
Suppose we want to change \( \dfrac{21}{4} \) to a mixed fraction.
\(21\) divided by \(4\) is \(5\) with remainder \(1\).
So:
\( \dfrac{21}{4} = 5\frac{1}{4} \)
To add/subtract fractions:
**If they are not, this can be done by rewriting one/both of the fractions by multiplying both numerator and denominator by the same value.
Example
\( \dfrac{1}{2} \) is the same as \( \dfrac{2}{4} \) because \( \dfrac{1}{2} = \dfrac{1\times 2}{2\times 2} = \dfrac{2}{4} \)
For simplicity, we aim to multiply both fractions separately to ensure each denominator becomes the least common multiple (LCM) of the two original denominators. See Example 2.
Suppose we want to add: \( \dfrac{3}{12} + \dfrac{5}{12} \).
The denominators are the same, so we add the numerators and simplify:
\begin{align} \dfrac{3}{12}+\dfrac{5}{12} &= \dfrac{8}{12} \\ &= \dfrac{8 \div 4}{12 \div 4} \\ &= \dfrac{2}{3} \end{align}
The above calculation can be visualised as:
Suppose we want subtract: \( \dfrac{2}{5} - \dfrac{1}{4} \).
The denominators are not the same so we must rewrite one/both of the fractions.
The least common multiple of \(5\) and \(4\) is \(20\), so we need to multiply each fraction above and below so that their denominator becomes \(20\).
\begin{align} \dfrac{2}{5} &= \dfrac{2\times4}{5\times4} \\ &= \dfrac{8}{20} \end{align}
and
\begin{align} \dfrac{1}{4} &= \dfrac{1\times5}{4\times5} \\ &= \dfrac{5}{20} \end{align}
Now that the denominators are the same, the fractions can be subtracted as usual:
\begin{align} \dfrac{2}{5}-\dfrac{1}{4} &= \dfrac{8}{20}-\dfrac{5}{20} \\ &= \dfrac{3}{20} \end{align}
The above calculation can be visualised as:
To multiply fractions:
In other words, remember the rule "Top by top, Bottom by bottom".
Example
\begin{align} \dfrac{3}{5} \times \dfrac{2}{7} &= \dfrac{3\times2}{5\times7} \\ &= \dfrac{6}{35} \end{align}
To divide fractions:
In other words, remember the rule "Keep, Change, Flip".
Example
\begin{align} \dfrac{2}{9} \div \dfrac{3}{4} &= \dfrac{2}{9} \times \dfrac{4}{3} \qquad \qquad \qquad \text{(1)} \\ &= \dfrac{2\times4}{9\times3} \\ &= \dfrac{8}{27} \end{align}
(1) Keep the \( \dfrac{2}{9} \) , Change the division to multiplication, Flip the \( \dfrac{3}{4} \)