A proper fraction is a fraction whose numerator is smaller than its denominator. For example,
\[ \frac12, \frac45, \frac37 \]
Consider the proper fraction \(\frac45\) can be visualised as:
An improper fraction is a fraction whose numerator is greater than its denominator. For example,
\[ \frac32, \frac74,\frac95 \]
The improper fraction \(\frac74\) can be visualised as:
A mixed fraction is a combination of a whole number and a proper fraction. For example,
\[ 3\frac12, 5\frac25, 1\frac34 \]
The mixed fraction \(1\frac34\) means there is one whole part plus \(\frac34\) of a whole part. In fact, \(1\frac34 = 1 + \frac34\) and can be visualised as follows:
Notice that the shaded region in the circles representing \(\frac74\) and \(1\frac34\) are the same! In fact, \(\frac74 = 1\frac34\). They are different ways of representing the same thing!
Switching from mixed to improper:
To switch from mixed to improper, multiply the whole number by the denominator of the proper fraction and add to the numerator of the proper fraction:
For example:
Switching from improper to mixed:
To switch from improper to mixed, use long division to divide the denominator into the numerator, finding the quotient and the remainder. The improper fraction is mixed form is
\[ \text{Quotient}\frac{\text{Remainder}}{\text{Denominator}} \]
For example, suppose we want to change \(\frac{21}{4}\) to a mixed fraction. \(21\) divided by \(4\) is \(5\) with remainder \(1\). so
\[ \frac{21}{4} = 5\frac14 \]
To add fractions, take the following steps:
For example, suppose we want to add
\[ \frac{3}{12} + \frac{5}{12} \]
The denominators are the same, so we add the numerators and simplify.
\[ \frac{3}{12} + \frac{5}{12} = \frac{8}{12} = \frac23 \]
The calculation above can be visualised as follows:
If the denominators are different, you must take the following steps to make them the same:
For example, suppose we want to add
\[\frac25 - \frac14\]
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 \(20\).
\[\frac25 = \frac{2\times 4}{5\times 4} = \frac{8}{20} \]
\[\frac14 = \frac{1\times 5}{4\times5} = \frac{5}{20} \]
Now that the denominators are the same, the fractions can be added as before:
\[\frac25 - \frac14 = \frac{8}{20} - \frac{5}{20} = \frac{3}{20} \]
See the picture below for a visual explanation of what was done.
See the video below for another example of adding and subtracting fractions.
To multiply fractions, just remember the following rule:
Top by top, bottom by bottom
For example,
\[ \frac35 \times \frac27 = \frac{3 \times 2}{5\times 7} = \frac{6}{35} \]
To divide fractions, you just need to flip the dividing fraction and then multiply:
Flip and multiply
For example,
\[ \frac29 \div \frac34 = \frac29 \times \frac43 = \frac{2 \times 4}{9 \times 3} = \frac{8}{27} \]
Notice that the \(\dfrac34\) gets flipped to \(\dfrac43\) and multiplied instead.