Math help from the Learning Centre

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:

• \(\displaystyle 4\tfrac23 = \frac{4\times3 + 2}{3} = \frac{12+2}{3} = \frac{14}{2}\)
• \(\displaystyle 5\tfrac14 = \frac{5\times4 + 1}{4} = \frac{20+1}{4} = \frac{21}{4}\)
• \(\displaystyle 2\tfrac37 = \frac{2\times7 + 3}{7} = \frac{14+3}{7} = \frac{17}{7}\)

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:

1. Make sure the denominators are the same.
2. Add the numerators and put that answer over the denominator.
3. Simplify the fraction if necessary.

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:

1. Find the least common multiple of the denominators (see Prime Factorisation and Least Common Multiple)
2. Multiply each fraction above and below so that the each denominator becomes the least common multiple.
3. Add the fractions as shown above.

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.