Improper fractions

Before learning about improper fractions, please make sure you've read the primer on fractions.

Now consider examples of those fractions we tend to use in everyday language - proper fractions. A proper fraction is one in which the numerator is less than the denominator. For instance, 1/4 of a cake, 1/2 a can of beans, 3/4 of a tank of gas.


So what about fractions in which the numerator is greater than (or equal to) the denominator? These are improper fractions and examples include

5/2            4/4            7/4            11/5            17/8

But what do these numbers mean? While it’s easy to imagine what a 1/4 (one fourth or one quarter) of something looks like, it’s more difficult to visualize say 5/2.

So let's take a closer look at 5/2

Remember the denominator is the total number of pieces which make up the 'whole' and the numerator is the number of pieces taken from the 'whole'.

In 5/2, the denominator is 2 and the numerator is 5 - but how can you take five pieces from a 'whole' made up of only two pieces?

The answer - there must be more than one 'whole'!

We can split 5/2 to show this is the case

To visualize this, consider pumpkin pies sliced into halves. So each ‘whole’ or pie is made up of 2 pieces (denominator 2 in the fraction 5/2). If there are two and a half ‘wholes’ or pies, we can take 5 pieces (numerator 5 in the fraction 5/2)

Now let's split 7/4

To visualize this, now consider pumpkin pies sliced into quarters. So each ‘whole’ or pie is made up of 4 pieces (denominator 4 in the fraction 7/4). If there are one and three quarter ‘wholes’ or pies, we can take 7 pieces (numerator 7 in the fraction 7/4)

Another method for splitting improper fractions

There is a method which doesn't involve breaking down the fraction into individual halves, thirds, quarters etc.

e.g.   11/5

First express as the sum of two fractions

11/5     =     _ / _     +     _ / _

- don't worry - we'll fill in the gaps!

Keep the denominator the same throughout

11/5     =     _ /5     +     _ /5

Next work out the highest multiple of the denominator (5) whose value is less than the numerator (11). Take this number (10) and the remainder (11 - 10 = 1) and fill in the final gaps

11/5     =     10/5     +     1/5


11/5     =     2     +     1/5

11/5     =     2  1/5   (two and one fifth)

In reality your working would look like this

11/5     =     10/5     +     1/5

11/5     =     2     +     1/5

11/5     =     2  1/5

i.e. you only need to put down enough information to show that you've followed the steps in the above method.


Now try yourself applying this method to 17/8

You should get the answer 2 1/8.

These are examples of converting to mixed numbers - you can find more on these types of numbers (also known as mixed fractions) and how to convert them back to improper fractions here


It's time to wrap things up

Can you see that 2 1/2 (two and a half) is easier to visualize than 5/2 and 1 3/4 (one and three quarters) is easier to visualize than 7/4?

For instance, you can imagine what 2 1/2 or 1 3/4 pumpkin pies look like - but maybe not 5/2 or 7/4 pumpkin pies!

Remember when you see an improper fraction there is more than one ‘whole’, the only exception being when the numerator equals the denominator (then you’re dealing with the whole of a ‘whole’, for example, 4/4 - imagine taking four pieces of pumpkin pie sliced into quarters!).

For more on fractions click:

Math fractions

Equivalent fractions

Simplifying fractions

Adding fractions

How to subtract fractions

Multiplying fractions

How to divide fractions

Mixed fractions (mixed numbers)

Comparing fractions

Ordering fractions

Fraction to decimal

Fraction to decimal chart

Decimal to fraction

Fraction to percent

Percent to fraction

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