Improper and Mixed Fractions
Introduction to Improper and Mixed Fractions
The word fraction always conveys the idea of a smaller part of the larger whole. If you have completed studying 4 chapters out of the 5 chapters in the syllabus for your next Math examination, you have got \({4\over 5}\)ths of the job done. If 7 wickets have fallen in a cricket inning, \({7\over 10}\)ths of the team is out. In these fractions, the numerator is smaller than the denominator. However, often, quantities greater than one can also be fractional. Have you ever considered that possibility? Consider a pastry shop bakes a cake and serves it as pastries. One cake generates 8 pastries and you go to the shop and buy 10 pastries. In this case, you have bought \({10\over 8}\) or \({5\over 4}\) of the cake!
The Big Idea: Improper and Mixed Fractions
A Simple Idea: An Improper Fraction has two parts
To get improper fractions into a usable form, you need to just remember that an improper fraction is nothing but the sum of an integer and a proper fraction (a proper fraction is one which has the numerator smaller than the denominator). Let us consider a very simple fraction to start with –\({5 \over 4}\). If you look at this fraction, you will realize that the numerator can be divided by the denominator. But the division is not perfect, and you get a remainder. In \({5 \over 4}\), the division of 5 by 4 gives an integer 1, with a remainder 1. This becomes clearer when you write;
Mixed Fractions
When you converted \({5 \over 4}\) into \(\begin {align}1+\frac14\end {align}\), the new number you got was a mixed number or mixed fraction. It was the sum of an integer and a proper fraction. When you are given two mixed fractions and have to use an operator \({(+,\, ,\text{ x, } /)}\) between them, it is not always necessary or convenient to convert them to mixed numbers before carrying out the operation. But after you have solved a problem, and the final answer is an improper fraction, then it is suggested that you convert it to a mixed number.
So, you need to know how to convert an improper fraction into a mixed number, and also how to convert a mixed fraction into an improper fraction.
How to Convert a Mixed Fraction into an Improper Fraction
Let us work with the same mixed fraction you got earlier, \(\begin {align}1+\frac14\end {align}\), when you used the improper fraction \({5\over 4}\). These are the steps you need to follow in the array method:

Convert the integer into a fraction with the same denominator as the fraction. In this case, the integer is 1, and the denominator of the fraction \({1\over 4}\) is 4.

1 can be considered as \({1\over 1}\) (any integer written in fraction form is the integer divided by 1).

To get denominator 4, multiply both numerator and denominator by 4.

The integer 1 will therefore become \({4\over 4}\).

To add this fraction \({4\over 4}\) to \({1\over 4}\), you just need to add the numerators and divide the sum by the denominator common to both fractions.

So, you write it as \({(1\,+\,4) \over 4}\), and you get \({5\over 4}\). This is the improper fraction you were looking for.
We have shown you how to convert the mixed fraction \(\begin {align}5+\frac18\end {align}\) in this example.
\(\begin{align} & 5 = 5 \times 1 = 5 \times \left( {\frac{8}{8}} \right) \\& = \frac{{\left( {5 \times 8} \right)}}{8} = \frac{{40}}{8}\\\\ & 5 + \frac{1}{8} = \frac{{40}}{8} + \frac{1}{8} \\ & = \frac{{41}}{8} \end{align}\)
Have a look at this simple worksheet in order to cement your concepts regarding mixed and improper fractions:
How to Add and Subtract Improper Fractions
When you are given two mixed fractions, say \({5\over 4}\) and \({41\over 8}\), you just go ahead and add/subtract them as you would two proper fractions. You take the LCM of the two denominators, and make that LCM the denominator of the solution, and multiply the denominators accordingly.
\(\begin{align} & \frac{41}{8}\frac{5}{4} \\ & =\frac{1}{8}\times \left( \left( 41\times 1 \right)\left( 5\times 2 \right) \right) \\ & =\frac{1}{8}\times \left( 4110 \right) \\ & =\frac{31}{8} \\ \end{align}\)
After that you simply convert the answer \({31\over 8}\) into a mixed number as explained above. So, your final answer is \(\begin {align}3+\frac78\end {align}\)
Common mistakes or misconceptions
 Children often don’t see a mixed fraction and it’s improper form as equal numbers. Because mixed fractions have a whole number part, it is assumed to be larger.
 Children often can’t plot improper fractions and mixed fractions on the number line.
This happens because they have seen fractions only as shapes. They may understand fractions as a part of the whole but don’t see them as numbers that can be plotted on the number line. Given that improper fractions are larger than 1, there is an additional challenge in plotting these numbers.
Tips and Tricks
 Tip: Using language that students can relate to makes the understanding of mixed and improper fractions easier. For example, describe \(\begin{align}\frac{7}{4}\end{align}\) as having 7 pieces from a cake shop that cuts their cakes only in 4 pieces each. This way students will make the connection that to have 7 pieces, that must have a full cake and 3 more pieces.
 Tip: Give a few examples of improper and mixed fractions and ask students to plot them on a number line. Even doing this for 56 examples conveys that idea that mixed and improper fractions are numbers that are greater than 1 and can be plotted on the number line.