Mixed Numbers and Fraction

How well do you understand Mixed Numbers and Fractions? To understand the relationship between the two we will look at how they are used to represent the same value, and how to convert from one form to another. Like all concepts in math, it is important to visualize the situation in different way. To do this we will use both fraction tiles and number lines to help us understand the problems.

Mixed numbers and fractions

Let’s look at the number 4$\frac{2}{3}$. We call this a mixed number, but why?

Simply put, a mixed number is a mixture of a whole number and a fraction. The 4 represents a whole number and the $\frac{2}{3}$ is a fraction that lets us know that we are dealing with a number that is little more than the original 4. In fact, we could write this as 4 + $\frac{2}{3}$
Therefore, we are looking at a number between 4 and 5 on a number line.

But where exactly would it lie on a number line?

Looking at the denominator of our fraction, we see that the whole was divided into thirds. So, we need to divide the spaces in between each number into three equal parts. Now we need to mark where two thirds would be located.

Is there any reason we couldn’t have just written the number as a fraction? Not really.

Counting each space on the number line, we can see that 1 is the same as $\frac{3}{3}$, 2 is the same as $\frac{6}{3}$, 3 is the same as $\frac{9}{3}$, and 4 is the same as $\frac{12}{3}$. Now we need to add the additional $\frac{2}{3}$, giving us $\frac{14}{3}$.

So, the mixed number 4$\frac{2}{3}$ represents the same amount as $\frac{14}{3}$.

Some people call $\frac{14}{3}$ an improper fraction, but there is nothing improper about this fraction. Depending on the situation, sometimes it is more useful to think of this as a mixed number and sometimes it is more useful to think of it as fraction.

So, now that we know what a mixed number is:

How do we convert mixed numbers to fractions?

Let’s start with 2$\frac{3}{5}$.

We can think of this as 2 + $\frac{3}{5}$, or 1 + 1 + $\frac{3}{5}$. One is the same as $\frac{5}{5}$ so this would be $\frac{5}{5}$ + $\frac{5}{5}$ + $\frac{3}{5}$. For a total of $\frac{13}{5}$. We could change any mixed number to a fraction by repeated addition, but that may become a problem if the whole number was much larger.

Instead of repeated addition, we could think of the problem as $\frac{5}{5}$ two times.
2 + \tfrac{3}{5} &= \\
\left(1 * 2\right)+ \tfrac{3}{5} &= \\
\left(\tfrac{5}{5}*2\right)+ \tfrac{3}{5} &= \\ \left(\tfrac{5}{5} * \tfrac{2}{1}\right) + \tfrac{3}{5} &= \\ \tfrac{10}{5} + \tfrac{3}{5} &= \tfrac{13}{5}

Now let’s look at the reverse situation:

What if we had a fraction and we wanted to turn that into a mixed number?

$\frac{23}{7}$ = ?

We could using receptive adding until we got to $\frac{23}{7}$

\tfrac{7}{7} + \tfrac{7}{7} + \tfrac{7}{7} + \tfrac{2}{7} & = \tfrac{23}{7} \\
1 + 1 + 1 + \tfrac{2}{7} &= 3\tfrac{2}{7}
OR we can remember that a fraction also represents division.
$\frac{23}{7} = 23 \div 7 = 3\tfrac{2}{7}$