Study Guide

Converting Between Mixed Numbers & Improper Fractions

Converting Between Mixed Numbers & Improper Fractions

Sometimes we'll be asked to convert mixed numbers into improper fractions, and vice versa. Once we get the hang of it, we'll be doing it in our sleep.

Let's start by looking at mixed numbers and improper fractions visually.

This is one way to draw the mixed number 2 3/4: it's 2 whole circles plus another of a circle.

diagram of 2 3/4

If we were to divide each whole circle into fourths, we could count the number of blue fourths: 4 fourths in each whole circle, and 3 fourths in the last circle. That makes 4 + 4 + 3 = 11 blue fourths in total. Shazam! 2 3/4 is equal to 11/4.

Here's what the improper fraction 10/3 looks like.

diagram of 10/3

Since it takes 3 pieces to make a whole, we can make 3 whole circles with 1 piece remaining. So 10/3 is really equal to .

We can also think of mixed numbers as sums of whole numbers and fractions. For example is really .

3 1/3 = 3 + 1/3

3/1 + 1/3

3/1 x 3/3 + 1/3

9/3 + 1/3

10/3

How to Change a Mixed Number into an Improper Fraction

The number 2 3/ 4 is called a mixed number because it has two parts: a whole number and a fraction mixed together. And it's coming to pieces. The question is, how many pieces? We need to know how many pieces are in the whole number and how many pieces are in the fraction.

The denominator tells us how many pieces are in a whole. Since there are 4 pieces in a whole and we have 2 wholes, there are 2 Ă— 4 or 8 pieces in the whole number part.

We've also got 3 pieces in the fraction part of 2 3/ 4 . We add those to the 8 pieces from the whole number part, and that makes 8 + 3 = 11 pieces in the mixed number. Finally, we place that big ol' number on top of the unchanged denominator and presto!

2 3/ 4 11/4

We could've also thought of 2 3/ 4 as the sum 2 + 3/4 . Then we convert the whole number 2 to a fraction and add 'em together.

2 3/4 = 2 + 3/4

2/1 +3/4

2/1 x 4/4 + 3/4

8/4 + 3/4

11/4

But all that was just to show you the ropes. We've got a quicker method to convert from mixed numbers to improper fractions, and we're not afraid to use it. Here are the steps and a few examples:

  1. Multiply the denominator by the whole number. 
  2. To that, add the numerator. 
  3. Place that number over the original denominator.

Example 1

changing mixed number to improper fraction

Example 2

changing mixed number to improper fraction

What if we've got a plain old whole number with no fraction but we want it to be a fraction? With a little fancy accessorizing and an up-do, we can totally make that whole number a fraction. Just give it a denominator of 1.

Let's say a 3 walks into a spa and asks to be a fraction. The chiropractor adjusts it and gives it a denominator of 1 so it now looks like this: . Since 3 divided by 1 is still 3, we can turn any whole number into an improper fraction by just sticking a 1 in the denominator.

How to Change an Improper Fraction into a Mixed Number

Now let's go backwards. To change an improper fraction into a mixed number, we need to know how many wholes we can make out of all the pieces in the numerator. Since the denominator tells us how many pieces it takes to make a whole, we can divide the numerator by the denominator. This tells us how many whole numbers we have. The remainder tells us how many pieces are left over and it becomes the numerator. The denominator doesn't change. Lucky. Here are the steps:

  1. Divide the denominator into the numerator. 
  2. The result will be a whole number with a remainder. The remainder becomes the numerator of the fraction part. The denominator remains the same.
Example 1

19/5

5 goes into 19 three times with a remainder of 4.

19/5

3 is the whole number. 4 is the numerator. 5 is the denominator, which stays the same.

Example 2

19/5

2 goes into 11 five times with a remainder of 1.

19/5

5 is the whole number. 1 is the numerator. The denominator stays the same.