How to Convert Between a Mixed Number and an Improper Fraction

Convert Between Mixed Number and Improper Fraction

An extremely common task in math is converting between what we call a mixed number and an improper fraction. This is typically even taught in the lower grades. Rather than to remember rules and formulas, it is far more important to just understand what is taking place. Then you will always be able to get the correct answer, and you won't have to doubt yourself. This article shows you the steps.

Instructions

1. First of all, a mixed number is a number that contains a whole number portion and a fraction portion. See the example at left. Many students don't realize that there is really an imaginary plus sign between the whole number and the fraction. If you say that you ate "two and a half" slices of pizza, what you really mean is that you ate two whole slices, plus half of a third slice.

2. An improper fraction is a fraction whose numerator (top number) is larger than its denominator (bottom number). An example would be 31/8.

   The format that we choose for our answer depends upon the context of the problem, as well as any specific instructions given. The steps below show you how to convert from one format to the other.

3. 

   To convert from a mixed number such as 2 3/4 to an improper fraction without using a shortcut, we must first remember that we are dealing with addition, as mentioned. We first convert the whole number into a fraction by putting it over a denominator of 1, which doesn't change its value. We must then multiply top and bottom by the same number, so that the fraction will have a denominator that matches the second fraction. Then we can add the fractions. See the picture for these steps.

4. 

   If that step seemed like a mouthful, you'll be glad to know that there is a shortcut. Rather than memorize the shortcut, it is far more important to just understand it. If you think about it, our whole number simply got multiplied by the denominator of the original fraction. We then added that result to the numerator of the original fraction, and put that sum over the original denominator. That is precisely what the shortcut tells us to do. There was never really the need to go through the motions of putting the whole number over 1, and all the other steps that we did.

5. 

   For practice, let's convert 7 2/5 to a mixed number using the shortcut. 7 times 5 is 35, plus 2 gives us 37, and we put that over the original denominator of 5, giving us an improper fraction of 37/5.

6. 

   Now let's convert from an improper fraction back to a mixed number. This is actually easier. The first thing we do is determine how many times the denominator divides into the numerator, and that becomes our whole number. Remember that a fraction is really a division problem. After dividing, our remainder becomes the numerator of the fractional part, and we keep the same denominator.

   For example, to convert 17/5 into a mixed number, note that 17 divided by 5 is 3 remainder 2. The 3 becomes our whole number, and we put the remainder of 2 over 5, giving us 3 2/5 as our mixed number. We essentially worked backwards. To check, convert the mixed number back to an improper fraction to see if you get what you started with.

7. One more exercise for practice. To convert 19/3 into a mixed number, divide 19 by 3, giving us 6R1. That gives us a mixed number of 6 1/3. Note that if you ever have an improper fraction such as 20/4 which yields no remainder, then there is no fractional part to your mixed number. It simply converts into 5.

8. That is all there is to it. Students should make sure that they are fully comfortable with these procedures, and more importantly, that they understand why they work as they do.

Resources

• Math with Larry (Free Online Math Help)