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How to Divide Fractions in Five Easy Steps

Updated on April 3, 2016

Dividing By Fractions is Easy

I think many people are just scared by the thought of dividing by fractions. The logic goes something like this; when we learn arithmetic we start with addition and subtraction and then go on to multiplication which is much harder, and then the final step is division which is by far the hardest. Adding and subtracting fractions was a new challenge, multiplying fractions was not too bad but now you're asking us to divide by fractions! Aaarrgh! Wait! There's no need to run screaming from the room. If you can do simple multiplication you can divide by fractions.


To divide by a fraction you do not have to actually divide! You just flip the dividing the fraction. You move the top number (the numerator) to the bottom and the bottom number (the denominator) to the top and then just multiply. At first this may not make sense. You could just learn the method but some people like to understand why a method works. I'm sure if you understand the logic behind it you're unlikely to ever forget how to divide by fractions.

Why You Need to Flip a Fraction and then Multiply in order to Divide by it.

This is a neat mathematical proof and shows the power and appeal of mathematics. It shows how you can use reasoning and a step by step approach to arrive at a proof to simplify seemingly complex problems. It shows how mathematicians think. Confession; previously I'd just known the rule, I didn't know why it worked, I had to look it up. I found the proof satisfying. I guess that it always bugged me that I knew how without knowing why.

Here's the proof:-


Surely This Article Should Say "2 Easy Steps - Flip and Multiply"?

Yes and no. Yes dividing by fractions is really straightforward, as we've seen above, you just flip the dividing fraction and multiply. That's the guts of it. However most mathematics tests introduce a twist or two. When you are asked to divide fractions you are quite likely to also need to know about mixed numbers, improper fractions and the need to simplify. So it's 5 steps, and not 2 steps, in order to cover all eventualities.

Here are the five easy steps using the example of:-


Step 1 Change any Mixed Numbers to Improper Fractions

A mixed number is a combination of a whole number and a fraction and an improper fraction is where the number at the top of a fraction (the numerator) is greater than the number at the bottom of a fraction (the denominator). Therefore using our example we have:

Step 2 "Flip" the dividing number and convert to a multiplication

As discussed above (and as shown in the proof), you "flip" the dividing fraction so that numerator becomes the denominator and the denominator becomes the numerator. You then change the divide sign into a multiply sign:-

Step 3 Simplify (if possible)

It's not always possible but where you can you should simplify before you move to the next step. It makes everything, how can I put this? I know, simpler!

Step 4 Multiply The Numerators and the Denominators

Just multiply across as shown in the example below:-

Step 5 If Necessary Convert An Improper Fraction to a Mixed Number

This is not always necessary, but if your answer is an improper fraction (i.e. the top number or numerator is greater then the bottom number or denominator) then you need to convert it to a mixed number as shown below:-

Summary

Here are the 5 steps again:-

How Easy Is It to Divide By Fractions?

Is Dividing By Fractions Easier Than You First Thought?

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