Most people imagine that dividing by a fraction will be very complicated but it’s really quite straightforward. You just have to “flip” (a technical maths term) the dividing fraction by swapping the top number (the numerator) with the bottom number (the denominator). You then just have to multiply, which we’ve already seen is simple.

You don’t need to know this, but the following proof explains why flipping the dividing fraction and then multiplying works. It’s not difficult to follow and if you understand it, you’ll never forget how to divide by a fraction. It’s also, I think, more satisfying to know why something works rather than just know how to do it. The proof is also quite elegant (mathematicians sometimes say that proofs are “beautiful”, that’s a bit over the top). Anyway, here’s the proof:-

### Proof: To Divide by a Fraction, Just Flip it and Multiply

Just as saw when we were multiplying fractions, you will probably have to deal with mixed numbers and improper fractions. This video walks through the five simple steps that will cover all possibilites:-

The video looks at the following division:-

These are the five steps that are shown in the video and that can be used to answer divide by a fraction question:-

### Step 1:- Convert any Mixed Numbers to Improper Fractions

A mixed number is a combination of an integer (or whole number) and a fraction. Any mixed numbers need to be converted to an improper fraction, a fraction where the top number (numerator) is greater then the bottom number (denominator):-

### Step 2:- Take the dividing number and switch or flip its numerator and denominator.

The proof shown above shows why this works. Don’t forget to change the divide sign into the multiply sign:-

### Step 3:- If it’s possible simplify any of the fractions

Again, this is not always possible, but if you can simplify it makes the remaining steps easier:-

### Step 4:- Multiply

Now we’re ready to just multiply the numerators and the denominators:-

### Step 5:- If Required Change your answer from an Improper Fraction to a Mixed Number

Your initial answer may or may not be an improper fraction (numerator greater than the denominator) but if it is you should convert it to a mixed number. You may lose marks (or to be more positive gain extra marks if you do) if you don’t complete this final step:-

## Re-cap: How to Divide by a Fraction in Five Simple Steps

Here’s all five steps summarized:-