Dividing fractions seems like such a challenging task. There are so many numbers flying about that you may think it can’t be done without a calculator.
Fortunately, this couldn’t be further from the truth. If you want to learn how to divide fractions, you just have to follow a simple three-step process that will work with almost all fraction problems.
It’s the key to unlocking the mystery that is dividing fractions. In fact, after reading this article, you may find that dividing fractions is incredibly easy.
But, there are a few instances when the three-step process won’t work, which we’ll also cover in this article. Let’s jump in.
Dividing fractions is a straightforward process that involves three easy steps. First, turn the second fraction upside down to create its reciprocal. The second step is to change the division sign into a multiplication sign. Thirdly, multiply the first fraction by the reciprocal of the second fraction to give you your final answer. Lastly, if applicable, simply the fraction to its simplest form. There you have it. You have now learned how to divide by a fraction – yes, it really is that simple.
This process is often referred to as ‘Keep it, change it, flip it’, where you keep the first fraction as it is, change the division sign to a multiplication sign, and flip the second fraction to create its reciprocal. You can then solve the problem by multiplying the fractions as you would normally. The order in which you flip the second fraction and change the division sign doesn’t matter, as long as these steps are done before you begin solving the problem.
You may be asking, why do we turn the fraction upside down? Well, it’s due to the relationship between multiplication and dividing.
A fraction tells us to multiply by the numerator (top number) and divide by the denominator (bottom number). When dividing by a fraction, we are inversing this; dividing by the numerator and multiplying by the bottom number.
To give you an example, dividing a number by 2/3 is the same as multiplying it by 3/2. Multiplying fractions is far easier than dividing them and will save you a lot of hassle, particularly when dealing with complex fractions. Therefore it makes much more sense to flip the second fraction over and create a multiplication problem. Let’s look at some examples to see how this works in practice.
Suppose we have to solve the question 1/3 ÷ 1/5. We will use the keep it, change it, and flip it process that we mentioned earlier.
- The first step is to keep the first fraction as it is.
- The second step involves changing the division sign into a multiplication sign.
- Finally, we can flip the numerator and denominator of the second equation to create 5/1.
This means that we have changed our original question. Instead of 1/3 ÷ 1/5, our new question now is as follows:
1/3 × 5/1
When we multiply fractions, we simply have to multiply the two fractions together; both numerators can be multiplied straight across, and both denominators can be multiplied straight across.
- This means that our numerator would be: 1 × 5 = 5
- Doing the same with the denominators, it would be: 3 × 1 = 3
The result is 5/3. We cannot simplify this fraction any further, which means we have reached our final answer. Therefore, we can conclude that the answer to our original problem is as follows:
1/3 ÷ 1/5 = 5/3
An improper fraction is when the value of the numerator is greater than the denominator. This can be pretty confusing, especially when you’re used to seeing fractions as 1/x. But luckily for us, the same three-step process of keep it, change it, flip it still applies to improper fractions. Let’s take a look at this in practice.
Dividing improper fractions example
Suppose we have to solve the question 1/2 ÷ 6/4.
- The first step is to keep the first fraction as it is.
- The second step involves changing the division sign into a multiplication sign.
- Finally, we can flip the numerator and denominator of the second equation to create 4/6.
This means that we have changed our original question. Instead of 1/2 ÷ 6/4, our new question now is as follows:
1/2 × 4/6
Again, both numerators can be multiplied straight across, and both denominators can be multiplied straight across.
- This means that our numerator would be: 1 × 4 = 4
- Doing the same with the denominators, it would be: 2 × 6 = 12
The result is 4/12. Although it can be tempting to think this is the final answer, we’re not quite done yet. There is still one step remaining, which is to simplify the answer. We know that the number 2 goes into 4 and 12, but it’s not the greatest common factor. The greatest common factor is 4 since both 4 and 12 can be divided by 4.
- 4 ÷ 4 = 1
- 12 ÷ 4 = 3
Therefore, we can conclude that the answer to our original problem is as follows:
1/2 ÷ 6/4 = 1/3
Before we explore how to divide mixed fractions, let’s quickly recap what mixed fractions are.
Mixed fractions mean that there is a whole number alongside the fraction. An example of this is 2½ – we have the whole number 2 alongside the fraction ½.
It can be tricky if this is your first time trying to divide mixed numbers. However, as with all division of fractions, it’s not as complicated as it looks. We can still follow the keep it, change it, and flip it process, but you just have to change the mixed fraction into an improper fraction first. Before we look at how to divide mixed fractions, let’s first explain how to change a mixed number into an improper one.
Using 2½ again, this would equate to 5/2. But why? Well, it follows a two-step process:
- Multiply the denominator by the whole number
- Add the numerator
Putting this into action, the denominator multiplied by the whole number equals 4 (2 × 2). Adding the numerator equals 5 (4 + 1). The denominator remains the same, which creates 5/2.
Dividing mixed fractions example
Suppose we have to solve the question 2/7 ÷ 3¼.
First, we must change the mixed fraction into an improper one.
- Multiply the denominator by the whole number (4 × 3 = 12)
- Add the numerator (12 + 1 = 13)
This creates the improper fraction 13/4, which we can then substitute into the original question:
2/7 ÷ 13/4
Now, we can move on to the keep it, change it, and flip it process.
- The first step is to keep the first fraction as it is.
- The second step involves changing the division sign into a multiplication sign.
- Finally, we can flip the numerator and denominator of the second equation to create 4/13.
This means that we have changed our original question again. Instead of 2/7 ÷ 3¼, our new question now is as follows:
2/7 × 4/13
As before, both numerators can be multiplied straight across, and both denominators can be multiplied straight across.
- This means that our numerator would be: 2 × 4 = 8
- Doing the same with the denominators, it would be: 7 × 13 = 91
The result is 8/91. We cannot simplify this fraction any further, which means we have reached our final answer. Therefore, we can conclude that the answer to our original problem is as follows:
2/7 ÷ 3¼ = 8/91
Dividing fractions with whole numbers is much easier than dividing with a mixed number. Before beginning the division process, you must convert the whole number into a fraction.
To do so, simply transform the whole number into a numerator, and place the value 1 as the denominator. For instance, 4 would change to 4/1, 7 would change to 7/1, and 16 would change to 16/1.
Once you have completed this step, you can move on to the division process as you’ve done in previous examples.
Dividing fractions with whole numbers example
Suppose we have to solve the question 4/9 ÷ 3.
First, we must transform the whole number into a fraction. In this case, 3 would change to 3/1, which we can then substitute into the original question:
4/9 ÷ 3/1
Now, we can move on to the keep it, change it, and flip it process.
- The first step is to keep the first fraction as it is.
- The second step involves changing the division sign into a multiplication sign.
- Finally, we can flip the numerator and denominator of the second equation to create 1/3.
This means that we have changed our original question again. Instead of 4/9 ÷ 3, our new question now is as follows:
4/9 × 1/3
Both numerators can be multiplied straight across, and both denominators can be multiplied straight across.
- This means that our numerator would be: 4 × 1 = 4
- Doing the same with the denominators, it would be: 9 × 3 = 27
The result is 4/27. We cannot simplify this fraction any further, which means we have reached our final answer. Therefore, we can conclude that the answer to our original problem is as follows:
4/9 ÷ 3 = 4/27
So far, we’ve used the keep it, change it, and flip it process to divide fractions. The only time that it doesn’t work is when both denominators are the same. Since the denominators will cancel each other out, you don’t need to find the reciprocal of the second fraction and multiply. You can simply divide the numerators and denominators straight across.
Dividing fractions with the same denominator example
Suppose we have to solve the question 6/13 ÷ 3/13
- This means that our numerator would be: 6 ÷ 3 = 2
- This means that our denominator would be: 13 ÷ 13 = 1
The result is 2/1, which we can simplify to 2. Therefore, we can conclude that the answer to our original problem is as follows:
6/13 ÷ 3/13 = 2
As you can see, dividing fractions is far easier than it looks. Remember the keep it, change it, flip it process, and you will be able to solve almost all division of fractions examples. You do have to bear in mind that all fractions that are not in a proper or improper format will have to be changed first before you can move ahead with the division. The only time this process does not work is when dividing fractions with the same denominator. For these, you can go ahead and divide straight across.
