What is a fraction?
A fraction is a number used to represent a part or several parts of something. For example, your class may be familiar with foods such as ½ a sandwich, ¼ of a pizza, or a whole cake (yum yum).
It’s essential to remember that there are two primary components of a fraction. The top number is called the numerator. It is used to show how many parts of something there are. The bottom number is called the denominator. This part shows how many features something has to be divided into. Going back to the kitchen to find an example, ½ of a cake would mean one piece of a whole cake has been split in two.
How to Simplify Fractions
Knowing how to simplify fractions need not be a challenge! First, you can simplify fractions by using a common factor that both parts of the fraction (the numerator and denominator) can divide into.
How to simplify fractions using a common factor
When it comes to fractions, you ideally want to have the simplest form of that fraction. A fraction can be simplified if the numerator and the denominator are divisible by the same number (known as a common factor). It makes it easier to solve calculations where you have to add, subtract, divide or multiply fractions together.
Let’s see how this works by taking a look at an example: Simplify 2/8
Let’s work through the process together to simplify 2/8. We can see that both numbers that make this fraction are even, meaning that two would be a common factor. When simplifying fractions, we must ensure that whatever we do to the top, we also do to the bottom. With this in mind, we can divide both fraction parts by 2.
2÷ 2 = 1
8÷ 2 = 4
And there you have your new simplified numerator and denominator, ready to embrace the simple life as 1/4.
How to simplify fractions using a Highest Common Factor
As with most maths methods, there’s more than one. You can also simplify your fractions by finding the highest common factor. To try our previous example of 2/8 this way, you start by placing both the numerator and denominator over the numerator to see if we can create whole numbers:
- As you can see, 2/2 gives us 1, and 8/2 gives us 4.
- Now that we’ve divided these down as far they can go, we can put the fraction back together.
- As a result, our final answer is that 1/4 is the simplest form of this fraction.
Sometimes, we can’t simplify the numerator down to 1, as not all numerators and denominators can divide perfectly. In this case, we must find the highest number that both can divide into. It is known as the Highest Common Factor (HCF). Let’s try an example of this together:
- We’re going to see if we can simplify the fraction 12/16.
- As a start, we know that this fraction can be simplified by dividing it down by 2, seeing as both the numerator and denominator are even numbers.
- By doing this, we come to 6/8. But we still have two even numbers, so this isn’t the simplest form. So let’s divide it by two again.
- Finally, we’ve come to 3/4. 3 and 4 don’t have any common factors, so 3/4 is our final answer.
That was a long-winded way to find our fraction. We could have used the highest common factor to save ourselves a step. So let’s try again, but with HCFs in mind.
- We’re going to see if we can simplify the fraction 12/16 (déjà vu, anyone?)
- We will identify the common factors of both numbers within our fraction. For example, 12 and 16 are multiples of 2 and 4. 4 is the most common factor.
- Divide the numerator and the denominator by 4 to get your simplified fraction.3/4 is our final answer.
How to Simplify Improper Fractions
Improper fractions, also known as top-heavy fractions, are fractions in which the numerator (the number above the line) is more significant than the denominator (the number beneath the line).
For example, 12/10, 15/3, and 4/2 are all examples of improper fractions.
We generally follow the same steps as common fractions when simplifying improper fractions. An all-essential rule (so important we’re saying it twice) is that whatever you do to the top, you must do to the bottom!
- Identify a common factor (a number that multiplies with another to make your given number) of both the numerator and denominator.
- Divide the numerator and the denominator by that same factor.
- You have a simplified fraction. Congratulations. But wait, could your fraction be simplified further?
Ideally, you would use the highest common factor (HCF) to simplify your fraction. But when working with more significant numbers or if you lack confidence with certain times tables, that won’t always be possible. So instead, you can repeat the above steps until your fraction either contains two prime numbers or the numerator and denominator no longer share a factor higher than 1.
How to simplify improper fractions examples:
- Let’s take 14/7 as our first example.
14 and 7 are both in the 7 times tables, so 7 is a factor of both. We divide both numbers by 7 to get our new simplified fraction: 2/1. It cannot be simplified further.
- Now let’s try 16/8.
We may quickly spot that as both numbers are even, 2 would be a possible factor to divide both numbers by. Let’s try that. That gives us 8/4. Yes, it’s more straightforward, but it isn’t in its simplest form. That’s because we didn’t use the highest common factor to divide by. We must simplify it once (if dividing by 4) or even twice more (if dividing by 2) to find this fraction in its simplest form.
Let’s go back to the start and see if we can find the HCF to save us some time. Both numbers can be found in the 2, 4, and 8 times tables. The highest number there is 8 so let’s go with that. Dividing the numerator and denominator by 8 gives us 2/1, a fraction in its simplest form, in just 1 step.
- For our last example, the answer isn’t going to be 2/1! We have 35/12.
They aren’t even numbers, so they can’t be divided by 2. Moreover, they don’t share any common factors other than 1; how unhelpful! In this case, 35/12 is already in its simplest form.
How to Simplify Algebraic Fractions
Teaching simple fractions and how to simplify them can be challenging on their own. When algebra is thrown into the mix, it’s natural for your class to feel nervous. However, simplifying algebraic fractions isn’t too challenging once it’s broken down, we promise.
Let’s work through an example to see how it’s done:
²ˣ⁄₄ₓ
It may look very complex, but simplifying algebraic fractions shouldn’t be too terrifying if you know how to simplify fractions. To simplify this fraction, we must find a number that can divide into 2 and 4. Thankfully the numerator of this fraction is 2, which makes our job much easier. It leads to:
²ˣ⁄₄ₓ ÷ 2 = ˣ⁄₂ₓ
That already looks much better, but we can simplify it even further. Much like we usually look for a number, we can divide the numerator and denominator by; we can do the same with x. When we divide our example by x into both sides, we get:
ˣ⁄₂ₓ ÷ x = ½
It may take some getting used to, but it does make sense! However, more complicated fractions can’t be solved in so few steps. This one looks even more daunting, but we’ll show you just how you manage it:
²⁽ˣ⁺¹⁾⁄₄₍ₓ₊₁₎
You may be scratching your head, wondering where you start. You’ll get this without bothering if you know the basics of simplifying algebraic fractions. Your immediate thought may be to divide both sides by 2 as we’ve done before. Instead, however, we can divide both sides by x+1, which gives us the following:
²⁽ˣ⁺¹⁾⁄₄₍ₓ₊₁₎÷ x+1 = ²⁄₄
See, we knew you had it in you! Now, all there is to do is simplify this fraction even further by dividing the numerator and denominator by 2 to get:
²⁄₄ ÷ 2 = ½
You may find yourself simplifying algebraic fractions that have numerous expressions and brackets on them, such as:
² ⁽ˣ⁺¹⁾ ⁽ˣ⁺²⁾⁄₄ ₍ₓ₊₁₎ ₍ₓ₊₂₎
However, as long as these expressions are the same on the top and the bottom, the process is the same as in our earlier example.
