How to Find Equivalent Fractions
Equivalency of 1/2 and 2/4
What are equivalent fractions?
Equivalent fractions are two or more fractions which contain different numbers from each other but are actually the same amount. For example, we can see in the image above that 1/2 and 2/4 are the same amount, even though they are written differently. In this guide, we will look at why equivalent fractions are useful and how to find them.
Why are equivalent fractions useful?
Equivalent fractions come in really useful in several scenarios. One is when trying to write a fraction in a simpler form, e.g. does 8/16 or 1/2 make more sense? They are both exactly the same fraction, but we are more used to using 1/2, and it carries more meaning in everyday use.
Another use is when comparing the size of different fractions. It may not be immediately obvious which is the largest out of 4/5 and 17/20, but equivalent fractions can be used to compare them. We will look at this particular example later.
The third use which we will look at here is when we add or subtract fractions. If we try to add 1/2 and 1/3 together, we will find this extremely difficult without first using equivalent fractions.
Another use is when comparing the size of different fractions. It may not be immediately obvious which is the largest out of 4/5 and 17/20, but equivalent fractions can be used to compare them. We will look at this particular example later.
The third use which we will look at here is when we add or subtract fractions. If we try to add 1/2 and 1/3 together, we will find this extremely difficult without first using equivalent fractions.
How to find equivalent fractions
Let's start with the fraction 2/3. Having 2/3 of something just means that you have split it into 3 equal parts (this is the number on the bottom of the fraction, otherwise known as the denominator), and you have taken 2 of these parts (this is the number on the top of the fraction, also known as the numerator).
To find an equivalent fraction to 2/3, let's look at the image below.
To find an equivalent fraction to 2/3, let's look at the image below.
Converting 2/3 into 4/6
Converting 2/3 into 4/6
The left-hand rectangle has been cut into thirds and two coloured in to represent 2/3. If we cut each of these thirds into two equal pieces, as shown in the right-hand rectangle, not only do we have twice as many pieces in total, but we also have twice as many pieces coloured in. This means we have doubled both the denominator and numerator to get 2/3 = 2×2 / 3×2 = 4/6.
We can see from the image that the amount of red hasn't changed; they are both definitely the same proportion of the rectangle. We can say, therefore, that 2/3 and 4/6 are equivalent to each other.
We can do this with other numbers as well. Have a look at the image below.
We can see from the image that the amount of red hasn't changed; they are both definitely the same proportion of the rectangle. We can say, therefore, that 2/3 and 4/6 are equivalent to each other.
We can do this with other numbers as well. Have a look at the image below.
Both rectangles show the same amount shaded in so 2/3 does equal 10/15. They are equivalent fractions.
In this example we have split the original rectangle up five times as much. We can see that we have five times as many blocks in total and five times as many shaded blocks, hence we have multiplied both numerator and denominator by 5 to get 2/3 = 2×5 / 3×5 = 10/15. Again we can confirm this is true by looking at the diagram. Both rectangles show the same amount shaded in so 2/3 does equal 10/15. They are equivalent fractions.
The quick way to find equivalent fractions
The diagrams we have looked at so far are great for illustrating how equivalent fractions work, but we don't need them for the actual calculations. All we need to do is multiply both the numerator and denominator by the same number to convert one fraction into an equivalent one.
Example 1: Find three fractions equivalent to 3/4
For each equivalent fraction, we just need to pick a number to multiply the numerator and denominator by. 2 is an easy one to start with.
3×2 / 4×2 = 6/8
Let's use 5 and 8 to find two more equivalent fractions.
3×5 / 4×5 = 15/20
3×8 / 4×8 = 24/32
So 3/4, 6/8, 15/20 and 24/32 are all equivalent.
3×2 / 4×2 = 6/8
Let's use 5 and 8 to find two more equivalent fractions.
3×5 / 4×5 = 15/20
3×8 / 4×8 = 24/32
So 3/4, 6/8, 15/20 and 24/32 are all equivalent.
Example 2: Convert 2/5 into an equivalent fraction over 20
To solve this, we need to work out what multiplies our current denominator of 5 to give the required denominator of 20.
20÷5 = 4
As the denominator is being multiplied by 4, we must do the same to the numerator.
2×4 = 8
Therefore 2/5 = 8/20.
20÷5 = 4
As the denominator is being multiplied by 4, we must do the same to the numerator.
2×4 = 8
Therefore 2/5 = 8/20.
Using equivalency to simplify fractions
We have seen how multiplying top and bottom by the same number produces equivalent fractions. By reversing the process, we can also see that dividing the top and bottom by the same will also produce equivalent fractions.
If we can divide top and bottom until we get to a point where there is no longer a shared factor to divide by (other than 1), then we have simplified the fraction.
If we can divide top and bottom until we get to a point where there is no longer a shared factor to divide by (other than 1), then we have simplified the fraction.
Example 3: Simplify 2/12
2 and 12 are both multiples of 2, so we can divide the numerator and denominator by this to simplify the fraction.
2/12 = 2÷2 / 12÷2 = 1/6.
2/12 = 2÷2 / 12÷2 = 1/6.
Example 4: Simplify 8/20
Sometimes, we may need to divide more than once. Let's simplify 8/20. Again, both numbers are even, so let's start by dividing them by 2.
8/20 = 8÷2 / 20÷2 = 4/10
4 and 10 are also both even, so we can halve again.
4/10 = 4÷2 / 10÷2 = 2/5.
2 and 5 do not share any factors other than 1, and so we have fully simplified them.
Therefore 8/20 = 2/5.
Of course, if we had noticed that our initial numbers of 8 and 20 are both multiples of 4, then we could have divided by this to get to our final answer in one go. It doesn't matter how many steps you take; you will still arrive at the same final simplified answer.
8/20 = 8÷2 / 20÷2 = 4/10
4 and 10 are also both even, so we can halve again.
4/10 = 4÷2 / 10÷2 = 2/5.
2 and 5 do not share any factors other than 1, and so we have fully simplified them.
Therefore 8/20 = 2/5.
Of course, if we had noticed that our initial numbers of 8 and 20 are both multiples of 4, then we could have divided by this to get to our final answer in one go. It doesn't matter how many steps you take; you will still arrive at the same final simplified answer.
Using equivalent fractions to compare fractions
At the start of this guide, we mentioned comparing 4/5 and 17/20. Which is bigger? By using equivalent fractions to get the denominators the same, we can solve this quickly.
5 goes into 20 4 times, so let's multiply the top and bottom of 4/5 by 4.
4/5 = 4×4 / 5×4 = 16/20.
It is clear that 17/20 > 16/20; hence, 17/20 is the bigger of our two fractions.
5 goes into 20 4 times, so let's multiply the top and bottom of 4/5 by 4.
4/5 = 4×4 / 5×4 = 16/20.
It is clear that 17/20 > 16/20; hence, 17/20 is the bigger of our two fractions.
Using equivalent fractions to add and subtract fractions
It only becomes possible to manually add or subtract fractions when we convert them into fractions with matching denominators.
For example, what is 1/4 + 3/8?
We can use what we have already learned to convert 1/4 into 2/8. We now have:
1/4 + 3/8 = 2/8 + 3/8 = 5/8.
Sometimes, we might need to convert both fractions in order to get the denominators to match.
For example, what is 1/4 + 3/8?
We can use what we have already learned to convert 1/4 into 2/8. We now have:
1/4 + 3/8 = 2/8 + 3/8 = 5/8.
Sometimes, we might need to convert both fractions in order to get the denominators to match.
Example 5: What is 1/4 + 3/8?
For example what is 1/4 + 3/8?
We can use what we have already learned to convert 1/4 into 2/8. We now have:
1/4 + 3/8 = 2/8 + 3/8 = 5/8.
We can use what we have already learned to convert 1/4 into 2/8. We now have:
1/4 + 3/8 = 2/8 + 3/8 = 5/8.
Example 6: What is 3/5 + 1/6?
Sometimes, we might need to convert both fractions in order to get the denominators to match. Usually, the quickest way to do this is to look for the lowest common multiple of the two denominators.
LCM(5,6) = 30, so we need to convert both fractions into thirtieths.
3/5 + 1/6 = 18/30 + 5/30 = 23/30.
LCM(5,6) = 30, so we need to convert both fractions into thirtieths.
3/5 + 1/6 = 18/30 + 5/30 = 23/30.
Equivalent fractions recap
- Equivalent fractions are fractions which are equal to each other.
- We can find equivalent fractions by multiplying (or dividing) both numerator and denominator by the same number.
- To compare or add/subtract fractions, first use equivalent fractions to ensure your fractions have the same denominator.
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