Subtracting fractions, particularly those with different denominators, may seem complex on face value. However, once you get to grips with some basic rules, it’s actually pretty straightforward.
First we need to understand the makeup of a fraction. There are two parts here – the numerator and the denominator – separated by a dividing line.
The denominator is the bottom figure. This shows us how many equal parts one whole has been separated into. The numerator is the top figure, and represents how many parts of the whole are present.
To successfully subtract one fraction from another, we need to ensure that the whole we’re dealing with in both cases is divided into the same number of equal parts, i.e. that the denominators are the same.
A step-by-step guide to subtracting fractions with different denominators
There are a couple of quick tricks you can use when subtracting fractions with different denominators. Below we explain the traditional method, since this is the one you’ll need to apply when asked to show your workings in any official exam.
The same rules apply whether you’re working with proper fractions (where the numerator is smaller than the denominator) or improper fractions (where the numerator is greater than the denominator).
Step 1: Find the least common denominator
The least common denominator (LCD) is the lowest common multiple of the two denominators you’re working with.
For example, if we’re asked to subtract 2/3 from 2/6, we know that 6 is a multiple of 3, so 6 is our LCD.
Step 2: Find the equivalent fraction
Once you’ve found the LCD, you need to keep the value of your fractions the same. So in changing the denominator, you need to apply the same change to the numerator.
Using the example above, to replace the denominator with 6, we’re multiplying 3 by 2. To keep a fraction of equal value, we must also multiply 2 (the numerator) by 2. This gives us an equivalent fraction of 4/6.
Repeat this process for the second fraction too. In the case of our example, we already have a denominator of 6, so no change is necessary.
We now have a revised equation of 4/6 - 2/6.
Step 3: Subtract the new numerators
The next stage is straightforward: simply subtract the numerators you have in your new equation. In this case, 4 - 2 = 2.
Take the resulting figure and place it over the common denominator. This gives us 2/6.
Step 4: Simplify the answer if necessary
The last step is to simplify the fraction if possible. To do this, you’ll need to find the highest common factor shared by both parts of the fraction and divide them by it.
In the case of 2/6, the highest common factor is 2. Since 2 ÷ 2 = 1, and 6 ÷ 2 = 3, our simplified fraction is 1/3.
2/3 - 2/6 = 1/3
Below you’ll find two example questions asking you to subtract fractions with different denominators. The first is expressed as a standard equation, the second as a more complex word problem.
Example question 1
What is 5/6 minus 13/25?
Start by finding the LCD. In this example it’s not quite as straightforward, so you might find it useful to write down multiples of the highest current denominator to help you: 25, 50, 75, 100, 125, 150…
Now we can see the first multiple of 25 divisible by 6 is 150, so this is our LCD.
Next we need to find our equivalent fractions:
6 x 25 = 150, so we need to use the same value to multiply the numerator, giving 5 x 25 = 125. Our first equivalent fraction is therefore 125/150.
Now move onto the second:
25 x 6 = 150 and 13 x 6 = 78. Our second equivalent fraction is 78/150.
Subtract the numerators and place the result over the LCD: 125/150 - 78/150 = 47/150.
This fraction is already in its simplest form.
Answer: 5/6 minus 13/25 = 47/150
Example question 2
Emma is training for a marathon and has set herself a target distance to reach by the end of the week. On Monday, she runs 7/15ths of the distance. On Wednesday, she runs 4/5ths of the distance.
How much further did Emma run on Wednesday compared to Monday?
To solve this problem, we need to subtract how far Emma ran on Monday from how far she ran on Wednesday, so would write our equation as: 4/5 - 7/15.
Next we find our LCD, which we can see is 15. We find the equivalent of our first fraction by multiplying both parts by the same value: 5 x 3 = 15 and 4 x 3 = 12.
Our equivalent fraction is therefore 12/15.
As our denominator is already 15 in the second fraction, no change is required here.
We can now subtract our numerators (12 - 7 = 5) and place our answer over the denominator: 5/15.
Finally, we identify the highest common factor of 5 and 15 as 5, and divide both parts of the fraction to simplify: 5 ÷ 5 = 1 and 15 ÷ 5 = 3.
Answer: 4/5 minus 7/15 = 1/3