The basic mathematical procedures of addition and subtraction are also relevant to fractions. Because we take distinct steps in each situation, it is vital to grasp whether the fractions have the same or different denominators in order to add and subtract fractions easily.
Tips for adding and subtracting fractions
Fractions are a component of a larger whole. Fractions can be solved using fundamental arithmetic methods. As a result, fractions can be added, subtracted, multiplied, and divided. Let’s review fractions before going on to the addition of fractions.
Like vs Unlike Denominators?
There are two types of quiz questions you can encounter when adding and subtracting fractions and it is primarily based on whether the fractions have like or unlike denominators. Quite simply:
- Fractions with the same denominators are called like fractions. 5/15, 3/15, 17/15, and 31/15, for example.
- Fractions with different denominators are called unlike fractions. 2/7, 9/11, 3/13, and 39/46 are some examples.
Don’t be afraid to simplify!
Later in this article we will discuss how to find common denominators, this process may yield large numbers that may seem overwhelming. As such, don’t be afraid to simplify. Questions may also be simplified to help you calculate them easier. For example:
18/36 + 14/70 looks incredibly challenging, but this question is merely asking what 1/2 + 1/5 is!
If you were to add 18/36 to 14/70, the answer would be 882/1260! This answer can be simplified, and your teacher may want to see your initiative. 882/1260 can be simplified to 7/10 if you divide the numerator and denominator by 126 (greatest common factor). Moral of the story – don’t be scared to simplify if you can!
How do I Add and Subtract Fractions with Like Denominators
Adding Like Fractions
Take, for example, the two fractions: 1/4 and 2/4. Both fractions have the same denominators in this situation. As such, these are Like Fractions. Adding and subtracting like fractions is simple as you do not need to convert the denominators and can simply add and subtract the numerators of the given fractions.
As such:
1/4 + 2/4 = 3/4
Visually, imagine a circle or even a pie. A complete pie can be represented as 4/4 as it takes 4 equal parts to create a whole.
1/4 is shown below in yellow and 2/4 is shown below in blue.
Out of the 4 equal parts, 3 in total are shaded. As such, 1/4 + 2/4 is 3/4.
Subtracting Like Fractions
Quiz question: What is 2/4 – 1/4?
Let’s use our circle or pie model to subtract the fractions 2/4 and 1/4. In this model, we’ll represent 2/4 by the two blue components. To illustrate eliminating 1/4, we’ll shade off 1 portion from our blue parts of the model.
As you can see, this means only 1 blue part is left. Therefore, 2/4 – 1/4 = 1/4
Because of the like denominators, all you have to do is add or subtract the numerators. You can ignore the /4 of both fractions and simply calculate 2 – 1. Ignoring the denominator of like fractions can make it visually easier for students who may be confused by them when adding and subtracting.
How do you add and subtract fractions with different denominators?
Adding Unlike Fractions
This is where it becomes more complex. Let us take a look at adding 1/2 to 1/3. To visualize this let us return to our circle or pie. 1/2 is represented in green, and 1/3 is represented in red.
If we take the remaining white segment and cut the circle equally, we find that it cuts equally into sixths.
If we then count the colored segments, we discover that these regions account for 5/6 of the circle. Therefore 1/2 + 1/3 = 5/6. But how do we calculate this mathematically? You have to find a common denominator and transform the unlike fractions into like fractions.
Finding the Common Denominator
The simplest way to find a common denominator is to multiply the given denominators, in this case that’s 2 and 3.
2 x 3 = 6 and so we know that we are working in sixths from now on. However, we need to adjust the numerators. To convert 1/2 into sixths we multiplied it by 3, and therefore we must multiply the numerator by 3:
1 x 3 = 3 and so 1/2 becomes 3/6.
Likewise with converting 1/3 into sixths we must multiply the numerator and denominator by 2.
1/3 becomes 2/6.
Calculating 1/2 + 1/3 is way more confusing than calculating 3/6 + 2/6. All we must do now is add the numerators.
3/6 + 2/6 = 5/6, just as our pie showed above!
Subtracting Unlike Fractions
To subtract unlike fractions, we repeat the identical techniques we used to add unlike fractions:
- Calculate the fraction’s lowest common denominator.
- Convert the provided fractions to equivalent fractions.
- Subtract the numerators from the denominators.
What if you were asked this quiz question: What is 1/2 – 1/3?
This is how it would visually look, with 1/2 being shaded green and 1/3 being the etched red lines.
If we take the remaining segment and cut the pie into equal parts, the remaining green segment is 1/6 of the pie.
Mathematically, this is how you would calculate 1/2 – 1/3.
As stated before, the simplest way to find a common denominator is to multiply the given denominators, 2 and 3.
2 x 3 = 6 so we know that we are working in sixths from now on. We can now adjust the numerators.
To convert 1/2 into sixths we multiplied it by 3, and therefore we must multiply the numerator by 3:
1 x 3 = 3 and so 1/2 becomes 3/6.
Likewise with converting 1/3 into sixths we must multiply the numerator and denominator by 2.
1/3 becomes 2/6.
Calculating 3/6 – 2/6 is far easier, as quite simply all we need to do it 3 – 2 = 1.
As such, 1/2 – 1/3 = 1/6
Adding and Subtracting Mixed Numbers
A mixed number, also called mixed fraction, is a fraction that contains a whole number and a proper fraction. An example of a question may be 3½ + 2¾ =?
Method One – Improper Fractions
One way to calculate these questions is to convert the mixed number into an improper fraction. To do this, you multiply the whole number by the denominator and add the numerator. Here is how to do it:
For 3½: 3 x 2 = 6, then +1 = 7, which as an improper fraction is 7/2
For 2¾: 2 x 4 = 8, then +3 = 11, which as an improper fraction is 11/4
So we must find out what 7/2 + 11/4 is. To do so we have to find the common denominator.
We can convert 7/2 into 28/8 by multiplying the numerator and denominator by 4, and we can convert 11/4 into 22/8 by multiplying both by 2. So we are now calculating the far easier question of 28/8 + 22/8 which equals 50/8.
To convert this back into a mixed number, we see how many 8’s go into 50. 8 x 6 is 48 which leaves a remainder of 2. Therefore 50/8 can be expressed as 6 2/8, or 6¼.
Therefore, 3½ + 2¾ = 6¼.
Method 2 – Mixed Numbers
The whole numbers will never change when you convert fractions. 7/2 and 28/8 will always become 3½. What this means is that we can almost ignore the whole numbers in the initial calculation.
As such, all we need to do is find a common denominator between ½ and ¾. We can convert 1/2 into 2/4 to have a common denominator of 4.
2/4 + 3/4 would then equal 5/4. Converting this into a mixed number leaves us with 1 ¼ and it is here that we calculate the whole numbers.
3 + 2 + 1 = 6
Putting it all back together leads us to the answer of 6¼. Essentially this method is a ‘divide and conquer’ as you tackle to fraction first, then the whole numbers.
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