# Addition and Subtraction of Fractions

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## Introduction to Addition and Subtraction of Fractions

It’s not enough to know what fractions are and how to represent them in their standard forms. The beauty and power of mathematics shines through when we start to look at how the operations that we used to carry out for whole numbers, extend to fractions. So, let’s get started!

## The Big Idea: Addition and Subtraction of Fractions

### A Simple Idea: Adding and Subtracting with Similar Denominators

We will take this one step at a time. We already agreed that if the denominator is similar, then you can go ahead and simply add or subtract. Let us first see how that works. Imagine that there are two identical cakes. You slice both into 12 equal pieces. From the first cake, you take 8 pieces and from the second you take 3. Remember, fractions represent parts of a whole, now that both the cakes represent the same whole, you simply add the numerators to get the result, that is $${(8\, + \,3 \,= \,11) \over 12}$$. Just see the image below to get a clearer picture of what happened.

If you had to subtract, you would do the same and get the right answer $${5 \over 12}$$. So

\begin{align} & \frac{8}{12}-\frac{3}{12}=\frac{5}{12}\ \\\\ & \qquad \text{and} \\\\ & \frac{8}{12}+\frac{3}{12}=\frac{11}{12} \\ \end{align}

### The Next Step: Adding and Subtracting with Different Denominators

Did you notice that the two fractions in the problem above were actually $${2 \over 3}$$ and $${1 \over 4}$$? All we did was to change the denominator of both fractions and change the numerator accordingly. So,$${2 \over 3}$$ became $${8\over 12}$$ and $${1 \over 4}$$ became $${3\over 12}$$.

Why did we choose 12 as the denominator and not any other number? This brings us to the simple rule to follow in adding or subtracting fractions with different denominators.

## How is it important?

### The LCM Rule

A question that never gets asked, but is on quite a few students’ minds is that, “Why are we taking the LCM of the denominators, and not the HCF?” It is a very valid question and the fact that it isn’t answered in the traditional approach, highlights one of its largest shortcomings, the failure to present the foundational concepts of operations.

The reason is quite straightforward. When two fractions have different denominators, they are called unlike fractions. Just like in your day to day lives, you can’t compare unlike objects, comparing (adding or subtracting) unlike fractions is not possible. Therefore, to be able to compare two unlikes, you convert them to like fractions, or make them part of the same whole by taking the LCM of the denominators, as the denominator determines the total number of pieces of any given fraction.

So, the question that one naturally arrives at is how does one visualise this? Well, have a look for yourself!

 Step 1 Calculate the LCM of 4 and 3, which is 12. Step 2 Multiply the numerator of each fraction by the same number that you multiplied to denominator with to reach the LCM. Step 3 Add the numerators to arrive at the result. Step 4 Reduce the resulting to fraction to the simplest form, if needed.

Since we at Cuemath are all about showing multiple approaches to the same problem, here’s another way of visualising the same problem:

### How Do I Calculate LCM?

The lowest common multiple of two or more numbers is the smallest number which leaves no remainder when divided by those numbers. Here is how you do it in two simple steps:

• The prime factors of both (or all) numbers are first listed.

• Multiply the highest powers of all the prime factors which would go into both (or all) the numbers.

### After the LCM Is Calculated

After this it is a simple 4 step process. Let us say the two fractions to be added are $${3 \over 4}$$ and $${1 \over 3}$$, while the $${1 \over 3}$$ is also to be subtracted from $${3 \over 4}$$.

The LCM of 4 and 3 needs to be calculated as shown above. The LCM in this case is 12. So, the respective steps and results are given below.

 Step 1 Multiply both numerators by the same number which the denominator would have to be multiplied by to give the LCM. For $${3 \over 4}$$ this means that the numerator 3 needs to be multiplied by 3, because the denominator 4 should be multiplied by 3 to give 12. So, the numerators of $${3 \over 4}$$ and $${1 \over 3}$$ would be multiplied to give 9 and 4 respectively. Step 2 For addition, add 9 and 4 to give 13. For subtraction, subtract 4 from 9 to give 5. Step 3 The addition would give $${13 \over 12}$$, and the subtraction would give $${5 \over 12}$$. Step 4 $${13 \over 12}$$ could also be written as a mixed fraction.

## Common mistakes or misconceptions

• Children often tend to add the numerators and denominators separately while adding two fractions.
This happens because they see a fraction as two whole numbers instead of one number. So they perform operations just the way they do with whole numbers. \begin{align}\frac{1}{2}+\frac{3}{4}=\frac{1+3}{2+4}=\frac{4}{6}\end{align} because it is addition of a bunch of whole numbers.
• Children subtract the numerators and denominators separately while subtracting two fractions.
This happens because they see a fraction as two whole numbers and perform operations just the way they do with whole numbers. So for them \begin{align}\frac{3}{4}-\frac{1}{2}=\frac{3-1}{4-2}=\frac{2}{2}\end{align} because it is subtraction of a bunch of whole numbers.
• When adding or subtracting unlike fractions, the first step is to find equivalent fractions that will form a like pair. To do this, students are expected to find the LCM of the denominators. If the reason behind this step is not clear, students may make various mistakes including taking LCM of the numerator or HCF of the denominator and get stuck.

## Tips and Tricks

• When adding or subtracting unlike fractions, it is not necessary to find the LCM of the denominators. Any common multiple will do. So simply multiplying the two denominators gives us a common multiple. This may lead to larger looking numbers but is often a quicker way to add or subtract.
E.g. \begin{align}\frac{2}{9}+\frac{1}{12}\end{align}. To do this, first, the LCM of 9 and 12 needs to be found. Instead just multiply the first fraction with 12 and second with 9 to get \begin{align}\frac{24}{108}+\frac{9}{108}=\frac{33}{108}=\frac{11}{36}\end{align}.