## Rules and steps: Addition and subtraction of fractions

A **fraction** is a number expressed as a quotient. Many numbers are represented this way, and it means they are not whole integers. The **quotient** is comprised of an upper number, the **numerator**, over a lower number, the **denominator**.

As we can see below, the numerator sits atop a horizontal line with the denominator under it. In mathematics, this horizontal line is equivalent to a division symbol. Therefore, the fraction simply represents the top number (the numerator) divided by the bottom number (the denominator).

### Fractions with the same denominator

When adding and subtracting fractions, there is one important rule to keep in mind: *If the fractions to be added or subtracted have the same denominators, their numerators can be added or subtracted while keeping the denominator constant. *This rule is the basis for all addition and subtraction operations on fractions.

Let us illustrate this process in a little more detail. Suppose we want to calculate $\frac{5}{9}-\frac{4}{9}$. Since the denominators are identical, we can just perform the subtraction on the numerators while keeping the denominator constant (i.e., denominator = 9). In other words, we perform 5 - 4 = 1 on the numerators. The final answer is $\frac{1}{9.}$The steps can be written as:

$\frac{5}{9}-\frac{4}{9}=\frac{5-4}{9}=\frac{1}{9}$

### Fractions with different denominators

Before adding or subtracting fractions with different denominators, we must manipulate the fractions so that they have the same denominators. To do this, we will first need to find the Lowest Common Denominator (LCD).

The lowest common denominator (LCD) of two fractions is the lowest possible denominator that can be shared by each fraction, while keeping the numerical value of each fraction the same.

To find the lowest common denominator of two fractions, it is important to first make sure that each fraction is in its most simplified form. This means making sure all common factors in the numerator and denominator have been factored out. The next step is to consider or list out all multiples of each denominator. We can then pick the lowest multiple shared by both lists. This is the lowest common denominator! Let's take a closer look at this process in the following example.

Find the lowest common denominator of the fractions $\frac{32}{64}$ and $\frac{7}{10}.$

**1. ** Make sure each fraction is in its most simplified form.

The first fraction we can see is not in its most simplified form. We can simplify this fraction by taking out a factor of 32 on the top and bottom.

$\frac{32}{64}=\frac{32\left(1\right)}{32\left(2\right)}=\frac{1}{2}$

The second fraction is already in its most simplified form, as there are no factors that can be taken out of both the top and bottom. This leaves us with the fractions $\frac{1}{2}$ and $\frac{7}{10}.$

**2.** List out the multiples of each denominator.

*Multiples of 2 are: 2, 4, 6, 8, 10, 12, 14...*

*Multiples of 10 are: 10, 20, 30, 40, 50, 60, 70...*

We can see from each of these lists that 10 is the lowest multiple common to both denominators. Therefore, it is the **lowest common denominator**.

After finding the LCD, the following procedure can be used to add or subtract fractions with unlike denominators:

**Step 1:** Set up the denominator of each term with the Lowest Common Denominator (LCD).

**Step 2:** Set the numerator of each term to$(originalnumerator)\times \frac{LCMofdenominators}{Originaldenominator}$.

**Step 3:** Now that all the denominators are the same, you may add or subtract the terms in the numerator together to obtain your answer.

Add the fractions $\frac{32}{64}$and $\frac{7}{10}.$

From our previous example, we know that the lowest common multiple of $\frac{32}{64}$ and $\frac{7}{10}$ is 10.

**1. **Set up the denominator of each term with the Lowest Common Denominator (LCD).

$\frac{a}{10}$ and $\frac{b}{10}$

**2. **Set the numerator of each term to $(originalnumerator)\times \frac{LCMofdenominators}{Originaldenominator}.$

$a=32\times \frac{10}{64}=5$

As the original denominator of the second fraction was 10 already, its numerator does not need to be converted.

The fractions we are left with are $\frac{5}{10}$ and $\frac{7}{10}.$

**3. **Now that all the denominators are the same, you may add the terms in the numerator together to obtain the answer.

$\frac{5}{10}+\frac{7}{10}=\frac{5+7}{10}\phantom{\rule{0ex}{0ex}}=\frac{12}{10}\phantom{\rule{0ex}{0ex}}=\frac{6}{5}$

This example used a longer way of performing the calculation; however, once you understand the basics, it can be much simpler to perform the calculation like this:

$\frac{1\times 5}{2\times 5}+\frac{7}{10}=\frac{5}{10}+\frac{7}{10}=\frac{12}{10}=\frac{6}{5}$

## Examples of addition and subtraction of fractions

**(1) **Evaluate

$\frac{5}{3}-\frac{7}{3}$

**Solution:**

Since the denominators are the same, we can directly subtract the numerators.

$\frac{5}{3}-\frac{7}{3}\phantom{\rule{0ex}{0ex}}=\frac{5-7}{3}\phantom{\rule{0ex}{0ex}}=-\frac{2}{3}$

**(2) **Evaluate

$\frac{25}{18}+\frac{14}{12}$

**Solution:**

The LCM of the denominators (18 and 12) is 36. So,

$\frac{25}{18}+\frac{14}{12}\phantom{\rule{0ex}{0ex}}=\frac{25\times 2+3\times 14}{36}\phantom{\rule{0ex}{0ex}}=\frac{92}{36}\phantom{\rule{0ex}{0ex}}=\frac{23}{9}(\mathrm{simplest}\mathrm{form})$

**(3) **Evaluate

$\frac{13}{20}-\frac{6}{12}+\frac{51}{30}-\frac{2}{3}$

**Solution:**

The LCM of the denominators (20, 12, 30, and 3) is 60.

$\frac{13}{20}-\frac{6}{12}+\frac{51}{30}-\frac{2}{3}\phantom{\rule{0ex}{0ex}}=\frac{13\times 3-6\times 5+51\times 2-2\times 30}{60}\phantom{\rule{0ex}{0ex}}=\frac{51}{60}\phantom{\rule{0ex}{0ex}}=\frac{17}{20}$

To add or subtract mixed fractions, first convert them into improper fractions and then continue with the standard process.

## Addition and subtraction of mixed fractions

A mixed fraction is a number that is represented as a whole number and a quotient, e.g. $5\frac{3}{4}.$

Adding and subtracting mixed fractions requires converting them to improper fractions. Then, we can carry out the standard process of addition and subtraction of fractions, as we did before. An improper fraction is a fraction whose numerator is greater than or equal to its denominator.

To convert a mixed fraction to an improper fraction, we must convert the whole number part of the mixed fraction to a fraction with the same denominator as the quotient part. Then, we simply add them. Let's take a look at an example.

Convert the following mixed fraction to an improper fraction.

$5\frac{3}{4}$

**Solution:**

**1. **Convert the whole number part of the improper fraction to a fraction with the same denominator as the quotient part.

$5=\frac{5\times 4}{4}=\frac{20}{4}$

**2.** Add this new fraction to the quotient part of the original mixed fraction to obtain the improper fraction.

$\frac{20}{4}+\frac{3}{4}=\frac{23}{4}$

And so we get the result:

$5\frac{3}{4}=\frac{23}{4}$

Evaluate

$4\frac{1}{3}+\frac{1}{4}-3\frac{5}{12}$

**Solution:**

Converting the mixed fractions to improper fractions we get:

$4\frac{1}{3}+\frac{1}{4}-3\frac{5}{12}\phantom{\rule{0ex}{0ex}}=\frac{13}{3}+\frac{1}{4}-\frac{41}{12}$

The LCM of the denominators (3, 4, and 12) is 12.

$\frac{13}{3}+\frac{1}{4}-\frac{41}{12}\phantom{\rule{0ex}{0ex}}=\frac{13\times 4+1\times 3-41\times 1}{12}\phantom{\rule{0ex}{0ex}}=\frac{14}{12}\phantom{\rule{0ex}{0ex}}=\frac{7}{6}$

## Addition and subtraction of positive and negative fractions

Much like any other number you will come across, fractions can be positive or negative. Luckily, the rules for adding and subtracting positive and negative fractions are just the same as for any other number! Let's look at some examples to see how this works.

**(1) **Evaluate

$\frac{3}{4}-\frac{-1}{4}$

**Solution:**

Subtracting a negative is the same as adding. So, our sum becomes:

$\frac{3}{4}+\frac{1}{4}=\frac{3+1}{4}=\frac{4}{4}=1$

**(2) **Evaluate

$-\frac{23}{19}+\frac{54}{19}$

**Solution:**

Since adding a negative is the same as subtracting, our sum becomes:

$\frac{54}{19}-\frac{23}{19}=\frac{54-23}{19}=\frac{31}{19}$

**(3) Evaluate**

**$\frac{-12}{5}-\frac{3}{5}$**

**Solution:**

When subtracting a negative from a negative, we add the numbers but keep the negative sign. So,

**$\frac{-(12+3)}{5}=\frac{-15}{5}=-3$**

## Addition and subtraction of decimal fractions

Decimal fractions are fractions where the denominator is some multiple of ten, e.g. $\frac{26}{100}$.

Decimal fractions are added and subtracted in much the same way as any other fractions previously discussed. First, they must be converted to a form with the lowest common denominator, and then the numerators may be added or subtracted as required. The handy thing about adding and subtracting decimal fractions is that the lowest common denominator is always the largest denominator in the sum! Let's take a look at some more examples.

**(1)** Evaluate

$\frac{5}{10}+\frac{23}{100}$

**Solution:**

First, we convert each to the lowest common denominator, which we can see is 100.

$\frac{5}{10}+\frac{23}{100}=\frac{10\times 5}{10\times 10}+\frac{23}{100}=\frac{50}{100}+\frac{23}{100}$

And then we carry out the addition.

$\frac{50}{100}+\frac{23}{100}=\frac{50+23}{100}=\frac{73}{100}$

**(1)** Evaluate

$\frac{234}{1000}-\frac{32}{100}$

**Solution:**

First, we convert each to the lowest common denominator, which we can see is 1000.

$\frac{234}{1000}-\frac{32}{100}=\frac{234}{1000}-\frac{32\times 10}{100\times 10}=\frac{234}{1000}-\frac{320}{1000}$

And then we carry out the subtraction.

$\frac{234}{1000}-\frac{320}{1000}=\frac{234-320}{1000}=\frac{-86}{1000}$

## Subtraction and addition of fractions - Key takeaways

- If the fractions to be added or subtracted have the same denominators, their numerators can simply be added or subtracted while keeping the denominator constant.
- If we have to add or subtract fractions with different denominators, we first manipulate the fractions so that they end up having the same denominators.
- To add or subtract mixed fractions, first convert them into improper fractions.
- Adding and subtracting positive and negative fractions works in just the same way as any other number.

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##### Frequently Asked Questions about Substraction and addition of fractions

How to add and subtract fractions?

If the fractions to be added or subtracted have the same denominators, the corresponding numerators can simply be added or subtracted keeping the denominator constant. If we have perform addition or subtraction on fractions with different denominators, we first manipulate the fractions so that they end up having the same denominators.

What is the rule for addition and subtraction of fractions?

If the fractions to be added or subtracted have the same denominators, the corresponding numerators can simply be added or subtracted keeping the denominator constant.

What is an example of addition and subtraction of fractions?

1/5+3/5=4/5

4/5-1/5=3/5

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