# Addition & Subtraction with Fractions

## Method 1: with a Formula

In this section we learn a method for adding and subtracting with fractions.

This is the first of two methods that we'll learn and relies upon a formula.

We start by learning the formula for addition and subtraction of fractions as well as watch a tutorial. We'll then work our way through some questions.

By the end of this section we should be comfortable adding, or subtracting any two fractions.

## Tutorial

In the following tutorial we review the method for adding and subtracting fractions and work through some well-explained examples.

## Formula for Addition & Subtraction

When adding two fractions, $$\frac{a}{b}$$ and $$\frac{c}{d}$$, we can use the following formula: $\frac{a}{b}+\frac{c}{d} = \frac{a\times d + b\times c}{b\times d}$

### Subtraction

When subtracting two fractions, $$\frac{a}{b} - \frac{c}{d}$$, we can use exactly the same formula. The only difference is that we replace the $$+$$ by a $$-$$ (since we're dealing with a subtraction).
So the formula for subtraction with fractions is: $\frac{a}{b}-\frac{c}{d} = \frac{a\times d - b\times c}{b\times d}$

Now that we know the method and the formula, we move-on to work our way through some exercises.
Make sure to try each of the following questions before looking at the answer.

## Exercise 1 - Adding Fractions

Using the method we've just learned, calculate each of the following and write your answer in its simplest form:

1. $$\frac{1}{4} + \frac{2}{3}$$

2. $$\frac{1}{8}+\frac{3}{4}$$

3. $$\frac{3}{8} + \frac{2}{5}$$

4. $$\frac{2}{5} + \frac{1}{3}$$

5. $$\frac{2}{7} + \frac{3}{5}$$

6. $$\frac{3}{8} + \frac{1}{2}$$

1. $$\frac{1}{4} + \frac{2}{3} = \frac{11}{12}$$

2. $$\frac{1}{8}+\frac{3}{4} = \frac{7}{8}$$

3. $$\frac{2}{7} + \frac{3}{5} = \frac{31}{35}$$

4. $$\frac{3}{8} + \frac{5}{12} = \frac{76}{96}$$

This fraction can also be simplified and written:

$$\frac{3}{8} + \frac{1}{2} = \frac{14}{16} = \frac{7}{8}$$

## Solution With Working

1. To calculate $$\frac{1}{4}+\frac{2}{3}$$ we use the formula: \begin{aligned} \frac{1}{4}+\frac{2}{3} & = \frac{1\times 3 + 4\times 2}{4\times 3} \\ & = \frac{3+8}{12} \\ & = \frac{11}{12} \end{aligned} It's always worth checking if we can simplify the fraction. To do that we look for the highest common factor, HCF, of $$11$$ and $$12$$. Since $$HCF(11,12) = 1$$ the fraction can't be simplified any further.

So the final answer is: $\frac{1}{4} + \frac{2}{3} = \frac{11}{12}$

2. To calculate $$\frac{1}{8}+\frac{3}{4}$$ we use the formula: \begin{aligned} \frac{1}{8}+\frac{3}{4} & = \frac{1\times 4 + 8 \times 3}{8\times 4} \\ & = \frac{4 + 24}{32} \\ & = \frac{28}{32} \end{aligned} We now check to see whether we can simplify this fraction.
The highest common factor of $$28$$ and $$32$$ is $$4$$, that's $$HCF(28,32)=4$$.
So we can simplify the fraction by cancelling-out the $$HCF$$: \begin{aligned} \frac{28}{32} &= \frac{4\times 7}{4\times 8}\\ & = \frac{7}{8} \end{aligned} And that's the final answer: $\frac{1}{8}+\frac{3}{4} = \frac{7}{8}$

3. To calculate $$\frac{3}{8} + \frac{2}{5}$$ we use the formula: \begin{aligned} \frac{3}{8}+\frac{2}{5} & = \frac{3\times 5 + 8 \times 2}{8\times 5} \\ & = \frac{15 + 16}{40} \\ & = \frac{31}{40} \end{aligned} Again, we check whether we can simplify this fraction.
Since the highest common factor of $$31$$ and $$40$$ is $$1$$, $$HCF(31,40)=1$$, the fraction can't be simplified any further.
The final answer is therefore: $\frac{3}{8}+\frac{2}{5} = \frac{31}{40}$

4. To calculate $$\frac{2}{5}+ \frac{1}{3}$$ we use the formula: \begin{aligned} \frac{2}{5}+ \frac{1}{3} & = \frac{2\times 3 + 5 \times 1}{5\times 3} \\ & = \frac{6+5}{5\times 3} \\ & = \frac{11}{15} \end{aligned} We now check whether we can simplify this fraction. The highest common factor of $$11$$ and $$15$$ is $$1$$, $$HCF(11,15)=1$$, so the fraction can't be simplified any further.
The final answer is therefore: $\frac{2}{5}+\frac{1}{3} = \frac{11}{15}$

5. To calculate $$\frac{2}{7} + \frac{3}{5}$$ we use the formula: \begin{aligned} \frac{2}{7} + \frac{3}{5} & = \frac{2\times 5 + 7 \times 2}{7\times 5} \\ & = \frac{10 + 14}{35} \\ & = \frac{24}{35} \end{aligned} We now check whether we can simplify this fraction. The highest common factor of $$24$$ and $$35$$ is $$1$$, $$HCF(24,35)=1$$, so the fraction can't be simplified any further.
The final answer is therefore: $\frac{2}{7}+\frac{3}{5} = \frac{24}{35}$

6. To calculate $$\frac{1}{3} + \frac{1}{4}$$ we use the formula: \begin{aligned} \frac{1}{3} + \frac{1}{4} & = \frac{1\times 4 + 3 \times 1}{3 \times 4} \\ & = \frac{4 + 3}{12} \\ & = \frac{7}{12} \end{aligned} We now check whether we can simplify this fraction. The highest common factor of $$7$$ and $$12$$ is $$1$$, $$HCF(7,12)=1$$, so the fraction can't be simplified any further.
The final answer is therefore: $\frac{1}{3}+\frac{1}{4} = \frac{7}{12}$

## Exercise 2 - Subtracting Fractions

Using the method we've just learned, calculate each of the following and write your answer in its simplest form:

1. $$\frac{3}{8} - \frac{1}{4}$$

2. $$\frac{11}{10} - \frac{3}{20}$$

3. $$\frac{7}{25} - \frac{5}{8}$$

4. $$\frac{2}{3} - \frac{1}{2}$$

5. $$\frac{7}{18} - \frac{2}{9}$$