# Adding & Subtracting with Fractions

## (Least Common Multiple - Least Common Denominator

We learn a second method for adding and subtracting with fractions. $\frac{a}{b}+\frac{c}{d} \quad \text{and} \quad \frac{a}{b}-\frac{c}{d}$ The trick is two write each of the two fractions over the same denominator. The denominator over which we write the fractions is equal to the least common multiple, LCM, of the denominators; this LCM is known as the least common denominator.

## Tutorial

In the following tutorial we learn the method for adding two fractions by finding their least common denominator.

## Method - Adding & Subtracting Fractions

The method we just learnt in the tutorial is summarized here in three steps.
To illustrate the three steps, we show how they work to calculate: $\frac{3}{8}+ \frac{5}{12}$

• Step 1: find the least Common Multiple, $$LCM$$, of the denominators.
The least common multiple of the denominators is known as the Least Common Denominator of the two fractions.

For $$\frac{3}{8}+ \frac{5}{12}$$, we find the $$LCM$$ of $$8$$ and $$12$$.

• the multiples of $$8$$ are: $$8,16,24,32,40,\dots$$

• the multiples of $$12$$ are: $$12,24,36,48,60,\dots$$

We can see that the $$LCM$$ of $$8$$ and $$12$$ is $$24$$ and $$24 = 8 \times 3$$ and $$12 = 12 \times 2$$.
So the least common denominator is $$24$$.

• Step 2: write each of the two fractions as their equivalent fractions over the least common denominator found in step 2.

We write both $$\frac{3}{8}$$ and $$\frac{5}{12}$$ as fractions over $$24$$: \begin{aligned} \frac{3}{8} & = \frac{3\times 3}{8\times 3} \\ & = \frac{9}{24} \end{aligned} and \begin{aligned} \frac{5}{12} & = \frac{5\times 2}{12\times 2} \\ & = \frac{10}{24} \end{aligned}

• Step 3: Add, or subtract, the fractions. Now that they are both written over the same denominator we can add, or subtract, the numerators directly.

\begin{aligned} \frac{3}{8}+ \frac{5}{12} & = \frac{9}{24} + \frac{10}{24} \\ & = \frac{9+10}{24} \\ & = \frac{19}{24} \end{aligned}

## Exercise 1

Using the method we've just seen, calculate each of the following additions with fractions:

1. $$\frac{2}{5} + \frac{3}{7}$$
2. $$\frac{4}{9} + \frac{1}{6}$$
3. $$\frac{1}{8} + \frac{2}{3}$$
4. $$\frac{3}{7} + \frac{1}{2}$$
5. $$\frac{3}{8} + \frac{5}{12}$$

1. $$\frac{2}{5} + \frac{3}{7} = \frac{29}{35}$$

2. $$\frac{4}{9} + \frac{1}{6} = \frac{11}{18}$$

3. $$\frac{1}{8} + \frac{2}{3} = \frac{19}{24}$$

4. $$\frac{3}{7} + \frac{1}{2} = \frac{13}{14}$$

5. $$\frac{3}{8} + \frac{5}{12} = \frac{19}{24}$$

## Exercise 2

Using the method we've just seen, calculate each of the following subtractions with fractions:

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

2. $$\frac{7}{8} - \frac{5}{12}$$

3. $$\frac{3}{4} - \frac{5}{8}$$

4. $$\frac{4}{5} - \frac{2}{3}$$

5. $$\frac{7}{9} - \frac{1}{6}$$

## Solution Without Working

1. $$\frac{3}{4} - \frac{2}{5} = \frac{11}{24}$$

2. $$\frac{7}{8} - \frac{5}{12} = \frac{7}{20}$$

3. $$\frac{3}{4} - \frac{5}{8} = \frac{1}{8}$$

4. $$\frac{4}{5} - \frac{2}{3} = \frac{2}{15}$$

5. $$\frac{7}{9} - \frac{1}{6} = \frac{11}{18}$$