# Operations With Fractions

## Fractions

In the previous lesson we reviewed fractions, equivalent fractions and conversions to decimal and percent. In this lesson, we will focus on operations with fractions.

When the denominators are the same, you can add/subtract the numerators and leave the denominators the same:

$$\cfrac{4}{15} + \cfrac{5}{15} = \cfrac{4+5}{15} = \cfrac{9}{15}$$

$$\cfrac{12}{17} - \cfrac{5}{17} = \cfrac{12-5}{17} = \cfrac{7}{17}$$

However, when the denominators are different we have to change the fractions to make the demoninators the same. We do this by using equivalent fractions. You cannot the numerators when the demoninators are different because the "size" of each peice is different. Let's say you have one pizza and cut it into 2. Each slice is $$\frac{1}{2}$$. Take another pizza and cut it into 4. Each slice is $$\frac{1}{4}$$ which is smaller than the other slice from the other pizza. The slices are different, so when you add them together it is not super obvious what fraction you have.

$$\cfrac{1}{2} + \cfrac{1}{4} = ?$$

To add these fractions together, let's find an equivalent fraction for $$\frac{1}{2}$$ that has a denominator of $$4$$.

$$\cfrac{1*?}{2*?} = \cfrac{?}{4} = ?$$

We will need to multiply the denominator by $$2$$ in order to get a $$4$$. That means, multiply the numerator by $$2$$ as well:

$$\cfrac{1*2}{2*2} = \cfrac{2}{4} = ?$$

This means the fractions $$\frac{1}{2}$$ and $$\frac{2}{4}$$ are equivalent. Now, let's rewrite the original addition problem and add now that the demoninators are the same:

$$\cfrac{1}{2} + \cfrac{1}{4}$$

$$= \cfrac{2}{4} + \cfrac{1}{4}$$

$$= \cfrac{2+1}{4}$$

$$= \cfrac{3}{4}$$

Typically, you want to find the lowest common denominator (LCD) between the two fractions and make both fractions over the LCD. To find the LCD, you want to find the smallest common multiple of the denominators. To find the multiples, just count by that number:

$$2: 2, 4, 6, 8, 10$$

$$4: 4, 8, 12, 16, 20$$

The LCD here is $$4$$ because that is the smallest common multiple.

Evalate $$\frac{7}{8} - \frac{6}{7}$$

## Multiplying Fractions

To multiply fractions, simply multiply across: multiply the numerators and then multiply the denominators. Don't forget your rules for integer multiplication!

$$\cfrac{3}{7} * \cfrac{-2}{3}$$

$$= \cfrac{3*(-2)}{7*3}$$

$$= \cfrac{-6}{21}$$

It is good practice to reduce the final answer. Here, we can divide both top and bottom by $$3$$:

$$= \cfrac{-6/3}{21/3}$$

$$= \cfrac{-2}{7}$$

## Dividing Fractions

To divide fractions, we are going to multiply by the reciprocal. A reciprocal is a fraction where you switch the top and bottom. Once you change the division question to multiplication by the reciprocal, use the rules to multiply fractions.

$$\cfrac{3}{7} \div \cfrac{-2}{3}$$

$$= \cfrac{3}{7} * \cfrac{3}{-2}$$

$$= \cfrac{3*3}{7*(-2)}$$

$$= \cfrac{9}{-14}$$

$$= \cfrac{-9}{14}$$

In this example, you cannot reduce the fraction any further. However, we typically put the negative sign on top.

Evaluate the following: $$\frac{2}{5} * \frac{-1}{6} \div \frac{4}{10}$$