## 0.1 Fractions

Chapter 0 Lecture Notes

### QUICK REFERENCE

• Convert the fractions so they have common denominators.
• Perform the addition or subtraction on the numerator and keep the common denominator.
• Simplify the answer (write the fraction in the lowest terms).

Multiplying and Dividing Fractions

• To multiply, multiply across the numerators and denominators.
• To divide, multiply the first number by the reciprocal of the second number.
• Simplify the answer (write the fraction in the lowest terms).

In order to add or subtract fractions, you first must make sure that the fractions have the same denominator. The denominator tells you how many pieces the whole has been broken into, and the numerator tells you how many of those pieces you are considering.

To find a common denominator you will determine the least common multiple. Remember that if a number is a multiple of another, you can divide them and have no remainder.

One way to find the least common multiple of two or more numbers is to first multiply each by 1, 2, 3, 4, etc.  For example, find the least common multiple of 2 and 5.

 First, list all the multiples of 2: Then list all the multiples of 5: $2\cdot 1 = 2$ $5\cdot 1 = 5$ $2\cdot 2 = 4$ $5\cdot 2 = 10$ $2\cdot 3 = 6$ $5\cdot 3 = 15$ $2\cdot 4 = 8$ $5\cdot 4 = 20$ $2\cdot 5 = 10$ $5\cdot 5 = 25$

The smallest multiple they have in common will be the common denominator.  The least common multiple of 2 and 5 is 10.

### Adding Fractions with Unlike Denominators

1. Find a common denominator.
2. Rewrite each fraction using the common denominator.
3. Add the numerators but keep the common denominator.
4. Simplify by canceling out all common factors in the numerator and denominator.

## Simplifying a Fraction

A common convention used in mathematics is writing a fraction in lowest terms.  The process of simplifying a fraction is often called reducing the fraction. We can simplify by canceling (dividing) the common factors in a fraction’s numerator and denominator.  This is possible because a fraction represents division (a part divided by the whole).

For example, to simplify $\frac{6}{9}$ you can rewrite 6 and 9 using the smallest factors possible as follows:

$\frac{6}{9}=\frac{2\cdot3}{3\cdot3}$

Since there is a 3 in both the numerator and denominator, and fractions can be considered division, we can divide the 3 in the top by the 3 in the bottom to reduce to 1.

$\frac{6}{9}=\frac{2\cdot\cancel{3}}{3\cdot\cancel{3}}=\frac{2\cdot1}{3}=\frac{2}{3}$

In the next example you are shown how to add two fractions with different denominators, then simplify the answer.

### Example

Add $\frac{2}{3}+\frac{1}{5}$. Simplify the answer.

### Example

Add $\frac{3}{7}+\frac{2}{21}$. Simplify the answer.

## Subtracting Fractions

When you subtract fractions, you will still find a common denominator, but the numerators will be subtracted. Below are some examples of subtracting fractions whose denominators are not alike.

### Example

Subtract $\frac{1}{5}-\frac{1}{6}$. Simplify the answer.

### Example

Subtract $\frac{5}{6}-\frac{1}{4}$. Simplify the answer.

## Multiplying Fractions

When you multiply a fraction by a fraction, you are finding a “fraction of a fraction.”  To multiply fractions you multiply across the numerators and denominators.

### Multiplying Two Fractions

$\frac{a}{b}\cdot \frac{c}{d}=\frac{a\cdot c}{b\cdot d}=\frac{\text{product of the numerators}}{\text{product of the denominators}}$

### Multiplying More Than Two Fractions

$\frac{a}{b}\cdot \frac{c}{d}\cdot \frac{e}{f}=\frac{a\cdot c\cdot e}{b\cdot d\cdot f}$

### Example

Multiply $\frac{2}{3}\cdot \frac{4}{5}$.

## Dividing Fractions

Dividing fractions requires using the reciprocal of a number or fraction. If you multiply two numbers together and get 1 as a result, then the two numbers are reciprocals. Here are some examples of reciprocals:

Original number Reciprocal Product
$\frac{3}{4}$ $\frac{4}{3}$ $\frac{3}{4}\cdot \frac{4}{3}=\frac{3\cdot 4}{4\cdot 3}=\frac{12}{12}=1$
$\frac{1}{2}$ $\frac{2}{1}$ $\frac{1}{2}\cdot\frac{2}{1}=\frac{1\cdot}{2\cdot1}=\frac{2}{2}=1$
$3=\frac{3}{1}$ $\frac{1}{3}$ $\frac{3}{1}\cdot \frac{1}{3}=\frac{3\cdot 1}{1\cdot 3}=\frac{3}{3}=1$
$2\frac{1}{3}=\frac{7}{3}$ $\frac{3}{7}$ $\frac{7}{3}\cdot\frac{3}{7}=\frac{7\cdot3}{3\cdot7}=\frac{21}{21}=1$

Sometimes we call the reciprocal the “flip” of the other number: flip $\frac{2}{5}$ to get the reciprocal $\frac{5}{2}$. Caution! Division by zero is undefined and so is the reciprocal of any fraction that has a zero in the numerator. For any real number a, $\frac{a}{0}$ is undefined. Additionally, the reciprocal of $\frac{0}{a}$ will always be undefined.

### Dividing is Multiplying by the Reciprocal

For all division, you can turn the operation into multiplication by using the reciprocal. Dividing is the same as multiplying by the reciprocal.

The same idea will work when the divisor (the thing being divided) is a fraction. If you have $\frac{3}{4}$ of a candy bar and need to divide it among 5 people, each person gets $\frac{1}{5}$ of the available candy:

$\frac{1}{5}\text{ of }\frac{3}{4}=\frac{1}{5}\cdot \frac{3}{4}=\frac{3}{20}$

Each person gets $\frac{3}{20}$ of a whole candy bar.

If you have a recipe that needs to be divided in half, you can divide each ingredient by 2, or you can multiply each ingredient by $\frac{1}{2}$ to find the new amount.

### Example

Find $\frac{2}{3}\div 4$.

### Example

Divide. $9\div\frac{1}{2}$.

## Dividing a Fraction by a Fraction

### Dividing with Fractions

1. Find the reciprocal of the number that follows the division symbol.
2. Multiply the first number (the one before the division symbol) by the reciprocal of the second number (the one after the division symbol).

Any easy way to remember how to divide fractions is the phrase “keep, change, flip.” This means to KEEP the first number, CHANGE the division sign to multiplication, and then FLIP (use the reciprocal) of the second number.

### Example

Divide $\frac{2}{3}\div \frac{1}{6}$.

### Example

Divide $\frac{3}{5}\div \frac{2}{3}$.