0.1 Fractions

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Adding and Subtracting Fractions

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).

Adding and Subtracting Fractions

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

Find a common denominator.
Rewrite each fraction using the common denominator.
Add the numerators but keep the common denominator.
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.

Show Solution

Since the denominators are not alike, find a common denominator by multiplying the denominators.

$3\cdot5=15$

Rewrite each fraction with a denominator of 15.

$\begin{array}{c}\frac{2}{3}\cdot \frac{5}{5}=\frac{10}{15}\\\\\frac{1}{5}\cdot \frac{3}{3}=\frac{3}{15}\end{array}$

Add the fractions by adding the numerators and keeping the denominator the same. Make sure the fraction cannot be simplified.

$\frac{10}{15}+\frac{3}{15}=\frac{13}{15}$

Answer

$\frac{2}{3}+\frac{1}{5}=\frac{13}{15}$

Example

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

Show Solution

Since the denominators are not alike, find the least common denominator by finding the least common multiple (LCM) of 7 and 21.

Multiples of 7: 7, 14, 21

Multiples of 21: 21

Rewrite each fraction with a denominator of 21.

$\begin{array}{c}\frac{3}{7}\cdot \frac{3}{3}=\frac{9}{21}\\\\\frac{2}{21}\end{array}$

Add the fractions by adding the numerators and keeping the denominator the same. Make sure the fraction cannot be simplified.

$\frac{9}{21}+\frac{2}{21}=\frac{11}{21}$

Answer

$\frac{3}{7}+\frac{2}{21}=\frac{11}{21}$

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.

Show Solution

The fractions have unlike denominators, so you need to find a common denominator. Recall that a common denominator can be found by multiplying the two denominators together.

$5\cdot6=30$

Rewrite each fraction as an equivalent fraction with a denominator of 30.

$\begin{array}{c}\frac{1}{5}\cdot \frac{6}{6}=\frac{6}{30}\\\\\frac{1}{6}\cdot \frac{5}{5}=\frac{5}{30}\end{array}$

Subtract the numerators. Simplify the answer if needed.

$\frac{6}{30}-\frac{5}{30}=\frac{1}{30}$

Answer

$\frac{1}{5}-\frac{1}{6}=\frac{1}{30}$

Example

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

Show Solution

Find the least common multiple of the denominators—this is the least common denominator.

Multiples of 6: 6, 12, 18, 24

Multiples of 4: 4, 8 12, 16, 20

12 is the least common multiple of 6 and 4.

Rewrite each fraction with a denominator of 12.

$\begin{array}{c}\frac{5}{6}\cdot \frac{2}{2}=\frac{10}{12}\\\\\frac{1}{4}\cdot \frac{3}{3}=\frac{3}{12}\end{array}$

Subtract the fractions. Simplify the answer if needed.

$\frac{10}{12}-\frac{3}{12}=\frac{7}{12}$

Answer

$\frac{5}{6}-\frac{1}{4}=\frac{7}{12}$

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

Show Solution

Multiply the numerators and multiply the denominators.

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

Simplify, if possible. This fraction is already in lowest terms.

$\frac{8}{15}$

Answer

$\frac{8}{15}$

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

Show Solution

Write your answer in lowest terms.

Dividing by 4 or $\frac{4}{1}$ is the same as multiplying by the reciprocal of 4, which is $\frac{1}{4}$ .

$\frac{2}{3}\div 4=\frac{2}{3}\cdot \frac{1}{4}$

Multiply numerators and multiply denominators.

$\frac{2\cdot 1}{3\cdot 4}=\frac{2}{12}$

Simplify to lowest terms by dividing numerator and denominator by the common factor 4.

$\frac{1}{6}$

Answer

$\frac{2}{3}\div4=\frac{1}{6}$

Example

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

Show Solution

Write your answer in lowest terms.

Dividing by $\frac{1}{2}$ is the same as multiplying by the reciprocal of $\frac{1}{2}$ , which is $\frac{2}{1}$ .

$9\div\frac{1}{2}=\frac{9}{1}\cdot\frac{2}{1}$

Multiply numerators and multiply denominators.

$\frac{9\cdot 2}{1\cdot 1}=\frac{18}{1}=18$

This answer is already simplified to lowest terms.

Answer

$9\div\frac{1}{2}=18$

Dividing a Fraction by a Fraction

Dividing with Fractions

Find the reciprocal of the number that follows the division symbol.
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 $\frac{2}{3}\div \frac{1}{6}$ .

Show Solution

Multiply by the reciprocal.

KEEP $\frac{2}{3}$

CHANGE $\div$ to $\cdot$

FLIP $\frac{1}{6}$

$\frac{2}{3}\cdot \frac{6}{1}$

Multiply numerators and multiply denominators.

$\frac{2\cdot6}{3\cdot1}=\frac{12}{3}$

Simplify.

$\frac{12}{3}=4$

Answer

$\frac{2}{3}\div \frac{1}{6}=4$

Example

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

Show Solution

Multiply by the reciprocal. Keep

$\frac{3}{5}$ , change

$\div$ to

$\cdot$ , and flip

$\frac{2}{3}$ .

$\frac{3}{5}\cdot \frac{3}{2}$

Multiply numerators and multiply denominators.

$\frac{3\cdot 3}{5\cdot 2}=\frac{9}{10}$

Chapter 0: Math Review

QUICK REFERENCE

Adding and Subtracting Fractions

Adding Fractions with Unlike Denominators

Simplifying a Fraction

Example

Answer

Example

Answer

Subtracting Fractions

Example

Answer

Example

Answer

Multiplying Fractions

Multiplying Two Fractions

Multiplying More Than Two Fractions

Example

Answer

Dividing Fractions

Dividing is Multiplying by the Reciprocal

Example

Answer

Example

Answer

Dividing a Fraction by a Fraction

Dividing with Fractions

Example

Answer

Example

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