Visualize Fractions

By the end of this section, you will be able to: find equivalent fractions, simplify fractions, multiply fractions, divide fractions, simplify expressions written with a fraction bar, and translate phrases to expressions with fractions.

Find equivalent fractions

Fractions are a way to represent parts of a whole. The fraction $\tfrac{1}{3}$ means that one whole has been divided into $3$ equal parts and each part is one of the three equal parts. The fraction $\tfrac{2}{3}$ represents two of three equal parts. In the fraction $\tfrac{2}{3}$ , the $2$ is called the numerator and the $3$ is called the denominator.

The circle on the left has been divided into $3$ equal parts. Each part is $\tfrac{1}{3}$ of the $3$ equal parts. In the circle on the right, $\tfrac{2}{3}$ of the circle is shaded (2 of the 3 equal parts).

Fraction. A fraction is written $\tfrac{a}{b}$ , where $b \neq 0$ and

$a$ is the numerator and $b$ is the denominator.

A fraction represents parts of a whole. The denominator $b$ is the number of equal parts the whole has been divided into, and the numerator $a$ indicates how many parts are included.

If a whole pie has been cut into $6$ pieces and we eat all $6$ pieces, we ate $\tfrac{6}{6}$ pieces, or, in other words, one whole pie. So $\tfrac{6}{6} = 1$ . This leads us to the property of one that tells us that any number, except zero, divided by itself is $1$ .

Property of one.

\frac{a}{a} = 1 \qquad (a \neq 0)

Any number, except zero, divided by itself is one.

If a pie was cut in $6$ pieces and we ate all $6$ , we ate $\tfrac{6}{6}$ pieces, or one whole pie. If the pie was cut into $8$ pieces and we ate all $8$ , we ate $\tfrac{8}{8}$ pieces, or one whole pie. We ate the same amount — one whole pie.

The fractions $\tfrac{6}{6}$ and $\tfrac{8}{8}$ have the same value, $1$ , and so they are called equivalent fractions. Equivalent fractions are fractions that have the same value.

Think of two same-size pizzas, one cut into $2$ equal pieces and the other cut into $8$ equal pieces. Shading $1$ of the $2$ pieces on the first pizza and $4$ of the $8$ pieces on the second pizza shades the same amount of pizza in both cases — this is a way to show that $\tfrac{1}{2}$ is equivalent to $\tfrac{4}{8}$ .

Equivalent fractions. Equivalent fractions are fractions that have the same value.

How can we use mathematics to change $\tfrac{1}{2}$ into $\tfrac{4}{8}$ ? How could we take a pizza that is cut into $2$ pieces and cut it into $8$ pieces? We could cut each of the $2$ larger pieces into $4$ smaller pieces! The whole pizza would then be cut into $8$ pieces instead of just $2$ . Mathematically, what we’ve described could be written as $\tfrac{1 \cdot 4}{2 \cdot 4} = \tfrac{4}{8}$ — cutting each half of a circle into $4$ pieces gives a circle cut into $8$ pieces, so $\tfrac{1 \cdot 4}{2 \cdot 4} = \tfrac{4}{8}$ .

This model leads to the following property.

Equivalent fractions property. If $a, b, c$ are numbers where $b \neq 0, c \neq 0$ , then

\frac{a}{b} = \frac{a \cdot c}{b \cdot c}

If we had cut the pizza differently, we could get

\frac{1 \cdot 2}{2 \cdot 2} = \frac{2}{4} \quad\text{so}\quad \frac{1}{2} = \frac{2}{4}

\frac{1 \cdot 3}{2 \cdot 3} = \frac{3}{6} \quad\text{so}\quad \frac{1}{2} = \frac{3}{6}

\frac{1 \cdot 10}{2 \cdot 10} = \frac{10}{20} \quad\text{so}\quad \frac{1}{2} = \frac{10}{20}

So, we say $\tfrac{1}{2}$ , $\tfrac{2}{4}$ , $\tfrac{3}{6}$ , and $\tfrac{10}{20}$ are equivalent fractions.

Example. Find three fractions equivalent to $\tfrac{2}{5}$ .

To find a fraction equivalent to $\tfrac{2}{5}$ , we multiply the numerator and denominator by the same number. We can choose any number, except for zero. Let’s multiply them by $2$ , $3$ , and then $5$ .

\frac{2 \cdot 2}{5 \cdot 2} = \frac{4}{10} \qquad \frac{2 \cdot 3}{5 \cdot 3} = \frac{6}{15} \qquad \frac{2 \cdot 5}{5 \cdot 5} = \frac{10}{25}

So, $\tfrac{4}{10}$ , $\tfrac{6}{15}$ , and $\tfrac{10}{25}$ are equivalent to $\tfrac{2}{5}$ .

Find three fractions equivalent to $\tfrac{3}{5}$ . Enter one of them.

Find three fractions equivalent to $\tfrac{4}{5}$ . Enter one of them.

Simplify fractions

A fraction is considered simplified if there are no common factors, other than $1$ , in its numerator and denominator. For example,

$\tfrac{2}{3}$ is simplified because there are no common factors of $2$ and $3$ .
$\tfrac{10}{15}$ is not simplified because $5$ is a common factor of $10$ and $15$ .

Simplified fraction. A fraction is considered simplified if there are no common factors in its numerator and denominator.

The phrase reduce a fraction means to simplify the fraction. We simplify, or reduce, a fraction by removing the common factors of the numerator and denominator. A fraction is not simplified until all common factors have been removed. If an expression has fractions, it is not completely simplified until the fractions are simplified.

We used the equivalent fractions property to find equivalent fractions. Now we’ll use the equivalent fractions property in reverse to simplify fractions. We can rewrite the property to show both forms together.

Equivalent fractions property. If $a, b, c$ are numbers where $b \neq 0, c \neq 0$ , then

\frac{a}{b} = \frac{a \cdot c}{b \cdot c} \qquad\text{and}\qquad \frac{a \cdot c}{b \cdot c} = \frac{a}{b}

Example. Simplify: $-\tfrac{32}{56}$ .

Rewrite the numerator and denominator showing the common factor: $-\tfrac{4 \cdot 8}{7 \cdot 8}$ . Simplify using the equivalent fractions property: $-\tfrac{4}{7}$ .

Notice that the fraction $-\tfrac{4}{7}$ is simplified because there are no more common factors.

Simplify: $-\tfrac{42}{54}$ .

Simplify: $-\tfrac{45}{81}$ .

Sometimes it may not be easy to find common factors of the numerator and denominator. When this happens, a good idea is to factor the numerator and the denominator into prime numbers. Then divide out the common factors using the equivalent fractions property.

Example — How to simplify a fraction. Simplify: $-\tfrac{210}{385}$ .

Step 1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into primes first: $-\tfrac{210}{385} = -\tfrac{2 \cdot 3 \cdot 5 \cdot 7}{5 \cdot 7 \cdot 11}$ .

Step 2. Simplify using the equivalent fractions property by dividing out the common factors $5$ and $7$ : $-\tfrac{2 \cdot 3}{11}$ .

Step 3. Multiply the remaining factors, if necessary: $-\tfrac{6}{11}$ .

Simplify: $-\tfrac{69}{120}$ .

Simplify: $-\tfrac{120}{192}$ .

We now summarize the steps you should follow to simplify fractions.

Simplify a fraction.

Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers first.
Simplify using the equivalent fractions property by dividing out the common factors.
Multiply any remaining factors, if needed.

Example. Simplify: $\tfrac{5x}{5y}$ .

Rewrite showing the common factors, then divide out the common factors: $\tfrac{5 \cdot x}{5 \cdot y} = \tfrac{x}{y}$ . Simplify: $\tfrac{x}{y}$ .

Simplify: $\tfrac{7x}{7y}$ . Use $x$ and $y$ in your answer.

Simplify: $\tfrac{3a}{3b}$ . Use $a$ and $b$ in your answer.

Multiply fractions

Many people find multiplying and dividing fractions easier than adding and subtracting fractions. Consider a model of $\tfrac{3}{4}$ : a rectangle divided into $4$ equal columns with $3$ of them shaded. Now take $\tfrac{1}{2}$ of $\tfrac{3}{4}$ : shading half of each of those $3$ shaded columns leaves the whole divided into $8$ equal parts, with $3$ of the $8$ shaded. Notice that now the whole is divided into $8$ equal parts. So $\tfrac{1}{2} \cdot \tfrac{3}{4} = \tfrac{3}{8}$ .

To multiply fractions, we multiply the numerators and multiply the denominators.

Fraction multiplication. If $a, b, c$ , and $d$ are numbers where $b \neq 0$ and $d \neq 0$ , then

\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}

To multiply fractions, multiply the numerators and multiply the denominators.

When multiplying fractions, the properties of positive and negative numbers still apply, of course. It is a good idea to determine the sign of the product as the first step.

Example. Multiply: $-\tfrac{11}{12} \cdot \tfrac{5}{7}$ .

The first step is to find the sign of the product. Since the signs are different, the product is negative. Determine the sign of the product, then multiply: $-\tfrac{11 \cdot 5}{12 \cdot 7} = -\tfrac{55}{84}$ . There are no common factors in the numerator and the denominator, so this is fully simplified.

Multiply: $-\tfrac{10}{28} \cdot \tfrac{8}{15}$ .

Multiply: $-\tfrac{9}{20} \cdot \tfrac{5}{12}$ .

When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, $a$ , can be written as $\tfrac{a}{1}$ . So, for example, $3 = \tfrac{3}{1}$ .

Example. Multiply: $-\tfrac{12}{5}(-20x)$ .

Determine the sign of the product — the signs are the same, so the product is positive. Write $-20x$ as a fraction: $\tfrac{12}{5}\left(\tfrac{20x}{1}\right)$ . Multiply, rewrite $20$ to show the common factor $5$ , and divide it out: $\tfrac{12 \cdot 4 \cdot 5x}{5 \cdot 1}$ . Simplify: $48x$ .

Multiply: $\tfrac{11}{3}(-9a)$ . Use $a$ in your answer.

Multiply: $\tfrac{13}{7}(-14b)$ . Use $b$ in your answer.

Divide fractions

Now that we know how to multiply fractions, we are almost ready to divide. Before we can do that, we need some vocabulary.

The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator. The reciprocal of $\tfrac{2}{3}$ is $\tfrac{3}{2}$ .

Notice that $\tfrac{2}{3} \cdot \tfrac{3}{2} = 1$ . A number and its reciprocal multiply to $1$ .

To get a product of positive $1$ when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign. The reciprocal of $-\tfrac{10}{7}$ is $-\tfrac{7}{10}$ , since $-\tfrac{10}{7}\left(-\tfrac{7}{10}\right) = 1$ .

Reciprocal. The reciprocal of $\tfrac{a}{b}$ is $\tfrac{b}{a}$ .

A number and its reciprocal multiply to one: $\tfrac{a}{b} \cdot \tfrac{b}{a} = 1$ .

To divide fractions, we multiply the first fraction by the reciprocal of the second.

Fraction division. If $a, b, c$ , and $d$ are numbers where $b \neq 0$ , $c \neq 0$ , and $d \neq 0$ , then

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}

To divide fractions, we multiply the first fraction by the reciprocal of the second.

We need to say $b \neq 0, c \neq 0$ , and $d \neq 0$ to be sure we don’t divide by zero.

Example. Divide: $-\tfrac{2}{3} \div \tfrac{n}{5}$ .

To divide, multiply the first fraction by the reciprocal of the second: $-\tfrac{2}{3} \cdot \tfrac{5}{n}$ . Multiply: $-\tfrac{10}{3n}$ .

Divide: $-\tfrac{3}{5} \div \tfrac{p}{7}$ . Use $p$ in your answer.

Divide: $-\tfrac{5}{8} \div \tfrac{q}{3}$ . Use $q$ in your answer.

Example. Find the quotient: $-\tfrac{7}{18} \div \left(-\tfrac{14}{27}\right)$ .

To divide, multiply the first fraction by the reciprocal of the second: $-\tfrac{7}{18} \cdot \left(-\tfrac{27}{14}\right)$ . Determine the sign of the product, and then multiply: $\tfrac{7 \cdot 27}{18 \cdot 14}$ . Rewrite showing common factors: $\tfrac{7 \cdot 9 \cdot 3}{9 \cdot 2 \cdot 7 \cdot 2}$ . Remove common factors: $\tfrac{3}{2 \cdot 2}$ . Simplify: $\tfrac{3}{4}$ .

Find the quotient: $-\tfrac{7}{27} \div \left(-\tfrac{35}{36}\right)$ .

Find the quotient: $-\tfrac{5}{14} \div \left(-\tfrac{15}{28}\right)$ .

There are several ways to remember which steps to take to multiply or divide fractions:

“To multiply fractions, multiply the numerators and multiply the denominators.”
“To divide fractions, multiply the first fraction by the reciprocal of the second.”

Another way to keep both procedures straight is to compare two everyday examples. One fourth of two pizzas is one half of a pizza — that’s multiplication: $2 \cdot \tfrac{1}{4} = \tfrac{2}{1} \cdot \tfrac{1}{4} = \tfrac{2}{4} = \tfrac{1}{2}$ . There are eight quarters in $2.00 — that’s division: $2 \div \tfrac{1}{4} = \tfrac{2}{1} \cdot \tfrac{4}{1} = 8$ .

The numerators or denominators of some fractions contain fractions themselves. A fraction in which the numerator or the denominator is a fraction is called a complex fraction.

Complex fraction. A complex fraction is a fraction in which the numerator or the denominator contains a fraction.

Some examples of complex fractions are

\cfrac{\frac{6}{7}}{3} \qquad \cfrac{\frac{3}{4}}{\frac{5}{8}} \qquad \cfrac{\frac{x}{2}}{\frac{5}{6}}

To simplify a complex fraction, we remember that the fraction bar means division. For example,

\cfrac{\frac{3}{4}}{\frac{5}{8}} \quad\text{means}\quad \frac{3}{4} \div \frac{5}{8}.

Example. Simplify the complex fraction

\cfrac{\frac{3}{4}}{\frac{5}{8}}.

Rewrite as division: $\tfrac{3}{4} \div \tfrac{5}{8}$ . Multiply the first fraction by the reciprocal of the second: $\tfrac{3}{4} \cdot \tfrac{8}{5}$ . Multiply: $\tfrac{3 \cdot 8}{4 \cdot 5}$ . Look for common factors: $\tfrac{3 \cdot \cancel{4} \cdot 2}{\cancel{4} \cdot 5}$ . Divide out common factors and simplify: $\tfrac{6}{5}$ .

Simplify the complex fraction: $\tfrac{2}{3}$ over $\tfrac{5}{6}$ .

Simplify the complex fraction: $\tfrac{3}{7}$ over $\tfrac{6}{11}$ .

Example. Simplify the complex fraction

\cfrac{\frac{x}{2}}{\frac{xy}{6}}.

Rewrite as division: $\tfrac{x}{2} \div \tfrac{xy}{6}$ . Multiply the first fraction by the reciprocal of the second: $\tfrac{x}{2} \cdot \tfrac{6}{xy}$ . Multiply: $\tfrac{x \cdot 6}{2 \cdot xy}$ . Look for common factors: $\tfrac{\cancel{x} \cdot 3 \cdot 2}{2 \cdot \cancel{x} \cdot y}$ . Divide out common factors and simplify: $\tfrac{3}{y}$ .

Simplify the complex fraction: $\tfrac{a}{8}$ over $\tfrac{ab}{6}$ . Use $a$ and $b$ in your answer.

Simplify the complex fraction: $\tfrac{p}{2}$ over $\tfrac{pq}{8}$ . Use $p$ and $q$ in your answer.

Simplify expressions with a fraction bar

The line that separates the numerator from the denominator in a fraction is called a fraction bar. A fraction bar acts as a grouping symbol. The order of operations then tells us to simplify the numerator and then the denominator. Then we divide.

To simplify the expression

\frac{5-3}{7+1},

we first simplify the numerator and the denominator separately. Then we divide:

\frac{5-3}{7+1} = \frac{2}{8} = \frac{1}{4}.

Simplify an expression with a fraction bar.

Simplify the expression in the numerator. Simplify the expression in the denominator.
Simplify the fraction.

Example. Simplify:

\frac{4 - 2(3)}{2^2 + 2}.

Use the order of operations to simplify the numerator and the denominator, then divide — a negative divided by a positive is negative:

\frac{4 - 2(3)}{2^2 + 2} = \frac{4-6}{4+2} = \frac{-2}{6} = -\frac{1}{3}.

Simplify: $\tfrac{6 - 3(5)}{3^2 + 3}$ .

Simplify: $\tfrac{4 - 4(6)}{3^2 + 3}$ .

Where does the negative sign go in a fraction? Usually the negative sign is in front of the fraction, but you will sometimes see a fraction with a negative numerator, or sometimes with a negative denominator. Remember that fractions represent division. When the numerator and denominator have different signs, the quotient is negative.

\frac{-1}{3} = -\frac{1}{3} \qquad \frac{\text{negative}}{\text{positive}} = \text{negative}

\frac{1}{-3} = -\frac{1}{3} \qquad \frac{\text{positive}}{\text{negative}} = \text{negative}

Placement of negative sign in a fraction. For any positive numbers $a$ and $b$ ,

\frac{-a}{b} = \frac{a}{-b} = -\frac{a}{b}

Example. Simplify:

\frac{4(-3) + 6(-2)}{-3(2) - 2}.

Multiply, simplify, and divide:

\frac{4(-3) + 6(-2)}{-3(2) - 2} = \frac{-12 + (-12)}{-6-2} = \frac{-24}{-8} = 3.

Simplify: $\tfrac{8(-2) + 4(-3)}{-5(2) + 3}$ .

Simplify: $\tfrac{7(-1) + 9(-3)}{-5(3) - 2}$ .

Translate phrases to expressions with fractions

Now that we have done some work with fractions, we are ready to translate phrases that would result in expressions with fractions.

The English words quotient and ratio are often used to describe fractions. Remember that “quotient” means division. The quotient of $a$ and $b$ is the result we get from dividing $a$ by $b$ , or $\tfrac{a}{b}$ .

Example. Translate the English phrase into an algebraic expression: the quotient of the difference of $m$ and $n$ , and $p$ .

We are looking for the quotient of the difference of $m$ and $n$ , and $p$ . This means we want to divide the difference of $m$ and $n$ by $p$ : $\tfrac{m-n}{p}$ .

Translate to an algebraic expression: the quotient of the difference of $a$ and $b$ , and $cd$ . Use $a$ , $b$ , $c$ , and $d$ in your answer.

Translate to an algebraic expression: the quotient of the sum of $p$ and $q$ , and $r$ . Use $p$ , $q$ , and $r$ in your answer.

Key terms

fraction — a way to represent parts of a whole, written $\tfrac{a}{b}$ where $b \neq 0$ ; $a$ is the numerator and $b$ is the denominator. equivalent fractions — fractions that have the same value. simplified fraction — a fraction with no common factors, other than $1$ , in its numerator and denominator. reciprocal — the fraction formed by inverting a given fraction; a number and its reciprocal multiply to $1$ . complex fraction — a fraction in which the numerator or the denominator itself contains a fraction. fraction bar — the line separating the numerator from the denominator, which acts as a grouping symbol.

This section is adapted from Elementary Algebra 2e, Section 1.5: Visualize Fractions by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the fraction-circle figure as an accessible inline graphic and described the pizza and quarters figures in prose; omitted the Be Prepared checklist, Manipulative Mathematics callouts, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.

Multiply and Divide Integers Add and Subtract Fractions