Multiply and Divide Fractions

4.2 Multiply and Divide Fractions

The topics covered in this section are:

  1. Simplify fractions
  2. Multiply fractions
  3. Find reciprocals
  4. Divide fractions

4.2.1 Simplify Fractions

In working with equivalent fractions, you saw that there are many ways to write fractions that have the same value, or represent the same part of the whole. How do you know which one to use? Often, we’ll use the fraction that is in simplified form.

A fraction is considered simplified if there are no common factors, other than $1$, in the numerator and denominator. If a fraction does have common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors.

SIMPLIFIED FRACTION

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

For example,

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

The process of simplifying a fraction is often called reducing the fraction. In the previous section, we used the Equivalent Fractions Property to find equivalent fractions. We can also use the Equivalent Fractions Property in reverse to simplify fractions. We 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

$\frac{a}{b} = \frac{a \cdot c}{b \cdot c}$ and $\frac{a \cdot c}{b \cdot c} = \frac{a}{b}$.

Notice that $c$ is a common factor in the numerator and denominator. Anytime we have a common factor in the numerator and denominator, it can be removed.

HOW TO: Simplify a fraction.

  1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.
  2. Simplify, using the equivalent fractions property, by removing common factors.
  3. Multiply any remaining factors.

Example 1

Simplify: $\frac{10}{15}$.

Solution

To simplify the fraction, we look for any common factors in the numerator and the denominator.

Notice that $5$ is a factor of both $10$ and $15$.$\frac{10}{15}$
Factor the numerator and denominator..
Remove the common factors..
Simplify.$\frac{2}{5}$

To simplify a negative fraction, we use the same process as in Example 1. Remember to keep the negative sign.

Example 2

Simplify: $- \frac{18}{24}$.

Solution

We notice that $18$ and $24$ both have factors of $6$.$- \frac{18}{24}$
Rewrite the numerator and denominator showing the common factor..
Remove common factors..
Simplify.$- \frac{3}{4}$

After simplifying a fraction, it is always important to check the result to make sure that the numerator and denominator do not have any more factors in common. Remember, the definition of a simplified fraction: a fraction is considered simplified if there are no common factors in the numerator and denominator.

When we simplify an improper fraction, there is no need to change it to a mixed number.

Example 3

Simplify: $- \frac{56}{32}$.

Solution

$- \frac{56}{32}$
Rewrite the numerator and denominator, showing the common factors, $8$..
Remove common factors..
Simplify.$- \frac{7}{4}$

HOW TO: Simplify a fraction.

  1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.
  2. Simplify, using the equivalent fractions property, by removing common factors.
  3. Multiply any remaining factors.

Sometimes it may not be easy to find common factors of the numerator and denominator. A good idea, then, is to factor the numerator and the denominator into prime numbers. (You may want to use the factor tree method to identify the prime factors.) Then divide out the common factors using the Equivalent Fractions Property.

Example 4

Simplify: $\frac{210}{385}$.

Solution

Use factor trees to factor the numerator and denominator.$\frac{210}{385}$
.
Rewrite the numerator and denominator as the product of the primes.$\frac{210}{385} = \frac{2 \cdot 3 \cdot 5 \cdot 7}{5 \cdot 7 \cdot 11}$
Remove the common factors..
Simplify.$\frac{2 \cdot 3}{11}$
Multiply any remaining factors.$\frac{6}{11}$

We can also simplify fractions containing variables. If a variable is a common factor in the numerator and denominator, we remove it just as we do with an integer factor.

Example 5

Simplify: $\frac{5xy}{15x}$.

Solution
$\frac{5xy}{15x}$
Rewrite numerator and denominator showing common factors.$\frac{5 \cdot x \cdot y}{3 \cdot 5 \cdot x}$
Remove common factors.$\frac{\cancel{5} \cdot \cancel{x} \cdot \cancel{y}}{\cancel{3} \cdot \cancel{5} \cdot \cancel{x}}$
Simplify.$\frac{y}{3}$

4.2.2 Multiply Fractions

A model may help you understand multiplication of fractions. We will use fraction tiles to model $\frac{1}{2} \cdot \frac{3}{4}$. To multiply $\frac{1}{2}$ and $\frac{3}{4}$, think $\frac{1}{2}$ of $\frac{3}{4}$.

Start with fraction tiles for three-fourths. To find one-half of three-fourths, we need to divide them into two equal groups. Since we cannot divide the three $\frac{1}{4}$ tiles evenly into two parts, we exchange them for smaller tiles.

A rectangle is divided vertically into three equal pieces. Each piece is labeled as one fourth. There is a an arrow pointing to an identical rectangle divided vertically into six equal pieces. Each piece is labeled as one eighth. There are braces showing that three of these rectangles represent three eighths.

We see $\frac{6}{8}$ is equivalent to $\frac{3}{4}$. Taking half of the six $\frac{1}{8}$ tiles gives us three $\frac{1}{8}$ tiles, which is $\frac{3}{8}$.

Therefore,

$\frac{1}{2} \cdot \frac{3}{4} = \frac{3}{8}$

Example 6

Us a diagram to model $\frac{1}{2} \cdot \frac{3}{4}$.

Solution

First shade in $\frac{3}{4}$ of the rectangle.

A rectangle is shown, divided vertically into four equal pieces. Three of the pieces are shaded.

We will take $\frac{1}{2}$ of this $\frac{3}{4}$, so we heavily shade $\frac{1}{2}$ of the shaded region.

A rectangle is shown, divided vertically into four equal pieces. Three of the pieces are shaded. The rectangle is divided by a horizontal line, creating eight equal pieces. Three of the eight pieces are darkly shaded.

Notice that $3$ out of the $8$ pieces are heavily shaded. This means that $\frac{3}{8}$ of the rectangle is heavily shaded.

Therefore, $\frac{1}{2}$ of $\frac{3}{4}$ is $\frac{3}{8}$, or $\frac{1}{2} \cdot \frac{3}{4} = \frac{3}{8}$

Look at the result we got from the model in Example 6. We found that $\frac{1}{2} \cdot \frac{3}{4} = \frac{3}{8}$. Do you notice that we could have gotten the same answer by multiplying the numerators and multiplying the denominators?

$\frac{1}{2} \cdot \frac{3}{4}$
Multiply the numerators, and multiply the denominators.$\frac{1}{2} \cdot \frac{3}{4}$
Simplify.$\frac{3}{8}$

This leads to the definition of fraction multiplication. To multiply fractions, we multiply the numerators and multiply the denominators. Then we write the fraction in simplified form.

FRACTION MULTIPLICATION

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

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

Example 7

Multiply, and write the answer is simplified form: $\frac{3}{4} \cdot \frac{1}{5}$.

Solution

$\frac{3}{4} \cdot \frac{1}{5}$
Multiply the numerators; multiply the denominators.$\frac{3 \cdot 1}{4 \cdot 5}$
Simplify.$\frac{3}{20}$

There are no common factors, so the fraction is simplified.

When multiplying fractions, the properties of positive and negative numbers still apply. It is a good idea to determine the sign of the product as the first step. In Example 8 we will multiply two negatives, so the product will be positive.

Example 8

Multiply, and write the answer is simplified form: $- \frac{5}{8} (- \frac{2}{3})$.

Solution

$- \frac{5}{8} (- \frac{2}{3})$
The signs are the same, so the product is positive. Multiply the numerators, multiply the denominators.$\frac{5 \cdot 2}{8 \cdot 3}$
Simplify.$\frac{10}{24}$
Look for common factors in the numerator and denominator. Rewrite showing common factors.$\frac{5 \cdot \cancel{2} }{12 \cdot \cancel{2} }$
Remove common factors.$\frac{5}{12}$

Another way to find this product involves removing common factors earlier.

$- \frac{5}{8} (- \frac{2}{3})$
Determine the sign of the product. Multiply.$\frac{5 \cdot 2}{8 \cdot 3}$
Show common factors and then remove them.$\frac{5 \cdot \cancel{2} }{4 \cdot \cancel{2} \cdot 3}$
Multiply remaining factors.$\frac{5}{12}$

We get the same result.

Example 9

Multiply, and write the answer is simplified form: $- \frac{14}{15} \cdot \frac{20}{21}$.

Solution

$- \frac{14}{15} \cdot \frac{20}{21}$
Determine the sign of the product; multiply.$- \frac{14}{15} \cdot \frac{20}{21}$
Are there any common factors in the numerator and the denominator?

We know that $7$ is a factor of $14$ and $21$, and $5$ is a factor of $20$ and $15$.
Rewrite showing common factors.$- \frac{2 \cdot \cancel{7} \cdot 4 \cdot \cancel{5} }{3 \cdot \cancel{5} \cdot 3 \cdot \cancel{7} }$
Remove the common factors.$- \frac{2 \cdot 4}{3 \cdot 3}$
Multiply the remaining factors.$- \frac{8}{9}$

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 $\frac{a}{1}$. So, $3 = \frac{3}{1}$, for example.

Example 10

Multiply, and write the answer in simplified form:

  1. $\frac{1}{7} \cdot 56$
  2. $\frac{12}{5} (-20x)$
Solution

Part 1.$\frac{1}{4} \cdot 56$
Write $56$ as a fraction.$\frac{1}{7} \cdot \frac{56}{1}$
Determine the sign of the product; multiply.$\frac{56}{7}$
Simplify.$8$
Part 2.$\frac{12}{5} (-20x)$
Write $-20x$ as a fraction.$\frac{12}{5} ( \frac{-20x}{1} )$
Determine the sign of the product; multiply.$- \frac{12 \cdot 20 \cdot x}{5 \cdot 1}$
Show common factors and then remove them.$- \frac{12 \cdot 4 \cdot \cancel{5} x}{\cancel{5} \cdot 1}$
Multiply remaining factors; simplify.$-48x$

4.2.3 Find Reciprocals

The fractions $\frac{2}{3}$ and $\frac{3}{2}$ are related to each other in a special way. So are $- \frac{10}{7}$ and $- \frac{7}{10}$. Do you see how? Besides looking like upside-down versions of one another, if we were to multiply these pairs of fractions, the product would be $1$.

$\frac{2}{3} \cdot \frac{3}{2} = 1$ and $- \frac{10}{7} (- \frac{7}{10} ) = 1$

Such pairs of numbers are called reciprocals.

RECIPROCAL

The reciprocal of the fraction $\frac{a}{b}$ is $\frac{b}{a}$, where $a \neq 0$ and $b \neq 0$,

A number and its reciprocal have a product of $1$.

$\frac{a}{b} \cdot \frac{b}{a} = 1$

To find the reciprocal of a fraction, we invert the fraction. This means that we place the numerator in the denominator and the denominator in the numerator.

To get a positive result when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.

“a” over “b” multiplied by “b” over “a” equals positive one.

To find the reciprocal, keep the same sign and invert the fraction. The number zero does not have a reciprocal. Why? A number and its reciprocal multiply to $1$. Is there any number $r$ so that $0 \cdot r = 1$? No. So, the number $0$ does not have a reciprocal.

Example 11

Find the reciprocal of each number. Then check that the product of each number and its reciprocal is $1$.

  1. $\frac{4}{9}$
  2. $- \frac{1}{6}$
  3. $- \frac{14}{5}$
  4. $7$
Solution

To find the reciprocals, we keep the sign and invert the fractions.

Part 1.
Find the reciprocal of $\frac{4}{9}$The reciprocal of $\frac{4}{9}$ is $\frac{9}{4}$.
Check:
Multiply the number and its reciprocal.$\frac{4}{9} \cdot \frac{9}{4}$
Multiply numerators and denominators.$\frac{36}{36}$
Simplify.$1$✓
Part 2.
Find the reciprocal of $- \frac{1}{6}$$- \frac{6}{1}$.
Simplify.$-6$
Check:$- \frac{1}{6} \cdot (-6)$
$1$✓
Part 3.
Find the reciprocal of $- \frac{14}{5}$$- \frac{5}{14}$.
Check:$- \frac{14}{5} \cdot (- \frac{5}{14} )$
$\frac{70}{70}$
$1$✓
Part 4.
Find the reciprocal of $7$
Write $7$ as a fraction.$\frac{7}{1}$
Write the reciprocal of $\frac{7}{1}$$\frac{1}{7}$
Check:$7 \cdot ( \frac{1}{7} )$
$1$✓

In a previous chapter, we worked with opposites and absolute values. The table below compares opposites, absolute values, and reciprocals.

OppositeAbsolute ValueReciprocal
has opposite signis never negativehas same sign, fraction inverts

Example 12

Fill in the chard for each fraction in the left column:

NumberOppositeAbsolute ValueReciprocal
$- \frac{3}{8}$
$\frac{1}{2}$
$\frac{9}{5}$
$-5$
Solution

To find the opposite, change the sign. To find the absolute value, leave the positive numbers the same, but take the opposite of the negative numbers. To find the reciprocal, keep the sign the same and invert the fraction.

NumberOppositeAbsolute ValueReciprocal
$- \frac{3}{8}$$\frac{3}{8}$$\frac{3}{8}$$- \frac{8}{3}$
$\frac{1}{2}$$- \frac{1}{2}$$\frac{1}{2}$$2$
$\frac{9}{5}$$- \frac{9}{5}$$\frac{9}{5}$$\frac{5}{9}$
$-5$$5$$5$$- \frac{1}{5}$

4.2.4 Divide Fractions

Why is $12 \div 3 = 4$? We previously modeled this with counters. How many groups of $3$ counters can be made from a group of $12$ counters?

Four red ovals are shown. Inside each oval are three grey circles.

There are $4$ groups of $3$ counters. In other words, there are four $3$’s in $12$. So, $12 \div 3 = 4$.

What about dividing fractions? Suppose we want to find the quotient: $\frac{1}{2} \div \frac{1}{6}$. We need to figure out how many $\frac{1}{6}$s there are in $\frac{1}{2}$. We can use fraction tiles to model this division. We start by lining up the half and sixth fraction tiles as shown in the figure below. Notice, htere are three $\frac{1}{6}$ tiles in $\frac{1}{2}$, so $\frac{1}{2} \div \frac{1}{6} = 3$.

A rectangle is shown, labeled as one half. Below it is an identical rectangle split into three equal pieces, each labeled as one sixth.

Example 13

Model: $\frac{1}{4} \div \frac{1}{8}$.

Solution

We want to determine how many $\frac{1}{8}$s are in $\frac{1}{4}$. Start with one $\frac{1}{4}$ tile. Line up $\frac{1}{8}$ tiles underneath the $\frac{1}{4}$ tile.

A rectangle is shown, labeled one fourth. Below it is an identical rectangle split into two equal pieces, each labeled as one eighth.

There are two $\frac{1}{8}$s in $\frac{1}{4}$.

So, $\frac{1}{4} \div \frac{1}{8} = 2$.

Example 14

Model: $2 \div \frac{1}{4}$.

Solution

We are trying to determine how many $\frac{1}{4}$s there are in $2$. We can model this as shown.

Two rectangles are shown, each labeled as 1. Below it are two identical rectangle, each split into four pieces. Each of the eight pieces is labeled as one fourth.

Because there are eight $\frac{1}{4}$s in $2$, $2 \div \frac{1}{4} = 8$.

Let’s use money to model $2 \div \frac{1}{4}$ in another way. We often read $\frac{1}{4}$ as a ‘quarter’, and we know that a quarter is one-fourth of a dollar as shown in the figure below. So w can think of $2 \div \frac{1}{4}$ as, “How many quarters are there in two dollars?” One dollar is $4$ quarters, so $2$ dollars would be $8$ quarters. So again, $2 \div \frac{1}{4} = 8$.

A picture of a United States quarter is shown.
The U.S. coin called a quarter is worth one-fourth of a dollar.

Using fraction tiles, we showed that $\frac{1}{2} \div \frac{1}{6} = 3$. Notice that $\frac{1}{2} \cdot \frac{6}{1} = 3$ also. How are $\frac{1}{6}$ and $\frac{6}{1}$ related? They are reciprocals. This leads us to the procedure for fraction division.

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, 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 15

Divide, and write the answer in simplified form: $\frac{2}{5} \div (- \frac{3}{7} )$.

Solution

$\frac{2}{5} \div (- \frac{3}{7} )$
Multiply the first fraction by the reciprocal of the second.$\frac{2}{5} (- \frac{7}{3} )$
Multiply. The product is negative.$- \frac{14}{15}$

Example 16

Divide, and write the answer in simplified form: $\frac{2}{3} \div (\frac{n}{5} )$.

Solution

$\frac{2}{3} \div (\frac{n}{5} )$
Multiply the first fraction by the reciprocal of the second.$\frac{2}{3} \cdot \frac{5}{n}$
Multiply.$\frac{10}{3n}$

Example 17

Divide, and write the answer in simplified form: $- \frac{3}{4} \div (- \frac{7}{8} )$.

Solution

$- \frac{3}{4} \div (- \frac{7}{8} )$
Multiply the first fraction by the reciprocal of the second.$- \frac{3}{4} \cdot (- \frac{8}{7} )$
Multiply. Remember to determine the sign first.$\frac{3 \cdot 8}{4 \cdot 7}$
Rewrite to show common factors.$\frac{3 \cdot \cancel{4} \cdot 2}{\cancel{4} \cdot 7}$
Remove common factors and simplify.$\frac{6}{7}$

Example 18

Divide, and write the answer in simplified form: $\frac{7}{18} \div (\frac{14}{27} )$.

Solution

$\frac{7}{18} \div (\frac{14}{27} )$
Multiply the first fraction by the reciprocal of the second.$\frac{7}{18} \cdot (\frac{14}{27} )$
Multiply.$\frac{7 \cdot 27}{18 \cdot 14}$
Rewrite showing common factors.$\frac{\cancel{7} \cdot \cancel{9} \cdot 3}{\cancel{9} \cdot 2 \cdot \cancel{7} \cdot 2}$
Remove common factors.$\frac{3}{2 \cdot 2}$
Simplify.$\frac{3}{4}$
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