# 2.10: Division of Rational Numbers

**Basic**Created by: CK-12

**Practice**Division of Rational Numbers

Suppose a box of cereal is \begin{align*}\frac{4}{5}\end{align*}

### Guidance

**Division of Rational Numbers**

Previously, you have added, subtracted, and multiplied rational numbers. It now makes sense to learn how to divide rational numbers. We will begin with a definition of **inverse operations.**

Inverse operations "undo" each other.

For example, addition and subtraction are inverse operations because addition cancels subtraction and vice versa. The additive identity results in a sum of zero. In the same sense, multiplication and division are inverse operations. This leads into the next property: The Inverse Property of Multiplication.

**The Inverse Property of Multiplication:** For every nonzero number \begin{align*}a\end{align*}

This means that the multiplicative inverse of \begin{align*}a\end{align*} is \begin{align*}\frac{1}{a}\end{align*}. The values of \begin{align*}a\end{align*} and \begin{align*}\frac{1}{a}\end{align*} are called also called **reciprocals.** In general, two nonzero numbers whose product is 1 are multiplicative inverses or reciprocals.

**Reciprocal:** The reciprocal of a nonzero rational number \begin{align*}\frac{a}{b}\end{align*} is \begin{align*}\frac{b}{a}\end{align*}.

Note: The number zero does not have a reciprocal.

**Using Reciprocals to Divide Rational Numbers**

When dividing rational numbers, use the following rule:

**“When dividing rational numbers, multiply by the ‘right’ reciprocal.”**

In this case, the “right” reciprocal means to take the reciprocal of the fraction on the right-hand side of the division operator.

#### Example A

*Simplify* \begin{align*}\frac{2}{9} \div \frac{3}{7}\end{align*}.

**Solution:**

Begin by multiplying by the “right” reciprocal.

\begin{align*}\frac{2}{9} \times \frac{7}{3} = \frac{14}{27}\end{align*}

#### Example B

*Simplify* \begin{align*}\frac{7}{3} \div \frac{2}{3}\end{align*}.

**Solution:**

Begin by multiplying by the “right” reciprocal.

\begin{align*}\frac{7}{3} \div \frac{2}{3} = \frac{7}{3} \times \frac{3}{2} = \frac{7 \cdot 3} {2 \cdot 3} = \frac{7}{2}\end{align*}

Instead of the division symbol \begin{align*}\div\end{align*}, you may see a large fraction bar. This is seen in the next example.

#### Example C

*Simplify* \begin{align*}\frac{\frac{2}{3}}{\frac{7}{8}}\end{align*}.

**Solution:**

The fraction bar separating \begin{align*}\frac{2}{3}\end{align*} and \begin{align*}\frac{7}{8}\end{align*} indicates division.

\begin{align*}\frac{2}{3} \div \frac{7}{8}\end{align*}

Simplify as in Example B:

\begin{align*}\frac{2}{3} \times \frac{8}{7} = \frac{16}{21}\end{align*}

### Guided Practice

1. Find the multiplicative inverse of \begin{align*}\frac{5}{7}\end{align*}.

2. Simplify \begin{align*} 5\div \frac{3}{2}\end{align*}.

**Solutions:**

1. The multiplicative inverse of \begin{align*}\frac{5}{7}\end{align*} is \begin{align*}\frac{7}{5}.\end{align*} We can see that by multiplying them together:

\begin{align*}\frac{5}{7}\times \frac{7}{5}= \frac{5\times 7}{7\times 5}=\frac{35}{35}=1.\end{align*}

2. When we are asked to divide by a fraction, we know we can rewrite the problem as multiplying by the reciprocal:

\begin{align*} 5\div \frac{3}{2}=5 \times \frac{2}{3}=\frac{5\times 2}{3}=\frac{10}{3}\end{align*}

### Practice

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Division of Rational Numbers (8:20)

- Define
*inverse.* - What is a multiplicative inverse? How is this different from an additive inverse?

In 3 – 11, find the multiplicative inverse of each expression.

- 100
- \begin{align*}\frac{2}{8}\end{align*}
- \begin{align*}-\frac{19}{21}\end{align*}
- 7
- \begin{align*}- \frac{z^3}{2xy^2}\end{align*}
- 0
- \begin{align*}\frac{1}{3}\end{align*}
- \begin{align*}\frac{-19}{18}\end{align*}
- \begin{align*}\frac{3xy}{8z}\end{align*}

In 12 – 20, divide the rational numbers. Be sure that your answer is in the simplest form.

- \begin{align*}\frac{5}{2} \div \frac{1}{4}\end{align*}
- \begin{align*}\frac{1}{2} \div \frac{7}{9}\end{align*}
- \begin{align*}\frac{5}{11} \div \frac{6}{7}\end{align*}
- \begin{align*}\frac{1}{2} \div \frac{1}{2}\end{align*}
- \begin{align*}- \frac{x}{2} \div \frac{5}{7}\end{align*}
- \begin{align*}\frac{1}{2} \div \frac{x}{4y}\end{align*}
- \begin{align*}\left ( - \frac{1}{3} \right ) \div \left ( - \frac{3}{5} \right )\end{align*}
- \begin{align*}\frac{7}{2} \div \frac{7}{4}\end{align*}
- \begin{align*}11 \div \left ( - \frac{x}{4} \right ) \end{align*}

Inverse Property of Multiplication

For every nonzero number , there is a**such that . This means that the**

*multiplicative inverse***of is .**

*multiplicative inverse*reciprocal

The reciprocal of a number is the number you can multiply it by to get one. The reciprocal of 2 is 1/2. It is also called the multiplicative inverse, or just inverse.Dividend

In a division problem, the dividend is the number or expression that is being divided.divisor

In a division problem, the divisor is the number or expression that is being divided into the dividend. For example: In the expression , 6 is the divisor and 152 is the dividend.identity element

An identity element is a value which, when combined with an operation on another number, leaves that other number unchanged. The identity element for addition is zero, the identity element for multiplication is one.Multiplicative Inverse

The multiplicative inverse of a number is the reciprocal of the number. The product of a real number and its multiplicative inverse will always be equal to 1 (which is the multiplicative identity for real numbers).Quotient

The quotient is the result after two amounts have been divided.Real Number

A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.### Image Attributions

Here you will learn how to use reciprocals to divide a number by a fraction, also known as a rational number.