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# 2.10: Division of Rational Numbers

Difficulty Level: Basic Created by: CK-12
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Suppose a box of cereal is $\frac{4}{5}$ full, and you want to divide the remaining cereal into portions so that each portion is $\frac{1}{5}$ of the full box. In this case, you would have to divide a fraction by a fraction to come up with the number of portions you could make. After completing this Concept, you'll be able to use reciprocals to perform division problems such as these.

### 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 $a$ , there is a multiplicative inverse $\frac{1}{a}$ such that $a \left ( \frac{1}{a} \right ) = 1$ .

This means that the multiplicative inverse of $a$ is $\frac{1}{a}$ . The values of $a$ and $\frac{1}{a}$ 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 $\frac{a}{b}$ is $\frac{b}{a}$ .

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 $\frac{2}{9} \div \frac{3}{7}$ .

Solution:

Begin by multiplying by the “right” reciprocal.

$\frac{2}{9} \times \frac{7}{3} = \frac{14}{27}$

#### Example B

Simplify $\frac{7}{3} \div \frac{2}{3}$ .

Solution:

Begin by multiplying by the “right” reciprocal.

$\frac{7}{3} \div \frac{2}{3} = \frac{7}{3} \times \frac{3}{2} = \frac{7 \cdot 3} {2 \cdot 3} = \frac{7}{2}$

Instead of the division symbol $\div$ , you may see a large fraction bar. This is seen in the next example.

#### Example C

Simplify $\frac{\frac{2}{3}}{\frac{7}{8}}$ .

Solution:

The fraction bar separating $\frac{2}{3}$ and $\frac{7}{8}$ indicates division.

$\frac{2}{3} \div \frac{7}{8}$

Simplify as in Example B:

$\frac{2}{3} \times \frac{8}{7} = \frac{16}{21}$

### Guided Practice

1. Find the multiplicative inverse of $\frac{5}{7}$ .

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

Solutions:

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

$\frac{5}{7}\times \frac{7}{5}= \frac{5\times 7}{7\times 5}=\frac{35}{35}=1.$

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

$5\div \frac{3}{2}=5 \times \frac{2}{3}=\frac{5\times 2}{3}=\frac{10}{3}$

### Practice

1. Define inverse.
2. What is a multiplicative inverse? How is this different from an additive inverse?

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

1. 100
2. $\frac{2}{8}$
3. $-\frac{19}{21}$
4. 7
5. $- \frac{z^3}{2xy^2}$
6. 0
7. $\frac{1}{3}$
8. $\frac{-19}{18}$
9. $\frac{3xy}{8z}$

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

1. $\frac{5}{2} \div \frac{1}{4}$
2. $\frac{1}{2} \div \frac{7}{9}$
3. $\frac{5}{11} \div \frac{6}{7}$
4. $\frac{1}{2} \div \frac{1}{2}$
5. $- \frac{x}{2} \div \frac{5}{7}$
6. $\frac{1}{2} \div \frac{x}{4y}$
7. $\left ( - \frac{1}{3} \right ) \div \left ( - \frac{3}{5} \right )$
8. $\frac{7}{2} \div \frac{7}{4}$
9. $11 \div \left ( - \frac{x}{4} \right )$

### Vocabulary Language: English Spanish

Inverse Property of Multiplication

Inverse Property of Multiplication

For every nonzero number $a$, there is a multiplicative inverse $\frac{1}{a}$ such that $a \left ( \frac{1}{a} \right ) = 1$. This means that the multiplicative inverse of $a$ is $\frac{1}{a}$.
reciprocal

reciprocal

The reciprocal of a nonzero rational number $\frac{a}{b}$ is $\frac{b}{a}$.

Basic

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Feb 24, 2012

Aug 21, 2014