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# 2.10: Division of Rational Numbers

Difficulty Level: Basic Created by: CK-12
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Practice Division of Rational Numbers

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Suppose a box of cereal is 45\begin{align*}\frac{4}{5}\end{align*} full, and you want to divide the remaining cereal into portions so that each portion is 15\begin{align*}\frac{1}{5}\end{align*} 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\begin{align*}a\end{align*}, there is a multiplicative inverse 1a\begin{align*}\frac{1}{a}\end{align*} such that a(1a)=1\begin{align*}a \left ( \frac{1}{a} \right ) = 1\end{align*}.

This means that the multiplicative inverse of a\begin{align*}a\end{align*} is 1a\begin{align*}\frac{1}{a}\end{align*}. The values of a\begin{align*}a\end{align*} and 1a\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 ab\begin{align*}\frac{a}{b}\end{align*} is ba\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 29÷37\begin{align*}\frac{2}{9} \div \frac{3}{7}\end{align*}.

Solution:

Begin by multiplying by the “right” reciprocal.

29×73=1427\begin{align*}\frac{2}{9} \times \frac{7}{3} = \frac{14}{27}\end{align*}

#### Example B

Simplify 73÷23\begin{align*}\frac{7}{3} \div \frac{2}{3}\end{align*}.

Solution:

Begin by multiplying by the “right” reciprocal.

73÷23=73×32=7323=72\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 2378\begin{align*}\frac{\frac{2}{3}}{\frac{7}{8}}\end{align*}.

Solution:

The fraction bar separating 23\begin{align*}\frac{2}{3}\end{align*} and 78\begin{align*}\frac{7}{8}\end{align*} indicates division.

23÷78\begin{align*}\frac{2}{3} \div \frac{7}{8}\end{align*}

Simplify as in Example B:

23×87=1621\begin{align*}\frac{2}{3} \times \frac{8}{7} = \frac{16}{21}\end{align*}

### Guided Practice

1. Find the multiplicative inverse of 57\begin{align*}\frac{5}{7}\end{align*}.

2. Simplify 5÷32\begin{align*} 5\div \frac{3}{2}\end{align*}.

Solutions:

1. The multiplicative inverse of 57\begin{align*}\frac{5}{7}\end{align*} is 75.\begin{align*}\frac{7}{5}.\end{align*} We can see that by multiplying them together:

57×75=5×77×5=3535=1.\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:

5÷32=5×23=5×23=103\begin{align*} 5\div \frac{3}{2}=5 \times \frac{2}{3}=\frac{5\times 2}{3}=\frac{10}{3}\end{align*}

### 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. 28\begin{align*}\frac{2}{8}\end{align*}
3. 1921\begin{align*}-\frac{19}{21}\end{align*}
4. 7
5. z32xy2\begin{align*}- \frac{z^3}{2xy^2}\end{align*}
6. 0
7. 13\begin{align*}\frac{1}{3}\end{align*}
8. 1918\begin{align*}\frac{-19}{18}\end{align*}
9. 3xy8z\begin{align*}\frac{3xy}{8z}\end{align*}

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

1. 52÷14\begin{align*}\frac{5}{2} \div \frac{1}{4}\end{align*}
2. 12÷79\begin{align*}\frac{1}{2} \div \frac{7}{9}\end{align*}
3. 511÷67\begin{align*}\frac{5}{11} \div \frac{6}{7}\end{align*}
4. 12÷12\begin{align*}\frac{1}{2} \div \frac{1}{2}\end{align*}
5. x2÷57\begin{align*}- \frac{x}{2} \div \frac{5}{7}\end{align*}
6. 12÷x4y\begin{align*}\frac{1}{2} \div \frac{x}{4y}\end{align*}
7. (13)÷(35)\begin{align*}\left ( - \frac{1}{3} \right ) \div \left ( - \frac{3}{5} \right )\end{align*}
8. 72÷74\begin{align*}\frac{7}{2} \div \frac{7}{4}\end{align*}
9. \begin{align*}11 \div \left ( - \frac{x}{4} \right ) \end{align*}

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### Vocabulary Language: English Spanish

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

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 $152 \div 6$, 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.

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