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# Division of Rational Numbers

## Multiply the first fraction by the reciprocal of the second fraction

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Division of Rational Numbers

Suppose a box of cereal is \begin{align*}\frac{4}{5}\end{align*} full, and you want to divide the remaining cereal into portions so that each portion is \begin{align*}\frac{1}{5}\end{align*} of the full box. How many portions should you make?

### Dividing Rational Numbers

Previously, you have added, subtracted, and multiplied rational numbers. It now makes sense to learn how to divide rational numbers.

#### Dividing Decimals

You can divide a decimal number by another decimal number using what you know about dividing whole numbers, but placing the decimal back into the quotient can become tricky. Simplify the process by change the divisor to a whole number.

#### Let's complete the following division problem:

\begin{align*}10.4 \div 2.6 = \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

The divisor is 2.6. To change a decimal number to a whole number, multiply the decimal number by a power of ten to move the decimal point. Multiply 2.6 by 10 to move the decimal point one space to the right.

\begin{align*}2.6 \times 10 = 26\end{align*}

Dividing by 26 is easier than dividing by 2.6. However, dividing 10.4 by 26 is not the same as dividing 10.4 by 2.6. If you change the divisor, you must also change the dividend.

Remember that a division problem can also be written as a fraction. The dividend is placed in the numerator and the divisor is placed in the denominator. In this problem, 10.4 is the dividend and is placed in the numerator. 2.6 is the divisor and is placed in the denominator.

\begin{align*}10.4 \div 2.6 = \frac{10.4}{2.6}\end{align*}

If you change the denominator, you must also change the numerator the same way in order for the fraction to have the same value. Multiply the numerator and denominator by 10.

\begin{align*}10.4 \div 2.6 = \frac{10.4 \times 10}{2.6 \times 10} = \frac{104}{26}\end{align*}

Now write the new fraction as a division problem and divide to find the quotient.

\begin{align*}104 \div 26 \quad \text{ or } \quad \overset{}{26 \overline{ ) {104 }}}\end{align*}

\begin{align*}\begin{array}{rcl} && \overset{\quad \ 4}{26 \overline{ ) {104 }}}\\ && \ \ \underline{-104}\\ && \qquad 0 \end{array}\end{align*}

\begin{align*}10.4 \div 2.6\end{align*} is the same as \begin{align*}104 \div 26\end{align*}. The quotient of 10.4 divided by 2.6 is 4.

#### Now, let's do a division problem where the divisor has two decimal places:

\begin{align*}\overset{}{0.45 \overline{ ) {1.44 }}}\end{align*}

First, change the divisor to a whole number by multiplying it by a power of ten. Multiply 0.45 by 100. Then, do the same thing to the dividend.

\begin{align*}\begin{array}{rcl} 0.45 \times 100 & = & 45\\ 1.44 \times 100 & = & 144 \end{array}\end{align*}

Here is the new division problem. Divide to find the quotient.

\begin{align*}\begin{array}{rcl} && \overset{\quad \ 3}{45 \overline{ ) {144 }}}\\ && \ \ \underline{-135\;}\\ && \qquad \ 9 \end{array}\end{align*}

Use zero placeholders to continue dividing.

\begin{align*}\begin{array}{rcl} && \overset{\quad \ 3.2}{45 \overline{ ) {144.{\color{red}0}}}}\\ && \ \ \underline{-135\;}\\ && \qquad \ 9{\color{red}0}\\ && \quad \ \ \underline{-90}\\ && \qquad \ \ \ 0 \end{array}\end{align*}

The quotient of 1.44 divided by 0.45 is 3.2.

Notice the pattern. You move the decimal the same number of spaces in the dividend as you do in the divisor.

\begin{align*}\begin{array}{rcl} 2.6 \rightarrow 2\underrightarrow{6} & & 0.45 \rightarrow \underrightarrow{45}\\ 10.4 \rightarrow 10\underrightarrow{4} & & 1.35 \rightarrow 1\underrightarrow{35} \end{array}\end{align*}

#### Dividing Fractions

To talk about dividing fractions, 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 states that for every nonzero number \begin{align*}a\end{align*}, there is a multiplicative inverse \begin{align*}\frac{1}{a}\end{align*} such that \begin{align*}a \left ( \frac{1}{a} \right ) = 1\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. The reciprocal of a nonzero rational number \begin{align*}\frac{a}{b}\end{align*} is \begin{align*}\frac{b}{a}\end{align*}.

Note that the number zero does not have a reciprocal.

#### Using Reciprocals to Divide Rational Numbers

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.

#### Let's simplify the following expressions:

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

Begin by multiplying by the “right” reciprocal.

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

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

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.

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

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

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

### Examples

#### Example 1

Earlier, you were told to that a box of cereal is \begin{align*}\frac{4}{5}\end{align*} full and that you want to divide the remaining cereal into portions so that each portion is \begin{align*}\frac{1}{5}\end{align*} of the full box. How many portions should you make?

To determine how portions to make, you need to divide the two fractions.

\begin{align*}\frac {4}{5}\div \frac{1}{5}\end{align*}

Rewrite the the problem as a multiplication problem with the reciprocal:

\begin{align*}\frac {4}{5}\div \frac{1}{5}=\frac {4}{5}\times \frac{5}{1}=\frac {4\times5}{5\times 1}=\frac{20}{5}=4\end{align*}

To split the portions so that each portion has exactly \begin{align*}\frac{1}{5}\end{align*} of the full box, you should make 4 equal portions from the cereal remaining in the box.

#### Example 2

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

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*}

#### Example 3

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

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*}

### Review

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. \begin{align*}\frac{2}{8}\end{align*}
3. \begin{align*}-\frac{19}{21}\end{align*}
4. 7
5. \begin{align*}- \frac{z^3}{2xy^2}\end{align*}
6. 0
7. \begin{align*}\frac{1}{3}\end{align*}
8. \begin{align*}\frac{-19}{18}\end{align*}
9. \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. \begin{align*}\frac{5}{2} \div \frac{1}{4}\end{align*}
2. \begin{align*}\frac{1}{2} \div \frac{7}{9}\end{align*}
3. \begin{align*}\frac{5}{11} \div \frac{6}{7}\end{align*}
4. \begin{align*}\frac{1}{2} \div \frac{1}{2}\end{align*}
5. \begin{align*}- \frac{x}{2} \div \frac{5}{7}\end{align*}
6. \begin{align*}\frac{1}{2} \div \frac{x}{4y}\end{align*}
7. \begin{align*}\left ( - \frac{1}{3} \right ) \div \left ( - \frac{3}{5} \right )\end{align*}
8. \begin{align*}\frac{7}{2} \div \frac{7}{4}\end{align*}
9. \begin{align*}11 \div \left ( - \frac{x}{4} \right ) \end{align*}

To see the Review answers, open this PDF file and look for section 2.10.

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### Vocabulary Language: English Spanish

TermDefinition
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.