Division of Rational Numbers (Definition, Types and Examples) - BYJUS

# Division of Rational Numbers

Rational numbers include all natural numbers, whole numbers, and integers. We have learned how to perform division operations on natural numbers, whole numbers, and integers. Here we will apply the same concept to divide two rational numbers. ...Read MoreRead Less

## Division of Rational Numbers

The dividend is the number that must be divided and the divisor is the number that is divided by the dividend. The result of dividing the dividend by the divisor is known as the quotient. When solving related problems, keep in mind that division by 0 is not defined.

Use the same rules for signs as you did for integers to divide rational numbers

• Dividing a positive rational number and a negative rational number will result in a negative quotient.
• Dividing two positive rational numbers will result in a positive quotient.
• Dividing two negative rational numbers will result in a positive quotient.

## Dividing Rational Numbers with the Same Sign

The quotient of two rational numbers with the same sign is positive.

For example, $$-\frac{1}{3}\div \left(-\frac{4}{3}\right)=?$$

$$-\frac{1}{3}\div \left(-\frac{4}{3}\right)=-\frac{1}{3}\times \left(-\frac{3}{4}\right)=\frac{1}{4}$$

## Dividing Rational Numbers with Different Signs

The quotient of two rational numbers with the different signs is negative.

For example,$$-\frac{1}{3}\div \left(-\frac{4}{3}\right)=?$$

$$-\frac{1}{3}\div \left(-\frac{4}{3}\right)=-\frac{1}{3}\times \left(\frac{3}{4}\right)=-\frac{1}{4}$$

## Using Complex Fractions to Divide the Rational Numbers

An expression with a fraction in the denominator or the numerator, or both, is called a complex fraction. A complex rational expression is a complex fraction that contains a variable.

For example, $$5\frac{1}{3}$$ is a complex fraction in which the numerator is 5 and the denominator is $$\frac{1}{3}$$

## Dividing Rational Numbers in the Form of Decimals

1. Find: -8.2 (-3.8) = ?

Solution: Since the decimals have the same sign, the quotient is positive.

First, we express the divisor as a whole number

$$\frac{-8.2}{-3.8}=\frac{8.2\times 10}{3.3\times 10}=\frac{82}{33}$$

Using long division to divide 82 and 38.

$$\underline{2.484}$$…

33)82.00

-66

———-

160

– 132

———-

280

– 264

———-

160

Since the remainder is getting repeated, it is a repeating decimal.

So that, $$-8.2\div (-3.3)=2.484…=2.\overline{48}$$

2. Find: -2.2 (3.3) = ?

Solution: Since the decimals have different signs, the quotient will be negative. Use long division to divide 2.2 as 22.00 and 3.4 as 34.

$$\frac{-2.2}{3.3}=\frac{-2.2\times 10}{3.3\times 10}=\frac{-22}{33}$$

$$\underline{0.66}$$…

34)22.00

– 0

_____

220

-198

______

220

-198

_____

22

Since the remainder is getting repeated, it is a repeating decimal.

So that, $$-2.2\div (3.3)=0.666…=-0.\overline{6}$$

3. Find: $$\frac{4}{5}\div \left(-\frac{2}{3}\right)=?$$

Solution: $$\frac{4}{5}\div \left(-\frac{2}{3}\right)=\frac{4}{5}\times \left(-\frac{3}{2}\right)$$              ( Multiplied by the reciprocal of $$-\frac{2}{3}$$)

$$=\frac{4\times (-3)}{5\times 2}$$                                                    (Multiply the numerators and denominators. And dividing out the common factor, 2)

$$=\frac{2\times(-3)}5$$

$$=\frac{-6}{5}$$

So that, $$\frac{4}{5}\div \left(-\frac{2}{3}\right)=\frac{-6}{5}$$

4. Find: $$-\frac{2}{5}\div\left(-\frac{3}{6}\right)=?$$

Solution: $$-\frac{2}{5}\left(-\frac{3}{6}\right)=-\frac{2}{5}\times \left(-\frac{6}{3}\right)$$          ( Multiplied by the reciprocal of $$-\frac{3}{6}$$)

$$=\frac{-2\times(-6)}{5\times 3}$$                                             (Multiply the numerators and denominators)

$$=\frac{12\div 3}{15\div 3}$$

$$=\frac{4}{5}$$

So that, $$-\frac{2}{5}\div \left(-\frac{3}{6}\right)=\frac{4}{5}$$

## Examples of Using Complex Fractions to Divide Rational Numbers

1. Evaluate: $$\frac{-\frac{9}{5}}{\frac{1}{5}+1}=?$$

Solution:

$$-\frac{9}{5}\div \left(-\frac{1}{5}+1\right)=-\frac{9}{5}\div \left(\frac{-1}{5}+\frac{5}{5}\right)$$            (Rewriting $$-\frac{1}{5}$$ as $$\frac{-1}{5}$$ and 1 as $$\frac{5}{5}$$)

$$=-\frac{9}{5}\div \frac{4}{5}$$                                                    (Adding the fractions)

$$=-\frac{9}{5}\times \frac{5}{4}$$                                                    (Multiply by the reciprocal of  $$\frac{4}{5}$$ )

$$=-\frac{9}{5}\times \frac{5}{4}$$

$$=-\frac{9}{4}\times$$

$$-\frac{9}{5}\div \left(-\frac{1}{5}+1\right)=-\frac{9}{4}$$                                 (Simplified)

2. Evaluate: $$\frac{-\frac{7}{5}}{\frac{9}{2}}=?$$

Solution: We have to rewrite the complex fraction as a divisional expression.

$$-\frac{7}{5}\div \left(\frac{9}{2}\right)=-\frac{7}{5}\div \left(\frac{9}{2}\right)$$

$$=-\frac{7}{5}\times \frac{2}{9}$$                                           (Multiply by the reciprocal of $$=\frac{9}{2}$$)

$$= -\frac{7\times 2}{5\times 9}$$

$$=-\frac{14}{45}$$

$$=-\frac{7}{5}\div \left(\frac{9}{2}\right)=-\frac{14}{45}$$

## Real-life Examples of Dividing Rational Numbers

1. A math test was conducted that comprised 10 questions. If an answer is right, the student is rewarded with +1 for each question, but if the answer is incorrect, the student is rewarded with -1 for that question. The scores for 8 students are: 8.25, -6.5, 0, -7.5, 4, -8.5, 6, and 3.5. Find the average of scores.

Solution:

Mean $$=\frac{8.25-6.5+0-7.5+4-8.5+6+ 3.5}{8}$$

$$=\frac{= 0+ 12.25 – 14 + 9.5 – 8.5}{8}$$

$$=\frac{-3}{32}$$ or -0.09375

Frequently Asked Questions on Division of Rational Numbers

There are three rules for dividing rational numbers:

• A positive rational number and a negative rational number have a negative quotient.
• Two positive rational numbers have a positive quotient.
• Two negative rational numbers have a positive quotient.

Let us understand this with the help of an example. Take $$=\frac{22}{7}$$. We know that both 22 and 7 are rational numbers. But $$=\frac{22}{7}$$ is Pi, which is 3.141592…. The digits in the decimal part are not getting repeated. Hence, it is not a repeating decimal and the quotient is not rational. Hence, when two rational numbers are divided, the quotient need not always be rational.