Operations on Rational Numbers

We have learned that when we divide an integer by another integer we might get an integer or a fraction. In all these cases the number is written in the form of a/b. For example, 10/ 5, 5 /2, 6/5, etc such numbers are called rational numbers. Remember that the denominator cannot be zero in any case.

A rational number may include any positive integer, a negative integer, a whole number, a decimal or a fraction. 

Now let us learn different arithmetic operations like addition, subtraction, multiplication, division on rational numbers.

Rational Number Definition

A rational number a/b is said to be in its standard form if a and b have no common factors other than 1. i.e., a and b are co-primes, where b is 0. For example, 4/5,6/7,2/5 etc., are in the standard form. We can also say that fractions are examples of rational numbers.

A number like 5/10 is not in the standard form of rational numbers, it can be reduced to 1/2. Now ½ is the standard form of a rational number.

To identify if a number is rational or not, it should satisfy the following conditions.

• A number should be represented in the form of a/b, where b ≠ 0.

• The ratio a/b can be further represented in decimal form.

Arithmetic Operations with Rational Numbers

We have carried out arithmetic operations like addition, subtraction, multiplication, and division on integers and fractions. Similarly, we can carry out these operations with rational numbers. Arithmetic operations on rational numbers with the same denominators are easy to calculate but in the case of rational numbers with different denominators, we have to operate after making the denominators the same. Now let us study different arithmetic operations with rational numbers.

Addition Operation on Rational Numbers

Addition of rational numbers has two possibilities.

Consider rational numbers having the same denominator 

For addition of rational numbers we can directly add the numerators.

For example Add 5/7 to 3/7

Here the denominator of both the rational number is 7. To add these numbers, we just keep the denominator the same and add the numerators.

\[= \frac{5}{7}  + \frac{3}{7} \]

\[= \frac{5+3}{7}  \]

\[= \frac{8}{7} \]

Consider Rational Numbers with Different Denominators

For the addition of rational numbers with different denominators first, we have to convert them into rational numbers with the same denominator.

To convert rational numbers with different denominators to the same denominators. We have to find the LCM of rational numbers.

Take the LCM of the denominators, then multiply both rational numbers by the LCM. we will get the rational numbers with the same denominator. Then add it.

For example Add \[= \frac{5}{6}\]  and \[\frac{3}{5} \]

Solution: To evaluate \[= \frac{5}{6}\]  and \[\frac{3}{5} \]

• Convert the rational numbers with the same denominators.

• Find LCM of 6 and 5 is 30

• Multiply 30 by both rational numbers

We get,

\[ \frac{\frac{5}{6}\times{30} + \frac{3}{5}\times{30}}{30} \]

\[ = \frac{25 + 18}{30} \]

\[ = \frac{43}{30} \]

Subtraction Operation on Rational Numbers

Subtracting rational numbers is similar to in addition.

Consider rational numbers having the same denominator 

For subtracting rational numbers with the same denominator we have to simply subtract the numerator.

For example Subtract \[ \frac{5}{7} \] to \[ \frac{3}{7} \]

Here the denominator of both the rational number is 7. To add these numbers, we just keep the denominator the same and add the numerators.

\[ = \frac{5}{7} - \frac{3}{7} \]

\[ = \frac{5-3}{7} \]

\[ = \frac{2}{7} \]

Consider Rational Numbers with Different Denominators

For subtracting rational numbers with different denominators first we have to convert them into rational numbers with the same denominator.

To convert rational numbers with different denominators to the same denominators we have to find the LCM of rational numbers.

Take the LCM of the denominators, then multiply both rational numbers by the LCM. we will get the rational numbers with the same denominator. Then subtract it.

For example, subtract \[= \frac{5}{6}\]  and \[\frac{3}{5} \]

Solution: To evaluate \[  \frac{5}{6} - \frac{3}{5} \]

• Convert the rational numbers with the same denominators.

• LCM of 6 and 5 is 30

• Multiply 30 by both rational numbers

We get,

\[ \frac{\frac{5}{6} \times{30} - \frac{3}{5} \times{30}}{30} \]

\[ = \frac{25 - 18}{30} \]

\[ = \frac{7}{30} \]

Multiplication Operation on Rational Number

For carrying our multiplication of rational numbers we don’t have to convert the different denominators into the same denominators.

Multiplication of rational numbers is equal to the product of numerators divided by the product of denominators. 

Product of rational numbers = product of numerators/ product of denominator

For example, Multiply \[ \frac{2}{8} \] and \[ \frac{(-5)}{6} \]

Solution: For finding multiplication of rational numbers, multiply the numerators by the multiplication of denominators

(\[ \frac{2}{8} \times \frac{(-5)}{6} \])

\[ \frac{2\times (-5)}{8\times6} \]

\[ \frac{-10}{48} \]

Division Operation on Rational Number

For carrying out division on rational numbers we have to multiply the first rational number with the reciprocal of the second rational number.

Reciprocal of a rational number means taking the inverse of the number that is taking the numerator in place of the denominator and the denominator in place of the numerator. For example, the reciprocal of  \[\frac{5}{6}\] is \[\frac{6}{5}\].

Example of division: Divide \[\frac{9}{2}\]by \[\frac{2}{3}\] 

Solution: take the reciprocal of a second rational number and multiply i.e \[\frac{2}{3}\] is \[\frac{3}{2}\]

\[ = \frac{9}{2} \times \frac{2}{3} \]

 \[ = \frac{{27}}{4} \]

Solved Examples

Example 1: Evaluate

\[ = \frac{5}{3} \times \frac{3}{4}\]

Solution:

\[ = \frac{5}{3} \times \frac{3}{4}\]

\[ = \frac{{5 \times 3}}{{2 \times 4}}\]

\[ = \frac{{15}}{8}\]

Example 2: 

\[\frac{{13}}{3} - \left( { - \frac{{24}}{9}} \right) + \frac{{17}}{6}\]

Solution:

\[\frac{{13}}{3} + \frac{{24}}{9} + \frac{{17}}{6}\]

LCM of rational numbers = 18

\[=  \frac{{13 \times 6}}{{3 \times 6}} + \frac{{24 \times 2}}{{9 \times 2}} + \frac{{17 \times 3}}{{6 \times 3}} \]

\[ = \frac{{78 + 48 + 51}}{{18}} \]

\[= \frac{{177}}{{18}} \]

Quiz Time

1. Simplify \[\frac{9}{4}\left( {\frac{1}{3} - \frac{5}{6} + \frac{1}{2}} \right) \div 5\]

2. Divide \[\left( {\frac{{28}}{5}} \right) \div \left( {\frac{{ - 30}}{7}} \right)\]

Properties of Rational Numbers

In general, rational numbers are those that can be written as p/q, where both p and q are integers and q is not zero. The following are the qualities of rational numbers:

• Closure Property

• Commutative Property

• Associative Property

• Distributive Property

• Identity Property

• Inverse Property

Closure Property

The outcomes of addition, subtraction, and multiplication operations for two rational integers say x and y, yield a rational number. When it comes to addition, subtraction, and multiplication, we may claim that they are closed under rational numbers. Consider the following example:

47/30 = (7/6)+(2/5)

Commutative Property

Addition and multiplication satisfy commutative property for rational numbers.

The commutative law of addition states that a+b = b+a.

ab = ba is a commutative multiplication law.

Associative Property

For addition and multiplication, rational numbers follow the associative property.

If x, y, and z are all rational, then the addition is as follows: x+(y+z)=(x+y)+z

x(yz)=(xy)z is the multiplication equation.

Identity Property

For rational numbers, 0 is an additive identity, whereas 1 is a multiplicative identity.

Examples:

1/2 + 0 equals 1/2 (Additive Identity)

1 x 1/2 = 1/2   (Identity Multiplication)

Inverse Property

The additive inverse of the rational number x/y is -x/y, while the multiplicative inverse is y/x.

Examples:

-1/4 is the additive inverse of 1/3. As a result, 1/4 + (-1/4) = 0.

1/3's multiplicative inverse is 3. As a result, 1/3 x 3 = 1