Operating rational numbers

ADDITION

Addition of rational numbers with same denominators: Consider two rational numbers 2/9 and 3/9, here both have different numerators but same denominators. To add these, we add both the numerators keeping common denominator.

Addition of rational numbers with different denominators:

To add rational numbers with different denominators convert them to rational numbers with same denominators and then add.

In order to convert rational numbers to numbers with same denominators take the LCM of the denominators. Then multiply both rational numbers by this LCM and we will get equivalent rational numbers with this LCM as the denominator.

Example 1: Add 7/6 and 2/9

Solution: To evaluate7/6+2/9

• Convert this to rational numbers with same denominators.
• LCM of 6 and 9 is 18
• Multiply 18 by both rational numbers
• Add 21/18 and 4/18

Additive Inverse Opposite of a number is the number with same magnitude of opposite sign. -1 is the opposite of 1. The sum of a number and its opposite is always zero.  This is same for rational numbers also. Two rational numbers are additive inverse of each other when their sum is zero. Thus, all opposite numbers are additive inverse to each other.

Subtraction of rational number

Subtraction of rational numbers with same and/or different denominators can be done like addition. Subtraction of rational numbers will be easy by transforming subtraction problem into addition problem. This can be done in two steps;

• Change the subtraction sign (-) into addition sign (+)
• Take the additive inverse of the rational number come after the sign.

Simply, take the additive inverse of the number after the sign and add them.

Example 2: Subtract  5/2 and -4/7

Solution: We have to subtract 5/2 and -4/7

For this, first convert subtraction problem into addition problem

Take  the additive inverse of the number after the sign. Here, -4/7  is the number after the sign.

Additive inverse of -4/7 is 4/7. Now add 5/2+4/7.

Here, LCM of 2 and 7 are 14.

Add 35/14  and 8/14

Multiplication of rational number

Multiplication of rational numbers is as simple as multiplication of integers. In general, product of two rational numbers will equal to the product of the numerators divided by the product of the denominators.

To find the product of a rational number and an integer, keep the denominator unchanged and multiply the numerators. To find the product of rational numbers, multiply the numerators and multiply denominators and write in the form given above.

Example 3: Multiply 3/8 and -9/11

Solution: To find the product, multiply the numerators and multiply denominators

Example 4: Divide 11/12 and 7

Solution:To find the product, multiply the numerators and keep the denominator unchanged.

We know, 7=1/7

Division of rational number

Divisions of rational numbers are similar to the divisions of fractions. To find the quotient of rational numbers, we need to recollect the concept of reciprocal of a rational number. Reciprocal of a rational number is nothing but the swapping of the numerator and denominator of that rational number i.e. invert the given the rational number will give you the reciprocal of that rational number. For example, 7/6 is the reciprocal of 6/7 . The product of a number and its reciprocal will always be 1.

In order to divide rational number, first change the division sign  to . Then take the reciprocal of the number after the sign and multiply.

Example 5: Divide 11/12 and  7/4

Solution: To evaluate 11/12÷7/4

• Change the division sign ÷ to x.
• Take the reciprocal of 7/4 which is equal to 4/7.
• Multiply