How do you Add and Subtract Rational Numbers?

How do you Add and Subtract Rational Numbers?

There are four basic operations on rational numbers :

  1. Addition
  2. Subtraction
  3. Multiplication
  4. Division.

Addition of Rational Numbers

If two rational numbers are to be added, we first express each one of them as rational number with positive denominator.
There are two possibilities :
(1) Either they have same denominators, or
(2) They have different denominators.

Adding Rational Numbers with Same Denominator:

Let us add \(\frac { 8 }{ 5 }\) and \(\frac { -6 }{ 5 }\)
Represent the numbers on the number line.
How do you Add and Subtract Rational Numbers 1
Here, the distance between two consecutive points is \(\frac { -6 }{ 5 }\). For  \(\frac { 8 }{ 5 }\), move 6 steps to the left of \(\frac { 8 }{ 5 }\) and we reach at \(\frac { 2 }{ 5 }\).
How do you Add and Subtract Rational Numbers 2
Example 1: Add : \(\frac { -5 }{ 9 }\) and \(\frac { -17 }{ 9 }\).
Solution:
Given rational numbers are \(\frac { -5 }{ 9 }\) and \(\frac { -17 }{ 9 }\).
Adding these two numbers, we have
How do you Add and Subtract Rational Numbers 3
Which is the required answer.

Example 2: Add : \(\frac { -23 }{ 28 }\) and \(\frac { 5 }{ -28 }\).
Solution:
We first express \(\frac { 5 }{ -28 }\) as a rational number with positive denominator.
How do you Add and Subtract Rational Numbers 4

Addition of Rational numbers with Different Denominators:
In this case, we convert the given rational numbers to a common denominator and then add.

Examples:

1. Add \(\frac { 8 }{ -5 }\) and \(\frac { 4 }{ -3 }\).
Solution:
The given rational numbers are \(\frac { 8 }{ -5 }\) and \(\frac { 4 }{ -3 }\). Clearly, they have different denominators. Here, first we express the given rational numbers into standard forms.
How do you Add and Subtract Rational Numbers 5
Which is the required answer.

2. Add : \(\frac { 8 }{ 10 }\), 3.
Solution: 
How do you Add and Subtract Rational Numbers 6

3. Simplify : \(\frac { 4 }{ 3 } +\frac { 3 }{ 5 } +\frac { -2 }{ 5 } +\frac { -11 }{ 3 }\)
Solution:
How do you Add and Subtract Rational Numbers 7

4. Add \(\frac { 7 }{ 9 }\) and \(\frac { -5 }{ 9 }\).
Solution:
How do you Add and Subtract Rational Numbers 8
In case, if denominator of the rational number is negative, first we make it (denominator) Positive and then add.

5. Add : \(\frac { 6 }{ -5 }\) and \(\frac { 4 }{ 5 }\).
Solution:
How do you Add and Subtract Rational Numbers 9

6. Find the sum of \(\frac { -8 }{ 5 }\) and \(\frac { -5 }{ 3 }\)
Solution:
How do you Add and Subtract Rational Numbers 10
Note : Addition of rational numbers is closure (the sum is also rational) commutative (a + b = b + a) and associative(a + (b + c)) = ((a + b) + c).

Additive inverse:
The negative of a rational number is called additive inverse of the given number.
Note: Zero is the only rational no. which is its even negative or inverse.

Subtraction of Rational Numbers

If we add the additive inverse of a rational number and other rational number then this is called subtraction of two rational numbers. So the subtraction is inverse process of addition and the term add the negative of use for subtraction.

Subtraction of Rational Numbers Problems with Solutions

1. Find value of \(\frac { 2 }{ 3 } -\frac { 4 }{ 5 }\).
Solution:
How do you Add and Subtract Rational Numbers 11

2. Find value of \(\frac { 2 }{ 7 } -\left( \frac { -5 }{ 3 }  \right)\).
Solution:
How do you Add and Subtract Rational Numbers 12

3. Subtract \(\frac { -5 }{ 8 }\) from \(\frac { -3 }{ 7 }\).
Solution:
How do you Add and Subtract Rational Numbers 13

4. Simplify : \(\frac { 1 }{ 6 } +\frac { -2 }{ 5 } -\frac { -2 }{ 15 }\).
Solution:
How do you Add and Subtract Rational Numbers 14

5. What number should be added to \(\frac { -5 }{ 8 }\) so that the sum is \(\frac { 5 }{ 9 }\)?
Solution:
How do you Add and Subtract Rational Numbers 15

6. What number should be subtracted from \(\frac { 27 }{ 11 }\) so as to get \(\frac { -5 }{ 33 }\)?
Solution:
How do you Add and Subtract Rational Numbers 16

7. The sum of two rational numbers is \(\frac { -3 }{ 5 }\) If one of them is \(\frac { -9 }{ 10 }\) Find the other.
Solution:
How do you Add and Subtract Rational Numbers 17

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