Before you understand this topic, you are adviced to read:
What are Negative Rational Numbers ?
What are Positive Rational Numbers ?
How to add Positive and Negative Integer ?
How to convert rational number in standard form ?
Positive Rational Numbers are of two types:
Rational Number with Positive Numerator and Denominator
Rational Number with Negative Numerator and Denominator
Negative Rational Number is of two types:
Rational Number with Negative Numerator
Rational Number with Negative Denominator
Based on above classification, you will find following situations:
Addition of Positive Rational Number (with positive numerator and denominator) and Addition of Negative Rational Number (with Negative Numerator), having same denominator
Example: (2/3) + (-1/3)
Addition of Positive Rational Number (with positive numerator and denominator) and Addition of Negative Rational Number (with Negative Denominator), having same denominator
Example: (3/5) + (7/-5)
Addition of Positive Rational Number (with negative numerator and denominator) and Addition of Negative Rational Number (with Negative Numerator), having same denominator
Example: (-1/-4) + (-6/4)
Addition of Positive Rational Number (with negative numerator and denominator) and Addition of Negative Rational Number (with Negative Denominator), having same denominator
Example: (-3/-9) + (4/-9)
Situation 1: Addition of Positive Rational Number (with positive numerator and denominator) and Addition of Negative Rational Number (with Negative Numerator), having same denominator
Steps of addition under this situation are:
Step 1: Since the denominators are same; so we keep the denominator same (i.e. common denominator)
Step 2: Add the numerators. Since numerators have one positive integer and one negative integer, so we will add numerators as we do addition of positive integer and negative integer.
Example 1: Add (2/3), (-1/3)
Solution: Add the given rational numbers and we get:
(2/3) + (-1/3)
Since the denominators are same; so we keep the denominator same (i.e. common denominator) as shown below:
= [(2) + (-1)] / 3
Add numerators as we do addition of positive integer and negative integer & we get:
= 1/3
Hence, (2/3) + (-1/3) = (1/3)
Situation 2: Addition of Positive Rational Number (with positive numerator and denominator) and Addition of Negative Rational Number (with Negative Denominator), having same denominator
Steps of addition under this situation are:
Step 1: Firstly we convert the rational numbers with negative denominator in standard form.
Step 2: Since the denominators are same; so we keep the denominator same (i.e. common denominator)
Step 3: Add the numerators. Since numerators have one positive integer and one negative integer, so we will add numerators as we do addition of positive integer and negative integer.
Example 2: Add (3/5), (7/-5)
Solution: In the given rational numbers there is one rational numbers which have negative denominator i.e. (7/-5). So firstly, convert this rational numbers in standard form and we get:
= (-7/5)
Now, add the rational numbers and we get:
(3/5) + (-7/5)
Since the denominators are same; so we keep the denominator same (i.e. common denominator) as shown below:
= [(3) + (-7)] / 5
Add numerators as we do addition of positive integer and negative integer & we get:
= (-4/5)
Hence, (3/5) + (7/-5) = (-4/5)
Situation 3: Addition of Positive Rational Number (with negative numerator and denominator) and Addition of Negative Rational Number (with Negative Numerator), having same denominator
Steps of addition under this situation are:
Step 1: Firstly we convert the rational numbers with negative denominator in standard form.
Step 2: Since the denominators are same; so we keep the denominator same (i.e. common denominator)
Step 3: Add the numerators. Since numerators have one positive integer and one negative integer, so we will add numerators as we do addition of positive integer and negative integer.
Example 3: Add (-1/-4), (-6/4)
Solution: In the given rational numbers there is one rational numbers which have negative denominator i.e. (-1/-4). So firstly, convert this rational numbers in standard form and we get:
= (1/4)
Now, add the rational numbers and we get:
(1/4) + (-6/4)
Since the denominators are same; so we keep the denominator same (i.e. common denominator) as shown below:
= [(1) + (-6)] / 4
Add numerators as we do addition of positive integer and negative integer & we get:
= (-5/4)
Hence, (-1/-4) + (-6/4) = (-5/4)
Situation 4: Addition of Positive Rational Number (with negative numerator and denominator) and Addition of Negative Rational Number (with Negative Denominator), having same denominator
Steps of addition under this situation are:
Step 1: Firstly we convert the rational numbers with negative denominator in standard form.
Step 2: Since the denominators are same; so we keep the denominator same (i.e. common denominator)
Step 3: Add the numerators. Since numerators have one positive integer and one negative integer, so we will add numerators as we do addition of positive integer and negative integer.
Example 4: Add (-3/-9), (4/-9)
Solution: In the given rational numbers both rational numbers have negative denominators. So firstly, convert this rational numbers in standard form and we get:
= (3/9), (-4/9)
Now, add the rational numbers and we get:
(3/9) + (-4/9)
Since the denominators are same; so we keep the denominator same (i.e. common denominator) as shown below:
= [(3) + (-4)] / 9
Add numerators as we do addition of positive integer and negative integer & we get:
= (-1/9)
Hence, (-3/-9) + (4/-9) = (-1/9)
Above examples 1, 2, 3 & 4 under different situations, must have given you the clarity on how to add a positive and a negative rational numbers having same denominators. Now, in the following examples you can now learn to add more than one; positive and negative rational numbers with same denominators.
Example 5: Add (10/11), (-9/11), (7/11), (-1/11)
Solution: Add the given rational numbers and we get:
(10/11) + (-9/11) + (7/11) + (-1/11)
Since the denominators are same; so we keep the denominator same (i.e. common denominator) as shown below:
= [ (10) + (-9) + (7) + (-1) ] / 11
Add numerators as we do addition of positive integer and negative integer & we get:
= 7/3
Hence, (10/11) + (-9/11) + (7/11) + (-1/11) = (7/3)
Example 6: Add (7/2), (3/-2), (5/2), (6/-2), (9/2)
Solution: In the given rational numbers there are few rational numbers which have negative denominator. So firstly, convert such rational numbers in standard form and we get:
(6/2), (-3/2), (5/2), (-6/2), (9/2)
Now, add the rational numbers and we get:
(6/2) + (-3/2) + (5/2) + (-6/2) + (9/2)
Since the denominators are same; so we keep the denominator same (i.e. common denominator) as shown below:
= [ (6) + (-3) + (5) + (-6) + (9) ] / 2
Add numerators as we do addition of positive integer and negative integer & we get:
= (11/2)
Hence, (6/2) + (3/-2) + (5/2) + (6/-2) + (9/2) = (11/2)
Example 7: Add (-8/-13), (-3/-13), (-2/13), (-4/13), (-11/13)
Solution: In the given rational numbers there are few rational numbers which have negative denominator. So firstly, convert such rational numbers in standard form and we get:
(8/13), (3/13), (-2/13), (-4/13), (-11/13)
Now, add the rational numbers and we get:
= (8/13) + (3/13) + (-2/13) + (-4/13) + (-11/13)
Since the denominators are same; so we keep the denominator same (i.e. common denominator) as shown below:
= [ (8) + (3) + (-2) + (-4) + (-11) ] / 13
Add numerators as we do addition of positive integer and negative integer & we get:
= (-6/13)
Hence, (-8/-13) + (-3/-13) + (-2/13) + (-4/13) + (-11/13) = (-6/13)
Example 8: Add (-2/-7), (3/-7), (-1/-7), (4/-7), (12/-7), (8/-7)
Solution: All the given rational numbers have negative denominators. So firstly, convert such rational numbers in standard form and we get:
(2/7), (-3/7), (1/7), (-4/7), (-12/7), (-8/7)
Now, add the rational numbers and we get:
(2/7) + (-3/7) + (1/7) + (-4/7) + (-12/7) + (-8/7)
Since the denominators are same; so we keep the denominator same (i.e. common denominator) as shown below:
= [ (2) + (-3) + (1) + (-4) + (-12) + (-8) ] / 7
Add numerators as we do addition of positive integer and negative integer & we get:
= (-24/7)
Hence, (-2/-7) + (3/-7) + (-1/-7) + (4/-7) + (12/-7) + (8/-7) = (-24/7)
|
