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 find LCM ?
How to convert rational number in standard form ?
Positive Rational Numbers are of two types:
Positive Rational Numbers with Positive Numerator and Denominator
Positive Rational Numbers 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 different denominator
Example: (2/3) + (-1/2)
Addition of Positive Rational Number (with positive numerator and denominator) and Addition of Negative Rational Number (with Negative Denominator), having different denominator
Example: (4/15) + (7/-5)
Addition of Positive Rational Number (with negative numerator and denominator) and Addition of Negative Rational Number (with Negative Numerator), having different denominator
Example: (-1/-4) + (-5/6)
Addition of Positive Rational Number (with negative numerator and denominator) and Addition of Negative Rational Number (with Negative Denominator), having different denominator
Example: (-3/-9) + (4/-6)
Situation 1: Addition of Positive Rational Number (with positive numerator and denominator) and Addition of Negative Rational Number (with Negative Numerator), having different denominator
Steps of addition under this situation are:
Step 1: Find LCM of denominators of given rational numbers
Step 2: LCM = common denominator of resultant rational number
Step 3: Divide common denominator by the denominator and multiply the quotient with the numerator of given respective rational numbers
Step 4: 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), (-1/2)
Solution: Add the given rational numbers and we get:
(2/3) + (-1/2)
Find LCM of denominators of given rational numbers and we get:
LCM of 3 and 2 = 6
LCM = common denominator of resultant rational number
And divide common denominator by the denominator and multiply the quotient with the numerator of given respective rational numbers; as shown below:
= [ (2 X 2) + (-1 X 3) ] / 6
Solve the multiplication expression in the brackets and we get;
= [ (4) + (-3) ] / 6
Add numerators as we do addition of positive integer and negative integer & we get:
= 1/6
Hence, (2/3) + (-1/2) = 1/6
Situation 2: Addition of Positive Rational Number (with positive numerator and denominator) and Addition of Negative Rational Number (with Negative Denominator), having different denominator
Steps of addition under this situation are:
Step 1: Firstly we convert the rational numbers with negative denominator in standard form.
Step 2: Find LCM of denominators of given rational numbers
Step 3: LCM = common denominator of resultant rational number
Step 4: Divide common denominator by the denominator and multiply the quotient with the numerator of given respective rational numbers
Step 5: 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 (4/15), (7/-5)
Solution: In the given rational numbers there is a rational number which have negative denominator i.e. (7/-5). So firstly, convert this rational number in standard form and we get:
= (-7/5)
Add the given rational number and we get:
= (4/15) + (-7/5)
Find LCM of denominators of given rational numbers and we get:
LCM of 15 and 5 = 5
LCM = common denominator of resultant rational number
And divide common denominator by the denominator and multiply the quotient with the numerator of given respective rational numbers; as shown below:
= [ (4 X 1) + (-7 X 3) ] / 15
Solve the multiplication expression in the brackets and we get;
= [ (4) + (-21) ] / 15
Add numerators as we do addition of positive integer and negative integer & we get:
= (-17/15)
Hence, (4/15) + (7/-5) = (-17/15)
Situation 3: Addition of Positive Rational Number (with negative numerator and denominator) and Addition of Negative Rational Number (with Negative Numerator), having different denominator
Steps of addition under this situation are:
Step 1: Firstly we convert the rational numbers with negative denominator in standard form.
Step 2: Find LCM of denominators of given rational numbers
Step 3: LCM = common denominator of resultant rational number
Step 4: Divide common denominator by the denominator and multiply the quotient with the numerator of given respective rational numbers
Step 5: 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) and (-5/6)
Solution: In the given rational numbers there is a rational number which have negative denominator i.e. (-1/-4). So firstly, convert this rational number in standard form and we get:
= (1/4)
Add the given rational number and we get:
= (1/4) + (-5/6)
Find LCM of denominators of given rational numbers and we get:
LCM of 4 and 6 = 12
LCM = common denominator of resultant rational number
And divide common denominator by the denominator and multiply the quotient with the numerator of given respective rational numbers; as shown below:
= [ (1 X 3) + (-5 X 2) ] / 12
Solve the multiplication expression in the brackets and we get;
= [ (3) + (-10) ] / 12
Add numerators as we do addition of positive integer and negative integer & we get:
= -7/12
Hence, (-1/-4) + (-5/6) = (-7/12)
Situation 4: Addition of Positive Rational Number (with negative numerator and denominator) and Addition of Negative Rational Number (with Negative Denominator), having different denominator
Steps of addition under this situation are:
Step 1: Firstly we convert the rational numbers with negative denominator in standard form.
Step 2: Find LCM of denominators of given rational numbers
Step 3: LCM = common denominator of resultant rational number
Step 4: Divide common denominator by the denominator and multiply the quotient with the numerator of given respective rational numbers
Step 5: 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) and (4/-6)
Solution: Both the given rational numbers have negative denominators. So firstly, convert such rational numbers in standard form and we get:
= (3/9) and (-4/6)
Add the given rational number and we get:
= (3/9) + (-4/6)
Find LCM of denominators of given rational numbers and we get:
LCM of 9 and 6 = 18
LCM = common denominator of resultant rational number
And divide common denominator by the denominator and multiply the quotient with the numerator of given respective rational numbers; as shown below:
= [ (3 X 2) + (-4 X 3) ] / 18
Solve the multiplication expression in the brackets and we get;
= [ (6) + (-12) ] / 18
Add numerators as we do addition of positive integer and negative integer & we get:
= (-6/18)
Divide both numerator and denominator by 6 and convert to standard form & we get:
= (-1/3)
Hence, (-3/-9) and (4/-6) = (-1/3)
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 different denominators. Now, in the following examples you can now learn to add more than one; positive and negative rational numbers with different denominators.
Example 5: Solve (1/2) + (-3/5) + (7/10)
Solution: Add the rational numbers and we get:
(1/2) + (-3/5) + (7/10)
Find LCM of denominators of given rational numbers and we get:
LCM of 2, 5 and 10 = 10
LCM = common denominator of resultant rational number
And divide common denominator by the denominator and multiply the quotient with the numerator of given respective rational numbers; as shown below:
= [ (1 X 5) + (-3 X 2) + (7 X 1) ]/ 10
Solve the multiplication expression in the brackets and we get;
= [ (5) + (-6) + (7) ] / 10
Add numerators as we do addition of positive integer and negative integer & we get:
= 6/10
Divide both numerator and denominator by 2 to convert the above rational number into standard form and we get:
= 3/5
Hence, (1/2) + (-3/5) + (7/10) = 3/5
Example 6: Add (2/-7), (1/3), (5/-6), (1/2)
Solution: In the given rational numbers there are rational numbers which have negative denominators i.e. (2/-7) and (5/-6). So firstly, convert these rational numbers in standard form and we get:
= (-2/7) and (-5/6)
Add the rational number and we get:
= (-2/7) + (1/3) + (-5/6) + (1/2)
Find LCM of denominators of given rational numbers and we get:
LCM of 7, 3, 6 and 2 = 42
LCM = common denominator of resultant rational number
And divide common denominator by the denominator and multiply the quotient with the numerator of given respective rational numbers; as shown below:
= [ (-2 X 6) + (1 X 14) + (-5 X 7) + (1 X 21) ] / 42
Solve the multiplication expression in the brackets and we get;
= [ (-12) + (14) + (-35) + (21) ] / 42
Add numerators as we do addition of positive integer and negative integer & we get:
= (-12/42)
Divide both numerator and denominator by 6 to convert the above rational number into standard form and we get:
= (-2/7)
Hence, (2/-7) + (1/3) + (5/-6) + (1/2) = (-2/7)
Example 7: Add (-2/4), (-3/-6), (-10/12), (-5/-3), (-9/2)
Solution: In the given rational numbers there are rational numbers which have negative denominators i.e. (-3/-6) and (-5/-3). So firstly, convert this rational number in standard form and we get:
= (3/6) and (5/3)
Add the rational numbers and we get:
= (-2/4) + (3/6) + (-10/12) + (5/3) + (-9/2)
Find LCM of denominators of given rational numbers and we get:
LCM of 4, 6, 12, 3 and 2 = 12
LCM = common denominator of resultant rational number
And divide common denominator by the denominator and multiply the quotient with the numerator of given respective rational numbers; as shown below:
= [ (-2 X 3) + (3 X 2) + (-10 X 1) + (5 X 4) + (-9 X 6) ] / 12
Solve the multiplication expression in the brackets and we get;
= (-6) + (6) + (-10) + (20) + (-54) / 12
Add numerators as we do addition of positive integer and negative integer & we get:
= (-44/12)
Divide both numerator and denominator by 4 to convert the above rational number into standard form and we get:
= (-11/3)
Hence, (-2/4) + (-3/-6) + (-10/12) + (-5/-3) + (-9/2) = (-11/3)
Example 8: Solve (3/-5) + (2/-10) + (-4/-2) + (7/-20)
Solution: All the given rational numbers have negative denominators. So firstly, convert such rational numbers in standard form and we get:
= (-3/5), (-2/10), (4/2), (-7/20)
Add the rational numbers and we get:
= (-3/5) + (-2/10) + (4/2) + (-7/20)
Find LCM of denominators of given rational numbers and we get:
LCM of 5, 10, 2 and 20 = 20
LCM = common denominator of resultant rational number
And divide common denominator by the denominator and multiply the quotient with the numerator of given respective rational numbers; as shown below:
= [ (-3 X 4) + (-2 X 2) + (4 X 10) + (-7 X 1) ] / 20
Solve the multiplication expression in the brackets and we get;
= [ (-12) + (-4) + (40) + (-7) ] / 20
Add numerators as we do addition of positive integer and negative integer & we get:
= 17/20
Hence, (3/-5) + (2/-10) + (-4/-2) + (7/-20) = 17/20
|
