Before you understand this topic, you are advice to read:
What are Negative Rational Numbers ?
How to find LCM ?
How to add Negative Integers ?
How to convert rational number into standard form ?
Negative Rational Number is of two types:
Rational Number with Negative Numerator
Rational Number with Negative Denominator
Addition of Negative Rational Numbers having Negative Numerator and whose denominators are different
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 numerators. The numerators are negative integers, so we will add numerators as we add negative integers.
Example 1: Add (-4/6), (-5/8)
Solution: Add the given rational numbers and we get:
= (-4/6) + (-5/8)
Find LCM of denominators of given rational numbers and we get:
LCM of 6 and 8 = 24
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 4) + (-5 X 3) / 24
Solve the multiplication expression in the brackets and we get;
= (-16) + (-15) / 24
Add numerators as we add negative integers and we get:
= (-31/24)
Hence, (-4/6) + (-5/8) = (-31/24)
Addition of Negative Rational Numbers having Negative Denominator and whose denominators are different
Steps of addition under this situation are:
Step 1: Since the denominators are negative, so firstly we convert the given rational numbers 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 numerators. The numerators are negative integers, so we will add numerators as we add negative integers.
Example 2: Add (8/-9), (7/-3)
Solution: Since the denominators are negative, so firstly we convert the given rational numbers in standard form and we get:
(-8/9), (-7/3)
Add the rational numbers and we get:
= (-8/9) + (-7/3)
Find LCM of denominators of given rational numbers and we get:
LCM of 9 and 3 = 9
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:
= (-8 X 1) + (-7 X 3) / 9
Solve the multiplication expression in the brackets and we get;
= (-8) + (-21) / 9
Add numerators as we add negative integers and we get:
= (-29/9)
Hence, (8/-9) + (7/-3) = (-29/9)
Addition of Negative Rational Numbers having different denominators, where one rational number have negative numerator and other have negative 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 numerators. The numerators are negative integers, so we will add numerators as we add negative integers.
Example 3: Add (2/-9) and (-7/6)
Solution: Convert the rational numbers (2/-9) in standard form and we get:
= (-2/9)
Add the rational numbers and we get:
= (-2/9) + (-7/6)
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:
= (-2 X 2) + (-7 X 3) / 18
Solve the multiplication expression in the brackets and we get;
= (-4) + (-21) / 18
Add numerators as we add negative integers and we get:
= (-25/18)
Hence, (2/-9) + (-7/6) = (-25/18)
Above examples 1, 2 & 3 under three different situations, must have given you the clarity how to add two negative rational numbers having different denominators.
Now, in the following examples you can now learn to add more than two rational numbers with different denominator.
Example 4: Add (-2/3), (-4/6), (-3/9)
Solution: Add the given rational numbers and we get:
= (-2/3) + (-4/6) + (-3/9)
Find LCM of denominators of given rational numbers and we get:
LCM of 3, 6 and 9 = 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:
= (-2 X 6) + (-4 X 3) X (-3 X 2) / 18
Solve the multiplication expression in the brackets and we get;
= (-12) + (-12) X (-6) / 18
Add numerators as we add negative integers and we get:
= (-30/18)
Divide both numerator and denominator by 6 and convert the resultant rational number into standard form. So we get:
= (-5/3)
Hence, (-2/3) + (-4/6) + (-3/8) = (-5/3)
Example 5: Add (1/-7), (2/-3), (5/-21)
Solution: Since the denominators are negative, so firstly we convert the given rational numbers in standard form and we get:
= (-1/7), (-2/3), (-5/21)
Add the rational numbers and we get:
= (-1/7) + (-2/3) + (-5/21)
Find LCM of denominators of given rational numbers and we get:
LCM of 7, 3 and 21 = 21
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) + (-2 X 7) + (-5 X 1) / 21
Solve the multiplication expression in the brackets and we get;
= (-3) + (-14) + (-5) / 21
Add numerators as we add negative integers and we get:
= (-22/21)
Hence, (1/-7) + (2/-3) + (5/-21) = (-22/21)
Example 6: Add (-3/4), (1/-2), (-7/3), (5/-6)
Solution: In the given rational numbers there are some rational numbers which have negative denominator.
So firstly, convert such rational numbers in standard form and we get:
= (-3/4), (-1/2), (-7/3), (-5/6)
Add the rational numbers and we get:
= (-3/4) + (-1/2) + (-7/3) + (-5/6)
Find LCM of denominators of given rational numbers and we get:
LCM of 4, 2, 3, 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:
= (-3 X 3) + (-1 X 6) + (-7 X 4) + (-5 X 2) / 12
Solve the multiplication expression in the brackets and we get;
= (-9) + (-6) + (-28) + (-10) / 12
Add numerators as we add negative integers and we get:
= (-53/12)
Hence, (-3/4) + (1/-2) + (-7/3) + (5/-6) = (-53/12)
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