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Home >> Numbers >> Real Numbers >> Rational Numbers >> Subtraction of Rational Numbers >> Positive & Negative Rational Numbers with Different Denominator >> Subtraction of Positive & Negative Rational Numbers with Different Denominator
Before you understand this topic, you are adviced to read:
What are Negative Rational Numbers ?
What are Positive Rational Numbers ?
How to convert rational number in standard form ?
Subtraction of Positive and Negative Integer
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:
Subtraction of Positive Rational Number (with positive numerator and denominator) and Negative Rational Number (with Negative Numerator), having different denominator
Example: Subtract (-1/2) from (2/3)
Subtraction of Positive Rational Number (with positive numerator and denominator) and Negative Rational Number (with Negative Denominator), having different denominator
Example : Subtract (7/-5) from (4/15)
Subtraction of Positive Rational Number (with negative numerator and denominator) and Negative Rational Number (with Negative Numerator), having different denominator
Example: Subtract (-5/6) from (-1/-4)
Subtraction of Positive Rational Number (with negative numerator and denominator) and Negative Rational Number (with Negative Denominator), having different denominator
Example: Subtract (4/-6) from (-3/-9)
Situation 1: Subtraction of Positive Rational Number (with positive numerator and denominator) and Negative Rational Number (with Negative Numerator), having different denominator
Steps of subtraction 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: Do subtraction of the numerators. The numerators are negative integers, so we do subtraction as we do subtraction of integers.
Example 1: Subtract (-1/2) from (2/3)
Solution: Subtract the given integers 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
Do subtraction of the numerators. The numerators are negative integers, so we do subtraction as we do subtraction of integers.
= 7/6
Hence, (2/3) - (-1/2) = 7/6
Situation 2: Subtraction of Positive Rational Number (with positive numerator and denominator) and Negative Rational Number (with Negative Denominator), having different denominator
Steps of subtraction 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: Do subtraction of the numerators. The numerators are negative integers, so we do subtraction as we do subtraction of integers.
Example 2: Subtract (7/-5) from (4/15)
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)
Subtract the rational numbers 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
Do subtraction of the numerators. The numerators are negative integers, so we do subtraction as we do subtraction of integers.
= 25/15
Divide both numerator and denominator by 6 and convert to standard form & we get:
= 5/3
Hence, (4/15) - (7/-5) = 5/3
Situation 3: Subtraction of Positive Rational Number (with negative numerator and denominator) and Negative Rational Number (with Negative Numerator), having different denominator
Steps of subtraction 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: Do subtraction of the numerators. The numerators are negative integers, so we do subtraction as we do subtraction of integers.
Example 3: Subtract (-5/6) from (-1/-4)
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)
Subtract the rational numbers 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
Do subtraction of the numerators. The numerators are negative integers, so we do subtraction as we do subtraction of integers.
= -13/12
Hence, (-1/-4) - (-5/6) = (-13/12)
Situation 4: Subtraction of Positive Rational Number (with negative numerator and denominator) and Negative Rational Number (with Negative Denominator), having different denominator
Steps of subtraction 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: Do subtraction of the numerators. The numerators are negative integers, so we do subtraction as we do subtraction of integers.
Example 4: Subtract (4/-6) from (-4/-9)
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)
Subtract the rational numbers and we get:
= (4/9) - (-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:
= (4 X 2) - (-4 X 3) / 18
Solve the multiplication expression in the brackets and we get;
= (8) - (-12) / 18
Do subtraction of the numerators. The numerators are negative integers, so we do subtraction as we do subtraction of integers.
= 20/18
Divide both numerator and denominator by 2 and convert to standard form & we get:
= 10/9
Hence, (-3/-9) - (4/-6) = 10/3
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