Before you study this concept, you are advice to read:
What are Rational Numbers ?
What are Integers ?
How to multiply Integers ?
Integers are of following two types:
Positive Integers
Negative Integers
So, with above mentioned types of integers, we get:
Multiplication of Rational Number and Positive Integer
Multiplication of Rational Number and Negative Integer
Multiplication of Rational Number and Positive Integers is similar to multiplication of Rational Number and Natural Number
Multiplication of Rational Number and Natural Number
Multiplication of Rational Number and Negative Integer : Under this multiplication, you will find the following situations:
Multiplication of Positive Rational Number (with positive integers) and Negative Integer
Example: (1/5) + (-2)
Multiplication of Positive Rational Number (with negative integers) and Negative Integer
Example: (-3/-5) + (-2)
Multiplication of Negative Rational Number (with negative numerator) and Negative Integer
Example: (-2/7) + (-5)
Multiplication of Negative Rational Number (with negative denominator) and Negative Integer
Example: (10/-11) + (-9)
Situation 1: Multiplication of Positive Rational Number (with positive integers) and Negative Integer
Method : 1
Step 1: Convert integer into rational number by putting 1 as its denominator
Step 2: Multiplication of numerators, divided by, multiplication of denominators
Note: For Multiplication follow the process of multiplication of integers
Method : 2
Multiply numerator of rational number with integer and keep denominator as such.
Example 1: Multiply (1/5) and (-2)
Solution:
With Method : 1
Write the given rational numbers and integer in Multiplication expression and we get:
(1/5) X (-2)
Convert the natural number into rational number by putting 1 as its denominator and we get:
= (1/5) X (-2/1)
Multiplication of numerators, divided by, multiplication of denominators and we get:
= (1 X -2) / (5 X 1)
Multiply the integer in the brackets.
Numerator has one positive integer and one negative integer, so follow process of multiplication of positive and negative integers
Denominator has positive integers, so follow process of multiplication of positive integers
And we get:
= (-2/5)
Hence, (1/5) X (-2)= (-2/5)
With Method: 2
Write the given rational numbers and integer in Multiplication expression and we get:
(1/5) X (-2)
Multiply numerator of rational number with integer and keep denominator as such & we get:
= (1 X -2) / 5
Multiply the integer in the brackets.
Numerator has one positive integer and one negative integer, so follow process of multiplication of positive and negative integers
And we get:
= (-2/5)
Hence, (1/5) X (-2)= (-2/5)
Situation 2: Multiplication of Positive Rational Number (with negative integers) and Negative Integer
Method : 1
Step 1: Convert integer into rational number by putting 1 as its denominator
Step 2: Multiplication of numerators, divided by, multiplication of denominators
Note: For Multiplication follow the process of multiplication of integers
Step 3: Since denominator has negative integer so convert it into standard form.
Method : 2
Step 1: Multiply numerator of rational number with natural number and keep denominator as such.
Step 2: Since denominator has negative integer so convert it into standard form.
Example 2: Multiply (-3/-5) and (-2)
Solution:
With Method : 1
Write the given rational numbers and integer in Multiplication expression and we get:
(-3/-5) X (-2)
Convert integer into rational number by putting 1 as its denominator and we get:
= (-3/-5) X (-2/1)
Multiplication of numerators, divided by, multiplication of denominators and we get:
= (-3 X -2) / (-5 X 1)
Multiply the integer in the brackets.
Numerator has negative integers, so follow process of multiplication of negative integers
Denominator has one positive integer and one negative integer, so follow process of multiplication of positive and negative integers
And we get:
= (6/-5)
Since the denominator has negative integer, so convert into standard form and we get:
= (-6/5)
Hence, (-3/-5) X (-2) = (-6/5)
With Method : 2
Write the given rational numbers in Multiplication expression and we get:
(-3/-5) X (-2)
Multiply numerator of rational number with natural number and keep denominator as such and
= (-3 X -2) / -5
Multiply the integer in the brackets.
Numerator has negative integers, so follow process of multiplication of negative integers
And we get:
= (6/-5)
Since the denominator has negative integer, so convert into standard form and we get:
= (-6/5)
Hence, (-3/-5) X (-2) = (-6/5)
Situation 3: Multiplication of Negative Rational Number (with negative numerator) and Negative Integer
Method : 1
Steps of multiplication under this situation are:
Step 1: Convert integer into rational number by putting 1 as its denominator
Step 2: Multiplication of numerators, divided by, multiplication of denominators
Note: For Multiplication follow the process of multiplication of integers
Method : 2
Multiply numerator of rational number with integer and keep denominator as such.
Example 2: Multiply (-2/7) and (-5)
Solution:
With Method : 1
Write the given rational numbers and integer in Multiplication expression and we get:
(-2/7) X (-5)
Convert the natural number into rational number by putting 1 as its denominator and we get:
= (-2/7) X (-5/1)
Multiplication of numerators, divided by, multiplication of denominators and we get:
= (-2 X -5) / (7 X 1)
Multiply the integer in the brackets.
Numerator has negative integers, so follow process of multiplication of negative integers
Denominator has one positive integer and one negative integer, so follow process of multiplication of positive and negative integers
And we get:
= 10/7
Hence, (-2/7) X (-5) = 10/7
With Method : 2
Write the given rational numbers and integer in Multiplication expression and we get:
(-2/7) X (-5)
Multiply numerator of rational number with integer and keep denominator as such & we get:
= (-2 X -5) / 7
Multiply the integer in the brackets.
Numerator has negative integers, so follow process of multiplication of negative integers
And we get:
= 10/7
Hence, (-2/7) X (-5) = 10/7
Situation 4: Multiplication of Negative Rational Number (with negative denominator) and Negative Integer
Method : 1
Step 1: Convert integer into rational number by putting 1 as its denominator
Step 2: Multiplication of numerators, divided by, multiplication of denominators
Note: For Multiplication follow the process of multiplication of integers
Step 3: Since denominator has negative integer so convert it into standard form.
Method : 2
Step 1: Multiply numerator of rational number with natural number and keep denominator as such.
Step 2: Since denominator has negative integer so convert it into standard form.
Example 4: Multiply (10/-11) and (-9)
Solution:
With Method : 1
Write the given rational numbers and integer in Multiplication expression and we get:
(10/-11) X (-9)
Convert integer into rational number by putting 1 as its denominator and we get:
= (10/-11) X (-9/1)
Multiplication of numerators, divided by, multiplication of denominators and we get:
= (10 X -9) / (-11 X 1)
Multiply the integer in the brackets.
Both Numerator and Denominator have one positive integer and one negative integer, so follow process of multiplication of positive and negative integers
And we get:
= (-90/-11)
Since the denominator has negative integer, so convert into standard form and we get:
= 90/11
Hence, (10/-11) X (-9) = 90/11
With Method : 2
Write the given rational numbers in Multiplication expression and we get:
(10/-11) X (-9)
Multiply numerator of rational number with natural number and keep denominator as such and
= (10 X -9) / -11
Multiply the integer in the brackets.
Numerator has negative integers, so follow process of multiplication of negative integers
And we get:
= (-90/-11)
Since the denominator has negative integer, so convert into standard form and we get:
= 90/11
Hence, (10/-11) X (-9) = 90/11
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