Multiplication and Division of Rational Numbers

Key Concepts

• Multiplication of rational numbers
• Division of rational numbers

Introduction:

• Understand multiplication and division of rational numbers.  
• Apply the multiplication rules for integers, decimals and fractions.  
• Identify equivalent expressions for two or more expressions.  
• Apply the division rules for integers, decimals and fractions to find the quotient.  
• Understand that the integers are divisible and the divisor is non-zero.  
• Understand that a negative fraction is written as a positive numerator and negative denominator or as a negative numerator and positive denominator. 

1.8 Multiplication of rational numbers 

Multiplication: 

When we multiply whole numbers, we usually replace repeated addition with the multiplication  
sign (×). 

Example: 

3+3+3+3 = 4×3 = 12 

How to Multiply Rational Numbers? 

Let us consider a/b,c/d to be rational numbers, then the product of rational numbers would be 

a / b×c / d= a×c / b×d 

Multiplication Rules: 

Rule 1: If the signs of the factors are the same, then the product is positive. 

Rule 2: If the signs of the factors are different, then the product is negative. 

Example:  

The cookies picture below shows the example 

Multiplying Decimals: 

We can multiply decimals in the same way like we multiply whole numbers with several digits.  

• Multiply normally, ignoring the decimal points. 
• Place the decimal point in the answer.  

The answer will have as many decimal places as the original numbers combined. 

Example: 

Multiply 3.1 × 2.9. 

Solution: 

3.1 × 2.9  

Multiplying without decimal points, we get 

31 × 29 = 899 

3.1 has 1 decimal places 

and 2.9 has 1 decimal place 

So, the product has 2 decimal places 

Placing the decimal point in the product, then the answer is 8.99 

Multiplication of Rational Numbers on a Number Line: 

The product of two rational numbers on the number line can be calculated in the following way: 

When we multiply − 2 / 7 By 3 on a number line, 

It means 3 jumps of − 2/7to the left from zero. Now We reach at  − 2/7. Thus we find

− 2/7 ×3= − 6/7, 

i.e., − 2/7 ×3 = − 2/7 × 3/1 =

 − 2 ×3 / 7 ×1  = − 6/7

1.8.1 Multiplication of a Negative number by a Positive Rational number 

Multiplying a negative number with a positive rational number, we simply multiply the integer with the numerator. The denominator remains the same, and the resultant rational number will be a negative rational number. 

a/b×−c=−ac / b

Example: 

Example 1: 

Find the product of 3.5 × (–1.2)  

Solution:  

3.5 × (–1.2) = –4.2 

1.8.2 Multiplication of a Positive number by a Negative Rational Number 

 Multiplying a positive number with a negative rational number, we simply multiply the numerator and the denominator, then the resultant rational number will be a negative rational number. 

−a / b×c / d=−ac / bd

Example: 

Find the product of −5/6 and 2/5. 

Solution: 

−5 / 6 × 2 / 5 = −5×2 / 6×5

 =−10 / 30

  =−1/3

1.8.3 Multiplication of a Negative number by a Negative Rational Number 

 Multiplying a negative number with a negative rational number, we simply multiply the numerator and the denominator, then the resultant rational number will be a positive rational number. 

−a / b×−c=ac / b

Example: 

Find the product of –0.3 and –11/30. 

Solution:  

−0.3×(−11 / 30) = −0.30×(−11 / 30)

Convert one of the rational numbers so that they are both fractions or decimals. 

⇒−3/10×(−11/30)

=−3 × (−11) / 10 × 30  

⇒33 / 300 or 0.11 

1.9 Division of Rational Numbers 

Introduction: 

What is division? 

Division is the inverse operation of multiplication. 

How to divide rational numbers? 

The following are the steps to solve the division of rational numbers problems. 

Step 1: Express the given rational numbers in the form of a fraction. 

Step 2: Keep the numerator part as it is and multiply with the reciprocal of the denominator in rational number. 

Step 3: Find the product of the rational numbers, which is nothing but the division of rational Numbers. 

Example 1: The following example shows the division of rational numbers in fraction form: 

Example 2: The following example shows the division of rational numbers in decimal form: 

Reciprocal: 

When the multiplier is multiplied with its reciprocal for the given rational number, we get the product of 1. 

Reciprocal of a / b is b/a

b/a

Product of Reciprocal 

If we multiply the reciprocal of the rational number with that rational number, then the product will always be 1. 

Example 

Division Rules: 

If the signs of the dividend and divisor are the same, then the quotient is positive. 

If the signs of the dividend and divisor are different, then the quotient is negative. 

1.9.1 Division of a Negative number by a Positive Rational number 

A rational number is said to be negative, when one of them is a positive rational number and the other is a negative rational number. 

Examples:  −2 / 5,3 / −5.−5 / 7,11 /−13,etc.

Example 1:  

Divide −3(3 / 5)÷6

Solution: 

Given

−3(3 / 5)÷6

=−18 / 5÷6

Reciprocal of 6 is 1/6

= −18 / 5×1 /6

=−18×1 / 5×6=−18 / 30=−3 / 5. 

1.9.2 Division of a Positive number by a Negative Rational number 

A rational number is said to be positive if its numerator and denominator are such that one of them is a negative integer and the other is a positive integer. 

Examples: 

2/−3,−3 / 5.−9 / 5,7 /−3,etc.

Example 1: Simplify 3x 2/3−2 /3

Given 3(2 / 3)÷−2 / 3

=11 / 3÷−2 / 3

Since the multiplicative inverse of −2/3 is −3/2, then we get 

=11 / 3×−3 / 2

=11×(−3) / 3×2=−33 / 6=−11 / 2=−5(1 /2).

1.9.3 Division of Rational numbers with the same sign 

When both the numerator and denominator of a rational number are either positive or negative, then the numbers are called positive rational numbers. 

Examples: 

5 / 7,−30 / −9.95,−7 / −3,etc.

Example 1: 

Divide −3 / 4÷−0.06

Solution:  

Given

−3 / 4÷−0.06

=(−0.75)÷(−0.06)

=−0.75−0.06=12.5.

Exercise:

       Multiply the following:

• 5/11 x 9/7
• (12.23)×(34.45)
• -1 5/6 x 6x 1/2
• (-0.2)×-5/6
• (-2.655)×(18.44)

Find the following quotients:

• (1 2/5)/((-1)/5)
• (-9/10)÷(-3/10)
• (-0.4)÷0.25
• 0.7÷-1(1/6)
• 7/12÷1/7

What have we learnt:

• Understand Multiplication and division of rational numbers.
• Apply the multiplication rules for integers , decimals and fractions.
• Identify equivalent expressions for two or more expressions.
• Apply the division rules for integers, decimals and fractions to find the quotient.
• Understand that the integers are divisible and the divisor is non-zero.
• Understand that a negative fraction is written as a positive numerator and negative denominator or as a negative numerator and positive denominator

Concept Map