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# Multiplication and Division of Rational Numbers

Sep 16, 2022

### 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

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