6th Grade - Multiplying and Dividing Rational Numbers

Introduction

• Multiplication and division of rational numbers are just like the multiplication and division of fractions. The only difference is that while multiplying and dividing rational numbers, integer rules of multiplication and division are followed.

Multiplying Rational Numbers

• To multiply rational numbers:

1. 1. Multiply the numerators by following the rules of multiplication of integers.
   2. Multiply the denominators by following the rules of multiplication of integers.
   3. Simplify to the lowest (simplest) term.

Click here to learn about the rules of the multiplication of integers.

 

Dividing Rational Numbers

• Rational numbers can be divided by changing the division expression into multiplication.
• Each division expression can be written as a multiplication expression by applying the rule " Keep, Change & Flip." This rule works from left to right.

Keep, Change, & Flip rule 1. Keep the first rational number of the expression the same. 2. Change the division sign to multiplication. 3. Flip the last rational number.

• To divide rational numbers:

1. 1. Convert the division expression to multiplication by following the rule "Keep, Change, & flip ."
   2. Multiply the numerators and follow the rules of multiplication of integers.
   3. Multiply the denominators and follow the rules of multiplication of integers.
   4. Simplify to the lowest (simplest) terms.

Cheat Sheet

• The product of two rational numbers is always a rational number.

Solved Examples

Example 1: Multiply: $\frac{-3}{4} \times \frac{2}{7}$

Solution: Multiply numerator with the numerator and denominator with the denominator by following the rules of multiplication of integers.

• Rule: When multiplying or dividing two integers with different signs, multiply/divide the integers and give the result a negative sign.

$=\frac{-6}{28}$

Simplify, by reducing to the lowest terms.

 $=\frac{-3}{14}$

 

Example 2: Multiply: $\frac{-9}{5} \times \frac{-8}{3}$

Solution: Multiply numerator with the numerator and denominator with the denominator by following the rules of multiplication of integers.

• Rule: When multiplying or dividing two integers with the same sign, multiply/divide the integers and give the result a positive sign.

$=\frac{72}{15}$

 

Example 3: Divide: $\frac{9}{2} ÷ \frac{-3}{5}$

Solution:Change the division expression to the multiplication expression by applying the Keep, Change and Flip rule.

$= \frac{9}{2} \times \frac{5}{-3}$

Multiply numerator with the numerator and denominator with the denominator by following the rules of multiplication of integers. 

$= -\frac{45}{6}$

Example 4: Divide: $-\frac{5}{7} ÷ -\frac{6}{7}$

Solution: Change the division expression to the multiplication expression by applying the Keep, Change and Flip rule.

$= -\frac{5}{7} \times -\frac{7}{6}$

Cross out the common factors.

$= -\frac{5}{\cancel{7}} \times -\frac{\cancel{7}}{6}$

$= \frac{5}{6}$