6th Grade - Multiplication and Division of Fractions

Introduction

• "Fraction" represents a relation of a part(s) to the whole, where the whole is divided into equal parts. 
• Fraction = $\frac{numerator}{denominator}= \frac{part\left(s\right)}{whole}$
• Any whole number can be written as a fraction. For example: $5 = \frac{5}{1}$. This is helpful when multiplying and dividing fractions.
• $\frac{2}{3}$ is a Proper Fraction (the denominator is greater than the numerator).
• $\frac{5}{3}$ is an Improper Fraction (the numerator is greater than the denominator).
• $1\frac{2}{3}$ is a Mixed Number/Fraction (whole number + proper fraction).

Multiplication of Fractions/Mixed Numbers

• To multiply fractions/mixed numbers.

1. 1. Convert the mixed numbers into an improper fraction. 
   2. Multiply the numerators.
   3. Multiply the denominators. 
   4. Simplify to the lowest (simplest) term.

• Occasionally, fractions can be simplified before multiplying.

Example 1: solve: $\frac{3}{4}·\frac{2}{5}$

Solution: $\frac{3}{4}·\frac{2}{5}=\frac{3\mathit{ }\times 2}{4\mathit{ }\times 5}$

           $=\frac{6}{20}$   (simplify the fraction to its lowest terms.)

           $=\frac{6 ÷ 2}{20 ÷ 2}$

           $=\frac{3}{10}$

Example 2: A recipe needs $2\frac{1}{3}$cups of sugar for a loaf of banana bread. Ms. Suzy wants to cut the sugar in half. How much sugar will Ms. Suzy use for the banana bread?  

Solution: Convert the mixed number to a proper fraction. 

$2\frac{1}{3} = 2\frac{+1}{\times 3} = \frac{7}{3}$

Ms. Suzy cuts the sugar in half.

$= \frac{7}{3} \times \frac{1}{2}$

$=\frac{7}{6}$

Change improper fraction to a mixed number by dividing the numerator by the denominator.

 $= 1\frac{1}{6}$           

Ms. Suzy uses $1\frac{1}{6}$cups of sugar to make a loaf of banana bread. 

Division of Fractions/Mixed Numbers

• Fractions can be divided by changing them 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 fraction of the expression the same.

2. Change the division sign to multiplication.

3. Flip the last fraction.

   Example: $\frac{1}{3} ÷\frac{2}{5}$

   $=\frac{1}{3} \times \frac{5}{2}$

• To divide fractions/mixed numbers.

1. 1. Convert each mixed number into an improper fraction. 
   2. Convert the division expression to multiplication by following the rule "Keep, Change, & flip ."
   3. Multiply the numerators.
   4. Multiply the denominators.
   5. Simplify to the lowest (simplest) form.

Example1: Solve $\frac{1}{2}÷2$.

Solution:  $\frac{1}{2}÷ 2$

$= \frac{1}{2} ÷ \frac{2}{1}$

Change division to multiplication by applying the "Keep, Change, & Flip" rule.

$=\frac{1}{2}\times \frac{1}{2}$

 Multiply across the numerators and denominators.

 $=\frac{1 \times 1}{2 \times 2}$

$=\frac{1}{4}$

 

Example2: Solve $\frac{7}{8}÷\frac{1}{4}$.

Solution: 

 $\frac{7}{8}÷\frac{1}{4}$

Change division to multiplication by applying the "Keep, Change, & Flip" rule.

$= \frac{7}{8} \times \frac{4}{1}$

Multiply across the numerators and denominators.

$=\frac{7 \times 4}{8 \times 1}$

$=\frac{28}{8}$         (simplify the fraction by reducing it to its lowest terms.)

$=\frac{28 ÷ 4}{8 ÷ 4}$

$=\frac{7}{2}$

Convert the improper fraction to the mixed number by dividing the numerator by the denominator.

$=3\frac{1}{2}$

 

Cheat Sheet

• Any whole number can be written as a fraction, such as 5 can be written as  $\frac{5}{1}$. 
• While creating an equivalent fraction, both the numerator and the denominator of the fraction must be multiplied/divided by the same number. 
• Always simplify the fraction whenever needed and possible. Sometimes a fraction can be simplified before multiplying/dividing. 

• A fraction multiplied by its reciprocal equals 1.

                                                           $\frac{1}{2}·\frac{2}{1}=1$

• The division is the inverse operation of multiplication; every division expression can be written as a multiplication expression by following the division rule of "keep, change, & flip."

                                                          $1 ÷ 7 = 1 \times \frac{1}{7}$

• The reciprocal of 3 is $\frac{1}{3}$ . The multiplication inverse of 3 is $\frac{1}{3}$.

Blunder Areas

• The word "of" means multiplication.

For example, Mr. Jim's house is $\frac{3}{4}$miles from the school. He walks $\frac{2}{3}$of the distance and then jogs the rest. How many miles does he walk?

        Solution: In this scenario, it's $\frac{2}{3}$of $\frac{3}{4}$miles. Therefore, $\frac{2}{3}\times \frac{3}{4}$ = $\frac{6}{12} = \frac{1}{2}$

• While multiplying two fractions, the numerator should be multiplied with the numerator and the denominator with the denominator.
• Always reduce the answer to its lowest term.