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# 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 =
• Any whole number can be written as a fraction. For example: . 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. 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:

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

$=\frac{3}{10}$

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.

Ms. Suzy cuts the sugar in half.

$=\frac{7}{6}$

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

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:

• To divide fractions/mixed numbers.
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:

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

$=\frac{1}{2}×\frac{1}{2}$

Multiply across the numerators and denominators.

$=\frac{1}{4}$

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.

Multiply across the numerators and denominators.

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

$=\frac{7}{2}$

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."

• 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, $\frac{2}{3}×\frac{3}{4}$

• 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.