## Fractions

 2.1 Multiplication of Fractions

Multiply proper fractions with proper fractions

• Multiply the numerator with the numerator.
• Multiply the denominator with the denominator.
• Give the answer in the simplest form.
 Question Fraction Multiplication Shading Method Cancellation Method $$\begin {array} {rr} \\ \quad\quad\small{\color{red}1}\quad\quad\small{\color{red}1} \\ \dfrac{2}{3}\times\dfrac{3}{4}=\dfrac{\cancel2}{\cancel3}\times\dfrac{\cancel3}{\cancel4}\\ \quad\quad\small{\color{red}1}\quad\quad\small{\color{red}1}\\ = \dfrac{1}{2}\quad\quad \end {array}$$ Steps to Multiply the Numerators and the Denominators $$\dfrac{3}{5}\times\dfrac{2}{7}=\dfrac{3\times2}{5\times7}\\ \quad\quad\quad\quad\quad =\dfrac{6}{35}$$

Multiply proper fractions with mixed numbers

• Convert the mixed numbers to improper fractions.
• Give the answer in the simplest form.
• The diagram for the larger fraction is drawn first.

Situation 1

Mr Leong has $$3\dfrac{1}{2}$$ packets of fertilizer. He wants to use $$\dfrac{2}{5}$$of the fertilizer to fertilize his hibiscus plant. How many packets of the fertilizer will he use to fertilize the hibiscus plant?

$$\dfrac{2}{5}\times3\dfrac{1}{2}=\dfrac{2}{5}\times\dfrac{7}{2}\\ \quad\quad\quad\quad=\dfrac{14}{10}\\ \quad\quad\quad\quad=1\dfrac{4\div2}{10\div2}\\ \quad\quad\quad\quad=1\dfrac{2}{5}$$

Multiply mixed numbers with mixed numbers

• A fraction multiplied by the inverse of the fraction will give the answer 1.

Situation 1

$$1\dfrac{1}{3}\times1\dfrac{1}{2}=$$

$$1\dfrac{1}{3}\times1\dfrac{1}{2}=(1\times1\dfrac{1}{2})+(\dfrac{1}{3}\times1\dfrac{1}{2})\\ \quad\quad\quad\quad=1\dfrac{1}{2}+(\dfrac{1}{3}\times\dfrac{3}{2})\\ \quad\quad\quad\quad=1\dfrac{1}{2}+\dfrac{3\div3}{6\div3}\\ \quad\quad\quad\quad=1\dfrac{1}{2}+\dfrac{1}{2}\\ \quad\quad\quad\quad=2$$

 2.2 Division of Fractions

Divide proper fractions by whole numbers

Situation 1

$$\begin {array} {rr}\\ \dfrac{4}{5}\div8=\dfrac{4}{5}\div\dfrac{8}{1}\quad\quad\quad\quad\quad\\ \quad\small{\color{red}1}\quad\quad\quad\quad\quad\quad\quad\quad\\ =\dfrac{\cancel4}{5}\times\dfrac{1}{\cancel8}\quad\quad\quad\quad\quad\\ \quad\quad\quad\small{\color{red}2}\quad\quad\quad\quad\quad\\ \quad=\dfrac{1}{5}\times\dfrac{1}{2}\quad\quad\quad\quad\quad\\ \quad=\dfrac{1}{10}\quad\quad\quad\quad\quad\quad \end {array}$$

Divide proper fractions by proper fractions

• Change division to multiplication.
• Inverse the divisor.
• Simplify the numerator and the denominator by cancellation (if any).

Situation 1

This is only $$\dfrac{3}{4}$$m. Cut the wood into a few pieces, each measuring $$\dfrac{1}{8}$$m. How many pieces of wood measuring $$\dfrac{1}{8}$$m were cut by Paramjit's father?

$$\dfrac{3}{4} m\div\dfrac{1}{8}m=$$

$$\quad\quad\quad\quad\quad\quad\small{\color{red}2}\\ \dfrac{3}{4}\div\dfrac{1}{8}=\dfrac{3}{\cancel4}\times\dfrac{\cancel8}{1}\\ \quad\quad\quad\quad\quad\small{\color{red}1}\\ \quad\quad\quad=6$$

Divide mixed numbers by whole numbers

Situation 1

$$1\dfrac{7}{9}\div4=\\ 1\dfrac{7}{9}\div4=1\dfrac{7}{9}\div\dfrac{4}{1}\\ \quad\quad\quad\quad\small{\color{red}4}\\ \quad\quad\quad=\dfrac{\cancel{16}}{9}\times\dfrac{1}{\cancel4}\\ \quad\quad\quad\quad\quad\quad\quad\small{\color{red}1}\\ \quad\quad\quad=\dfrac{4}{9}$$

Situation 2

$$6\dfrac{2}{7}\div22=\\ 6\dfrac{2}{7}\div22=6\dfrac{2}{7}\div\dfrac{22}{1}\\ \quad\quad\quad\quad\small{\color{red}2}\\ \quad\quad\quad=\dfrac{\cancel{44}}{7}\times\dfrac{1}{\cancel{22}}\\ \quad\quad\quad\quad\quad\quad\quad\small{\color{red}1}\\ \quad\quad\quad=\dfrac{2}{7}$$

Divide mixed numbers by proper fractions

Situation 1

How many $$\dfrac{1}{5}$$are there in $$1\dfrac{1}{2}$$?

$$1\dfrac{1}{2}\div\dfrac{1}{5}=\dfrac{3}{2}\times\dfrac{5}{1}\\ \quad\quad\quad=\dfrac{15}{2}\\ \quad\quad\quad=7\dfrac{1}{2}$$

 2.3 Solve the Problems

Situation 1

The recipe to make chocolate chip muffins is as shown. $$\dfrac{3}{4}$$cup of chocolate chips is needed to produce 12 muffins. Zara tried the recipe and used only half of the flour. How chocolate chips did Zara use?

 A Recipe for Chocolate Chip Muffins (12 pieces) 2 cups of flour 1/2 cup of granulated sugar 3 teaspoons of baking powder 1/2 teaspoon of milk 1/3 cup od cooking oil 1 egg 3/4 cup of chocolate chips 3 tablespoons of white sugar 2 tablespoons of brown sugar

Solution

$$\dfrac{1}{2}\times\dfrac{3}{4}=\dfrac{1\times3}{2\times4}\\ \quad\quad\quad=\dfrac{3}{8}$$

Situation 2

Damia has $$1\dfrac{1}{2} m$$  of ribbon. She uses $$\dfrac{1}{3}$$of the ribbon to decorate her mother's gift box. What is the length of the ribbon that she uses?

Solution

$$\quad\quad\quad\quad\quad\quad\small{\color{red}1}\\ \dfrac{1}{3}\times1\dfrac{1}{2}=\dfrac{1}{\cancel3}\times\dfrac{\cancel3}{2}\\ \quad\quad\quad\quad\quad\small{\color{red}1}\\ \quad\quad\quad\quad=\dfrac{1}{1}\times\dfrac{1}{2}\\ \quad\quad\quad\quad=\dfrac{1}{2}$$

## Fractions

 2.1 Multiplication of Fractions

Multiply proper fractions with proper fractions

• Multiply the numerator with the numerator.
• Multiply the denominator with the denominator.
• Give the answer in the simplest form.
 Question Fraction Multiplication Shading Method Cancellation Method $$\begin {array} {rr} \\ \quad\quad\small{\color{red}1}\quad\quad\small{\color{red}1} \\ \dfrac{2}{3}\times\dfrac{3}{4}=\dfrac{\cancel2}{\cancel3}\times\dfrac{\cancel3}{\cancel4}\\ \quad\quad\small{\color{red}1}\quad\quad\small{\color{red}1}\\ = \dfrac{1}{2}\quad\quad \end {array}$$ Steps to Multiply the Numerators and the Denominators $$\dfrac{3}{5}\times\dfrac{2}{7}=\dfrac{3\times2}{5\times7}\\ \quad\quad\quad\quad\quad =\dfrac{6}{35}$$

Multiply proper fractions with mixed numbers

• Convert the mixed numbers to improper fractions.
• Give the answer in the simplest form.
• The diagram for the larger fraction is drawn first.

Situation 1

Mr Leong has $$3\dfrac{1}{2}$$ packets of fertilizer. He wants to use $$\dfrac{2}{5}$$of the fertilizer to fertilize his hibiscus plant. How many packets of the fertilizer will he use to fertilize the hibiscus plant?

$$\dfrac{2}{5}\times3\dfrac{1}{2}=\dfrac{2}{5}\times\dfrac{7}{2}\\ \quad\quad\quad\quad=\dfrac{14}{10}\\ \quad\quad\quad\quad=1\dfrac{4\div2}{10\div2}\\ \quad\quad\quad\quad=1\dfrac{2}{5}$$

Multiply mixed numbers with mixed numbers

• A fraction multiplied by the inverse of the fraction will give the answer 1.

Situation 1

$$1\dfrac{1}{3}\times1\dfrac{1}{2}=$$

$$1\dfrac{1}{3}\times1\dfrac{1}{2}=(1\times1\dfrac{1}{2})+(\dfrac{1}{3}\times1\dfrac{1}{2})\\ \quad\quad\quad\quad=1\dfrac{1}{2}+(\dfrac{1}{3}\times\dfrac{3}{2})\\ \quad\quad\quad\quad=1\dfrac{1}{2}+\dfrac{3\div3}{6\div3}\\ \quad\quad\quad\quad=1\dfrac{1}{2}+\dfrac{1}{2}\\ \quad\quad\quad\quad=2$$

 2.2 Division of Fractions

Divide proper fractions by whole numbers

Situation 1

$$\begin {array} {rr}\\ \dfrac{4}{5}\div8=\dfrac{4}{5}\div\dfrac{8}{1}\quad\quad\quad\quad\quad\\ \quad\small{\color{red}1}\quad\quad\quad\quad\quad\quad\quad\quad\\ =\dfrac{\cancel4}{5}\times\dfrac{1}{\cancel8}\quad\quad\quad\quad\quad\\ \quad\quad\quad\small{\color{red}2}\quad\quad\quad\quad\quad\\ \quad=\dfrac{1}{5}\times\dfrac{1}{2}\quad\quad\quad\quad\quad\\ \quad=\dfrac{1}{10}\quad\quad\quad\quad\quad\quad \end {array}$$

Divide proper fractions by proper fractions

• Change division to multiplication.
• Inverse the divisor.
• Simplify the numerator and the denominator by cancellation (if any).

Situation 1

This is only $$\dfrac{3}{4}$$m. Cut the wood into a few pieces, each measuring $$\dfrac{1}{8}$$m. How many pieces of wood measuring $$\dfrac{1}{8}$$m were cut by Paramjit's father?

$$\dfrac{3}{4} m\div\dfrac{1}{8}m=$$

$$\quad\quad\quad\quad\quad\quad\small{\color{red}2}\\ \dfrac{3}{4}\div\dfrac{1}{8}=\dfrac{3}{\cancel4}\times\dfrac{\cancel8}{1}\\ \quad\quad\quad\quad\quad\small{\color{red}1}\\ \quad\quad\quad=6$$

Divide mixed numbers by whole numbers

Situation 1

$$1\dfrac{7}{9}\div4=\\ 1\dfrac{7}{9}\div4=1\dfrac{7}{9}\div\dfrac{4}{1}\\ \quad\quad\quad\quad\small{\color{red}4}\\ \quad\quad\quad=\dfrac{\cancel{16}}{9}\times\dfrac{1}{\cancel4}\\ \quad\quad\quad\quad\quad\quad\quad\small{\color{red}1}\\ \quad\quad\quad=\dfrac{4}{9}$$

Situation 2

$$6\dfrac{2}{7}\div22=\\ 6\dfrac{2}{7}\div22=6\dfrac{2}{7}\div\dfrac{22}{1}\\ \quad\quad\quad\quad\small{\color{red}2}\\ \quad\quad\quad=\dfrac{\cancel{44}}{7}\times\dfrac{1}{\cancel{22}}\\ \quad\quad\quad\quad\quad\quad\quad\small{\color{red}1}\\ \quad\quad\quad=\dfrac{2}{7}$$

Divide mixed numbers by proper fractions

Situation 1

How many $$\dfrac{1}{5}$$are there in $$1\dfrac{1}{2}$$?

$$1\dfrac{1}{2}\div\dfrac{1}{5}=\dfrac{3}{2}\times\dfrac{5}{1}\\ \quad\quad\quad=\dfrac{15}{2}\\ \quad\quad\quad=7\dfrac{1}{2}$$

 2.3 Solve the Problems

Situation 1

The recipe to make chocolate chip muffins is as shown. $$\dfrac{3}{4}$$cup of chocolate chips is needed to produce 12 muffins. Zara tried the recipe and used only half of the flour. How chocolate chips did Zara use?

 A Recipe for Chocolate Chip Muffins (12 pieces) 2 cups of flour 1/2 cup of granulated sugar 3 teaspoons of baking powder 1/2 teaspoon of milk 1/3 cup od cooking oil 1 egg 3/4 cup of chocolate chips 3 tablespoons of white sugar 2 tablespoons of brown sugar

Solution

$$\dfrac{1}{2}\times\dfrac{3}{4}=\dfrac{1\times3}{2\times4}\\ \quad\quad\quad=\dfrac{3}{8}$$

Situation 2

Damia has $$1\dfrac{1}{2} m$$  of ribbon. She uses $$\dfrac{1}{3}$$of the ribbon to decorate her mother's gift box. What is the length of the ribbon that she uses?

Solution

$$\quad\quad\quad\quad\quad\quad\small{\color{red}1}\\ \dfrac{1}{3}\times1\dfrac{1}{2}=\dfrac{1}{\cancel3}\times\dfrac{\cancel3}{2}\\ \quad\quad\quad\quad\quad\small{\color{red}1}\\ \quad\quad\quad\quad=\dfrac{1}{1}\times\dfrac{1}{2}\\ \quad\quad\quad\quad=\dfrac{1}{2}$$