Algebraic Fractions

Algebraic Techniques, SM-Bank 108

Simplify $\dfrac{3x}{2}\times \dfrac{8}{5x}\ ÷\ \dfrac{4}{x}$ giving your answer as an algebraic fraction in simplest form. (2 marks)

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$\dfrac{3x}{5}$

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$\dfrac{3x}{2}\times \dfrac{8}{5x}\ ÷\ \dfrac{4}{x}$	$=\dfrac{3x}{2}\times \dfrac{8}{5x}\times \dfrac{x}{4}$
	$=\dfrac{3x}{5}$

Algebraic Techniques, SM-Bank 107

Simplify the following quotients, giving your answer as an algebraic fraction in simplest form.

$\dfrac{5}{y}\ ÷\ \dfrac{4}{x}$ (1 mark)

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$\dfrac{9a}{7}\ ÷\ \dfrac{c}{b}$ (1 mark)

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$\dfrac{r}{3}\ ÷\ \dfrac{5}{4s}$ (2 marks)

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a. $\dfrac{5x}{4y}$

b. $\dfrac{9ab}{7c}$

c. $\dfrac{4rs}{45}$

Show Worked Solution

a.	$\dfrac{5}{y}\ ÷\ \dfrac{4}{x}$	$=\dfrac{5}{y}\times \dfrac{x}{4}$
		$=\dfrac{5x}{4y}$

b.	$\dfrac{9a}{7}\ ÷\ \dfrac{c}{b}$	$=\dfrac{9a}{7}\times\dfrac{b}{c}$
		$=\dfrac{9ab}{7c}$

c.	$\dfrac{r}{3}\ ÷\ \dfrac{5}{4s}$	$=\dfrac{r}{3}\times \dfrac{4s}{5}$
		$=\dfrac{4rs}{15}$

Algebraic Techniques, SM-Bank 106

Simplify the following quotients, giving your answer as an algebraic fraction in simplest form.

$\dfrac{5m}{8}\ ÷\ \dfrac{m}{4}$ (2 marks)

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$\dfrac{14x}{3}\ ÷\ \dfrac{7y}{6}$ (2 marks)

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a. $\dfrac{5}{2}$

b. $\dfrac{8x}{y}$

Show Worked Solution

a.	$\dfrac{5m}{8}\ ÷\ \dfrac{m}{4}$	$=\dfrac{5m}{8}\times \dfrac{4}{m}$
		$=\dfrac{20m}{8m}$
		$=\dfrac{5}{2}$

b.	$\dfrac{14x}{3}\ ÷\ \dfrac{7y}{6}$	$=\dfrac{14x}{3}\times \dfrac{6}{7y}$
		$=\dfrac{7x\times 2\times 3\times 2}{7y\times 3}$
		$=\dfrac{4x}{y}$

Algebraic Techniques, SM-Bank 105

Simplify the following products, giving your answer as an algebraic fraction in simplest form.

$\dfrac{2b}{3}\times \dfrac{9c}{5}$ (2 marks)

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$\dfrac{8x}{5}\times \dfrac{25y}{12}$ (2 marks)

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a. $\dfrac{6bc}{5}$

b. $\dfrac{10xy}{3}$

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a.	$\dfrac{2b}{3}\times \dfrac{9c}{5}$	$=\dfrac{2b\times 9c}{3\times 5}$
		$=\dfrac{18bc\ ÷\ 3}{15\ ÷\ 3}$
		$=\dfrac{6bc}{5}$

b.	$\dfrac{8x}{5}\times \dfrac{25y}{12}$	$=\dfrac{8x\times 25y}{5\times 12}$
		$=\dfrac{200xy\ ÷\ 20}{60\ ÷\ 20}$
		$=\dfrac{10xy}{3}$

Algebraic Techniques, SM-Bank 104

Simplify the following products, giving your answer as an algebraic fraction in simplest form.

$\dfrac{a}{2}\times \dfrac{1}{5}$ (1 mark)

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$\dfrac{3x}{5}\times \dfrac{2y}{7}$ (1 mark)

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$\dfrac{5r}{9}\times \dfrac{4s}{5}$ (2 marks)

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a. $\dfrac{a}{10}$

b. $\dfrac{6xy}{35}$

c. $\dfrac{4rs}{9}$

Show Worked Solution

$\dfrac{a}{2}\times \dfrac{1}{5}$

$=\dfrac{a\times 1}{2\times 5}=\dfrac{a}{10}$

$\dfrac{3x}{5}\times \dfrac{2y}{7}$

$=\dfrac{3x\times 2y}{5\times 7}=\dfrac{6xy}{35}$

$\dfrac{5r}{9}\times \dfrac{4s}{5}$

$=\dfrac{5r\times 4s}{9\times 5}=\dfrac{4rs}{9}$

Algebraic Techniques, SM-Bank 103

For the expression $\dfrac{a}{2}+\dfrac{2a}{3}-\dfrac{a}{4}$, simplify and write an equivalent algebraic fraction. (3 marks)

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$\dfrac{11a}{12}$

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$\dfrac{a}{2}+\dfrac{2a}{3}-\dfrac{a}{4}$	$=\dfrac{a}{2}\times \dfrac{6}{6}+\dfrac{2a}{3}\times \dfrac{4}{4}-\dfrac{a}{4}\times \dfrac{3}{3}$
	$=\dfrac{6a}{12}+\dfrac{8a}{12}-\dfrac{3a}{12}$
	$=\dfrac{11a}{12}$

Algebraic Techniques, SM-Bank 102

Ben recieved $$x$ for his birthday.

He spent $\dfrac{1}{2}$ of his money on shoes and $\dfrac{1}{3}$ on Gold Class movie tickets.

Write an algebraic fraction to represent the amount he spent on shoes. (1 mark)

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Write an algebraic fraction to represent the amount he spent on movie tickets. (1 mark)

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Using part (a) and (b) above, write a simplified algebraic fraction to represent the total amount of money he has spent so far. (2 marks)

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a. $\dfrac{$x}{2}$

b. $\dfrac{$x}{3}$

c. $\dfrac{$5x}{6}$

Show Worked Solution

a.	$\text{Shoes}$	$=\dfrac{1}{2}\times x$
		$=\dfrac{$x}{2}$

b.	$\text{Movie tickets}$	$=\dfrac{1}{3}\times x$
		$=\dfrac{$x}{3}$

c.	$\text{Total Spent}$	$=\dfrac{x}{2}+\dfrac{x}{3}$
		$=\dfrac{x}{2}\times \dfrac{3}{3}+\dfrac{x}{3}\times \dfrac{2}{2}$
		$=\dfrac{3x}{6}+\dfrac{2x}{6}$
		$=\dfrac{$5x}{6}$

Algebraic Techniques, SM-Bank 101

Yesterday Gareth walked $x$ kilometres in Kosciuszko National park.

He started at Thredbo village and walked up Merritts Nature Track to the Kosciuszko express chairlift. When he arrived at the chairlift he had complete $\dfrac{1}{5}$ of the total distance.

Gareth then joined a walking group and walked a further $\dfrac{1}{3}$ of the total distance before beginning his descent back to the village.

Write an algebraic fraction to represent the distance he walked to the Kosciuszko express chairlift. (1 mark)

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Write an algebraic fraction to represent the distance he walked with the walking group. (1 mark)

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Using part (a) and (b) above, write a simplified algebraic fraction to represent the total distance Gareth covered in the first two legs of his walk. (2 marks)

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a. $\dfrac{x}{5}\ \text{kilometres}$

b. $\dfrac{x}{3}\ \text{kilometres}$

c. $\dfrac{8x}{15}\ \text{kilometres}$

Show Worked Solution

a.	$\text{First leg}$	$=\dfrac{1}{5}\times x$
		$=\dfrac{x}{5}\ \text{kilometres}$

b.	$\text{Second leg}$	$=\dfrac{1}{3}\times x$
		$=\dfrac{x}{3}\ \text{kilometres}$

c.	$\text{Distance travelled}$	$=\dfrac{x}{5}+\dfrac{x}{3}$
		$=\dfrac{x}{5}\times \dfrac{3}{3}+\dfrac{x}{3}\times \dfrac{5}{5}$
		$=\dfrac{3x}{15}+\dfrac{5x}{15}$
		$=\dfrac{8x}{15}\ \text{kilometres}$

Algebraic Techniques, SM-Bank 100

For the sum $a+\dfrac{a}{4}$, simplify and write an equivalent algebraic fraction. (2 marks)

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$\dfrac{5a}{4}$

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$a+\dfrac{a}{4}$	$=\dfrac{a}{1}\times \dfrac{4}{4}+\dfrac{a}{4}$
	$=\dfrac{4a}{4}+\dfrac{a}{4}$
	$=\dfrac{5a}{4}$

Algebraic Techniques, SM-Bank 099

For the difference $\dfrac{a}{3}-\dfrac{b}{4}$, simplify and write an equivalent algebraic fraction. (3 marks)

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$\dfrac{4a-3b}{12}$

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$\dfrac{a}{3}-\dfrac{b}{4}$	$=\dfrac{a}{3}\times \dfrac{4}{4}-\dfrac{b}{4}\times \dfrac{3}{3}$
	$=\dfrac{4a}{12}-\dfrac{3b}{12}$
	$=\dfrac{4a-3b}{12}$

Algebraic Techniques, SM-Bank 098

For the difference $\dfrac{5r}{8}-\dfrac{3r}{8}$, simplify and write an equivalent algebraic fraction. (2 marks)

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$\dfrac{r}{4}$

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$\dfrac{5r}{8}-\dfrac{3r}{8}$	$=\dfrac{2r}{8}$
	$=\dfrac{r}{4}$

Algebraic Techniques, SM-Bank 097

For the difference $\dfrac{m}{2}-\dfrac{m}{7}$, simplify and write an equivalent algebraic fraction. (2 marks)

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$\dfrac{5m}{14}$

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$\dfrac{m}{2}-\dfrac{m}{7}$	$=\dfrac{m}{2}\times \dfrac{7}{7}-\dfrac{m}{7}\times \dfrac{2}{2}$
	$=\dfrac{7m}{14}-\dfrac{2m}{14}$
	$=\dfrac{5m}{14}$

Algebraic Techniques, SM-Bank 096

For the difference $\dfrac{a}{4}-\dfrac{a}{5}$, simplify and write an equivalent algebraic fraction. (2 marks)

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$\dfrac{a}{20}$

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$\dfrac{a}{4}-\dfrac{a}{5}$	$=\dfrac{a}{4}\times \dfrac{5}{5}-\dfrac{a}{5}\times \dfrac{4}{4}$
	$=\dfrac{5a}{20}-\dfrac{4a}{20}$
	$=\dfrac{a}{20}$

Algebraic Techniques, SM-Bank 095

For the difference $\dfrac{2x}{3}-\dfrac{x}{4}$, simplify and write an equivalent algebraic fraction. (2 marks)

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$\dfrac{5x}{12}$

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$\dfrac{2x}{3}-\dfrac{x}{4}$	$=\dfrac{2x}{3}\times \dfrac{4}{4}-\dfrac{x}{4}\times \dfrac{3}{3}$
	$=\dfrac{8x}{12}-\dfrac{3x}{12}$
	$=\dfrac{5x}{12}$

Algebraic Techniques, SM-Bank 094

For the sum $\dfrac{a}{2}+\dfrac{a}{3}+\dfrac{a}{4}$, simplify and write an equivalent algebraic fraction. (3 marks)

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$\dfrac{13a}{12}$

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$\dfrac{a}{2}+\dfrac{a}{3}+\dfrac{a}{4}$	$=\dfrac{a}{2}\times \dfrac{6}{6}+\dfrac{a}{3}\times \dfrac{4}{4}+\dfrac{a}{4}\times \dfrac{3}{3}$
	$=\dfrac{6a}{12}+\dfrac{4a}{12}+\dfrac{3a}{12}$
	$=\dfrac{13a}{12}$

Algebraic Techniques, SM-Bank 093

For the sum $\dfrac{b}{3}+\dfrac{5b}{6}$, simplify and write an equivalent algebraic fraction. (2 marks)

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$\dfrac{7b}{6}$

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$\dfrac{b}{3}+\dfrac{5b}{6}$	$=\dfrac{b}{3}\times \dfrac{2}{2}+\dfrac{5b}{6}$
	$=\dfrac{2b}{6}+\dfrac{5b}{6}$
	$=\dfrac{7b}{6}$

Algebraic Techniques, SM-Bank 092

For the sum $\dfrac{2x}{3}+\dfrac{x}{5}$, simplify and write an equivalent algebraic fraction. (2 marks)

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$\dfrac{13x}{15}$

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$\dfrac{2x}{3}+\dfrac{x}{5}$	$=\dfrac{2x}{3}\times \dfrac{5}{5}+\dfrac{x}{5}\times \dfrac{3}{3}$
	$=\dfrac{10x}{15}+\dfrac{3x}{15}$
	$=\dfrac{13x}{15}$

Algebraic Techniques, SM-Bank 091

For the sum $\dfrac{a}{4}+\dfrac{a}{2}$, simplify and write an equivalent algebraic fraction. (2 marks)

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$\dfrac{3a}{4}$

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$\dfrac{a}{4}+\dfrac{a}{2}$	$=\dfrac{a}{4}+\dfrac{a}{2}\times \dfrac{2}{2}$
	$=\dfrac{a}{4}+\dfrac{2a}{4}$
	$=\dfrac{3a}{4}$

Algebraic Techniques, SM-Bank 090

For the sum $\dfrac{m}{3}+\dfrac{m}{2}$, simplify and write an equivalent algebraic fraction. (2 marks)

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$\dfrac{5m}{6}$

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$\dfrac{m}{3}+\dfrac{m}{2}$	$=\dfrac{m}{3}\times \dfrac{2}{2}+\dfrac{m}{2}\times \dfrac{3}{3}$
	$=\dfrac{2m}{6}+\dfrac{3m}{6}$
	$=\dfrac{5m}{6}$

Algebraic Techniques, SM-Bank 089 MC

$\dfrac{2}{m}$ is equivalent to:

$\dfrac{4}{8m}$
$\dfrac{3m}{12}$
$\dfrac{6m}{3m^2}$
$\dfrac{1}{m}+\dfrac{1}{m}$

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$C$

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$\text{Consider Option C:}$

$\dfrac{6m}{3m^2}$	$=\dfrac{3\times 2\times m}{3\times m\times m}$
	$=\dfrac{2}{m}$

$\Rightarrow C$

Algebraic Techniques, SM-Bank 088 MC

$\dfrac{a}{4}$ is equivalent to:

$\dfrac{4a}{a}$
$\dfrac{3a}{12}$
$\dfrac{a^2}{4a^3}$
$\dfrac{a^4}{a^3}$

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$B$

Show Worked Solution

$\text{Consider Option B:}$

$\dfrac{3a}{12}$	$=\dfrac{3\times a}{3\times 4}$
	$=\dfrac{a}{4}$

$\Rightarrow B$

Algebraic Techniques, SM-Bank 087 MC

$\dfrac{x}{3}$ is equivalent to:

$\dfrac{2x}{6}$
$x+\dfrac{1}{3}$
$\dfrac{12x}{4}$
$\dfrac{x^3}{x^2}$

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$A$

Show Worked Solution

$\text{Consider Option A:}$

$\dfrac{2x}{6}$	$=\dfrac{2\times x}{2\times 3}$
	$=\dfrac{x}{3}$

$\Rightarrow A$

a.	\(\dfrac{5}{y}\ ÷\ \dfrac{4}{x}\)	\(=\dfrac{5}{y}\times \dfrac{x}{4}\)
		\(=\dfrac{5x}{4y}\)

b.	\(\dfrac{9a}{7}\ ÷\ \dfrac{c}{b}\)	\(=\dfrac{9a}{7}\times\dfrac{b}{c}\)
		\(=\dfrac{9ab}{7c}\)

c.	\(\dfrac{r}{3}\ ÷\ \dfrac{5}{4s}\)	\(=\dfrac{r}{3}\times \dfrac{4s}{5}\)
		\(=\dfrac{4rs}{15}\)

a.	\(\dfrac{5m}{8}\ ÷\ \dfrac{m}{4}\)	\(=\dfrac{5m}{8}\times \dfrac{4}{m}\)
		\(=\dfrac{20m}{8m}\)
		\(=\dfrac{5}{2}\)

b.	\(\dfrac{14x}{3}\ ÷\ \dfrac{7y}{6}\)	\(=\dfrac{14x}{3}\times \dfrac{6}{7y}\)
		\(=\dfrac{7x\times 2\times 3\times 2}{7y\times 3}\)
		\(=\dfrac{4x}{y}\)

a.	\(\dfrac{2b}{3}\times \dfrac{9c}{5}\)	\(=\dfrac{2b\times 9c}{3\times 5}\)
		\(=\dfrac{18bc\ ÷\ 3}{15\ ÷\ 3}\)
		\(=\dfrac{6bc}{5}\)

b.	\(\dfrac{8x}{5}\times \dfrac{25y}{12}\)	\(=\dfrac{8x\times 25y}{5\times 12}\)
		\(=\dfrac{200xy\ ÷\ 20}{60\ ÷\ 20}\)
		\(=\dfrac{10xy}{3}\)

\(\dfrac{a}{2}+\dfrac{2a}{3}-\dfrac{a}{4}\)	\(=\dfrac{a}{2}\times \dfrac{6}{6}+\dfrac{2a}{3}\times \dfrac{4}{4}-\dfrac{a}{4}\times \dfrac{3}{3}\)
	\(=\dfrac{6a}{12}+\dfrac{8a}{12}-\dfrac{3a}{12}\)
	\(=\dfrac{11a}{12}\)

a.	\(\text{Shoes}\)	\(=\dfrac{1}{2}\times x\)
		\(=\dfrac{$x}{2}\)