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Algebraic Techniques, SM-Bank 154

A rectangle has an area of $2pq^2+4p^2q$. By factorising the expression, find the length of the rectangle if the width is $2pq$. (2 marks)

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$q+2p$

Show Worked Solution

$2pq^2+4pq^2$	$=2pq\times q+2pq\times 2p\ \ \ \ (\text{HCF}=2pq)$
	$=2pq(q+2p)$

$\therefore\ \text{Length of rectangle}=q+2p$

Algebraic Techniques, SM-Bank 153

A rectangle has an area of $9a^2-6a$. By factorising the expression, find the width of the rectangle if the length is $3a$. (2 marks)

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$3a-2$

Show Worked Solution

$9a^2-6a$	$=3a\times 3a+3a\times (-2)\ \ \ \ (\text{HCF}=3a)$
	$=3a(3a-2)$

$\therefore\ \text{Width of rectangle}=3a-2$

Algebraic Techniques, SM-Bank 152

A rectangle has an area of $9a^2-6a$. By factorising the expression, find the width of the rectangle if the length is $3a$. (2 marks)

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$3a-2$

Show Worked Solution

$9a^2-6a$	$=3a\times 3a+3a\times (-2)\ \ \ \ (\text{HCF}=3a)$
	$=3a(3a-2)$

$\therefore\ \text{Width of rectangle}=3a-2$

Algebraic Techniques, SM-Bank 151

Fully factorise the following:

$4x^2-2x$ (2 marks)

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$-6ab+3a$ (2 marks)

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$-5q-10q^2$ (2 marks)

--- 2 WORK AREA LINES (style=lined) ---

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a. $2x(2x-1)$

b. $-3a(2b-1)$

c. $-5q(1+2q)$

Show Worked Solution

a.	$4x^2-2x$	$=2x\times 2x+2x\times (-1)\ \ \ \ (\text{HCF}=2x)$
		$=2x(2x-1)$

b.	$-6ab+3a$	$=-3a\times 2b-3a\times(-1)\ \ \ \ (\text{HCF}=-3a)$
		$=-3a(2b-1)$

c.	$-5q-10q^2$	$=-5q\times 1-5q\times 2q\ \ \ \ (\text{HCF}=-5q)$
		$=-5q(1+2q)$

Algebraic Techniques, SM-Bank 150 MC

Which of the following is the correct factorisation of $7y+14x$.

$7(y+2x)$
$y(7+14x)$
$7(y+14x)$
$7(y+7x)$

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$A$

Show Worked Solution

$7y+14x$	$=7\times y+7\times 2x\ \ \ \ (\text{HCF}=7)$
	$=7(y+2x)$

$\Rightarrow A$

Algebraic Techniques, SM-Bank 149 MC

Which of the following is the correct factorisation of $15m-20$.

$15(m-20)$
$5(m-4)$
$5(3m-4)$
$5(3m-20)$

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$C$

Show Worked Solution

$15m-20$	$=5\times 3m-5\times 4\ \ \ \ (\text{HCF}=5)$
	$=5(3m-4)$

$\Rightarrow C$

Algebraic Techniques, SM-Bank 148 MC

Which of the following is the correct factorisation of $3x-12$.

$3(x-12)$
$x(3-12x)$
$3(x-9)$
$3(x-4)$

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$D$

Show Worked Solution

$3x-12$	$=3\times x-3\times 4\ \ \ \ (\text{HCF}=3)$
	$=3(x-4)$

$\Rightarrow D$

Algebraic Techniques, SM-Bank 147

State the highest common factor of $36mn^2\ \text{and}\ 24m^2n$. (1 mark)

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$12mn$

Show Worked Solution

$\text{HCF of}\ 36\ \text{and}\ 24=12$

$\text{HCF of}\ mn^2\ \text{and}\ m^2n=mn$

$\therefore\ \text{HCF of}\ 36mn^2\ \text{and}\ 24m^2n=12mn$

Algebraic Techniques, SM-Bank 146

State the highest common factor of $27ab^2c\ \text{and}\ 3a^2b^2c$. (1 mark)

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$3ab^2c$

Show Worked Solution

$\text{HCF of}\ 27\ \text{and}\ 3=3$

$\text{HCF of}\ ab^2c\ \text{and}\ a^2b^2c=ab^2c$

$\therefore\ \text{HCF of}\ 27ab^2c\ \text{and}\ 3a^2b^2c=3ab^2c$

Algebraic Techniques, SM-Bank 145

State the highest common factor of $15xy\ \text{and}\ 20y$. (1 mark)

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$5y$

Show Worked Solution

$\text{HCF of}\ 15\ \text{and}\ 20=5$

$\text{HCF of}\ xy\ \text{and}\ y=y$

$\therefore\ \text{HCF of}\ 15xy\ \text{and}\ 20y=5y$

Algebraic Techniques, SM-Bank 144

List all the factors of $-6z$. (2 marks)

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$ 1\ ,\ 2\ ,\ 3\ ,\ 6\ ,-z\ ,-2z \ ,-3z\ ,-6z$

$ -1\ , -2\ , -3\ , -6\ ,\ z\ ,\ 2z \ ,\ 3z\ ,\ 6z$

Show Worked Solution

$\text{Listing factors in pairs}$

$(1\ ,-6z) (2\ ,-3z) (3\ ,-2z) (6\ ,-z)$

$(-1 ,\ 6z) (-2 , 3z) (-3 ,\ 2z) (-6 , z)$

$\therefore\ \text{Factors are:}$

$ 1\ ,\ 2\ ,\ 3\ ,\ 6\ ,-z\ ,-2z \ ,-3z\ ,-6z$

$ -1\ , -2\ , -3\ , -6\ ,\ z\ ,\ 2z \ ,\ 3z\ ,\ 6z$

Algebraic Techniques, SM-Bank 143

List all the factors of $-7q$. (2 marks)

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$ 1\ ,\ 7\ ,-q\ ,-7q \ , -1\ , -7\ ,\ q\ ,\ 7q$

Show Worked Solution

$\text{Listing factors in pairs}$

$\longrightarrow (1\ ,-7q) (7\ ,-q) (-1 ,\ 7q) (-7 , q) $

$\therefore\ \text{Factors are:}$

$ 1\ ,\ 7\ ,-q\ ,-7q \ , -1\ , -7\ ,\ q\ ,\ 7q$

Algebraic Techniques, SM-Bank 142

List all the factors of $-12y$. (2 marks)

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$ 1\ ,\ 2\ ,\ 3\ ,\ 4\ ,\ 6\ ,\ 12\ ,-y\ ,-2y \ ,-3y\ ,-4y\ ,-6y\ ,-12y$

$ -1\ , -2\ , -3\ , -4\ , -6\ , -12\ ,\ y\ ,\ 2y \ ,\ 3y\ ,\ 4y\ ,\ 6y\ ,\ 12y$

Show Worked Solution

$\text{Listing factors in pairs}$

$(1\ ,-12y) (2\ ,-6y) (3\ ,-4y) (4\ ,-3y) (6\ ,-2y) (12\ ,-y)$

$(-1 ,\ 12y) (-2 , 6y) (-3 , 4y) (-4 , 3y)(-6 , 2y) (-12 , y)$

$\therefore\ \text{Factors are:}$

$ 1\ ,\ 2\ ,\ 3\ ,\ 4\ ,\ 6\ ,\ 12\ ,-y\ ,-2y \ ,-3y\ ,-4y\ ,-6y\ ,-12y$

$ -1\ , -2\ , -3\ , -4\ , -6\ , -12\ ,\ y\ ,\ 2y \ ,\ 3y\ ,\ 4y\ ,\ 6y\ ,\ 12y$

Algebraic Techniques, SM-Bank 141

List all the factors of $10m$. (2 marks)

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$ 1\ ,\ 2\ ,\ 5\ ,\ 10\ ,\ m\ ,\ 2m \ ,\ 5m\ ,\ 10m$

$-1 , -2 , -5 , -10 , -m , -2m , -5m , -10m$

Show Worked Solution

$\text{Listing factors in pairs}$

$(1\ ,\ 10m) (2\ ,\ 5m) (5\ ,\ 2m) (10\ ,\ m) (-1 ,-10m) (-2 , -5m) (-5 , -2m) (-10,-m)$

$\therefore\ \text{Factors are:}$

$ 1\ ,\ 2\ ,\ 5\ ,\ 10\ ,\ m\ ,\ 2m \ ,\ 5m\ ,\ 10m$

$-1 , -2 , -5 , -10 , -m , -2m , -5m , -10m$

Algebraic Techniques, SM-Bank 140

List all the factors of $2x$. (2 marks)

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$\ 1\ ,\ 2\ ,\ x\ ,\ 2x \ , -1\ , -2 \ , -x\ , -2x$

Show Worked Solution

$\text{Listing factors in pairs}$

$\longrightarrow (1\ ,\ 2x)\ \ (2\ ,\ x)\ \ (-1\ ,-2x)\ \ (-2\ ,-x)$

$\therefore \text{Factors are:}\ 1\ ,\ 2\ ,\ x\ ,\ 2x \ , -1\ , -2 \ , -x\ , -2x$

Algebraic Techniques, SM-Bank 139

List all the factors of $4b$. (2 marks)

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$\ 1\ ,\ 2\ ,\ 4\ ,\ b\ ,\ 2b ,\ 4b\ , -1\ , -2\ , -4\ , -b\ , -2b\ , -4b$

Show Worked Solution

$\text{Listing factors in pairs}$

$(1\ ,\ 4b)\ \ (2\ ,\ 2b)\ \ (4\ ,\ b)$

$(-1\ ,-4b)\ \ (-2\ ,-2b)\ \ (-4\ ,-b)$

$\therefore\ \text{Factors are:}$

$\ 1\ ,\ 2\ ,\ 4\ ,\ b\ ,\ 2b ,\ 4b ,\ -1\ , -2\ , -4\ , -b\ , -2b\ , -4b$

Algebraic Techniques, SM-Bank 138

Use the rectangle below to prove that $(a+b)^2=a^2+2ab+b^2$. (3 marks)

--- 5 WORK AREA LINES (style=lined) ---

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$\text{See worked solution}$

Show Worked Solution

$\text{Length of large rectangle}$	$=a+b$
$\text{Width of large rectangle}$	$=a+b$

$\text{Area of large rectangle}$	$=(a+b)\times(a+b)$
	$=(a+b)^2$

$\text{Adding 4 areas inside large rectangle}$	$=a^2+a\times b+a\times b+b^2$
	$=a^2+2ab+b^2$

$\therefore (a+b)^2=a^2+2ab+b^2$

Algebraic Techniques, SM-Bank 137

The triangle below has a base of $(2+6\large m)$ and a perpendicular height of $\large m$.

Write an algebraic expression and expand it to represent the area of the triangle. (2 marks)

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$x^2+xy$

Show Worked Solution

$\text{Area of Triangle}$	$=\dfrac{1}{2}\times \text{base} \times \text{height}$
	$=\dfrac{1}{2}\times \Big(2+6m\Big)\times m$
	$=\dfrac{m}{2}\Big(2+6m\Big)$
	$=\dfrac{m}{2}\times 2 +\dfrac{m}{2}\times 6m$
	$=m+3m^2$

Algebraic Techniques, SM-Bank 136 MC

The expansion of $-2q(q-8)$ is:

$-2q+16$
$-2q^2+16q$
$-2q^2-16q$
$2q^2+16q$

Show Answers Only

$B$

Show Worked Solution

$-2q(q-8)$	$=-2q\times q-2q\times (-8)$
	$=-2q^2+16q$

$\Rightarrow B$

Algebraic Techniques, SM-Bank 135 MC

The expansion of $4x(5-y)$ is:

$9x-4xy$
$4x-5xy$
$20-4xy$
$20x-4xy$

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$D$

Show Worked Solution

$4x(5-y)$	$=4x\times 5+4x\times (-y)$
	$=20x-4xy$

$\Rightarrow D$

Algebraic Techniques, SM-Bank 134

Jimmy works $\large x$ hours every week and his hourly rate of pay is $(40+\large y)$.

Write an algebraic expression and expand it to represent Jimmy's weekly wage. (2 marks)

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$40x+xy$

Show Worked Solution

$x(40+y)$	$=x\times 40+x\times y$
	$=40x+xy$

Algebraic Techniques, SM-Bank 133

The rectangle below has a length of $(\large x+y)$ and a width of $\large x$.

Write an algebraic expression and expand it to represent the area of the rectangle. (2 marks)

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$x^2+xy$

Show Worked Solution

$x(x+y)$	$=x\times x+x\times y$
	$=x^2+xy$

Algebraic Techniques, SM-Bank 132

The rectangle below has a length of $(3+\large b)$ and a width of 4.

Write an algebraic expression and expand it to represent the area of the rectangle. (2 marks)

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$12+4b$

Show Worked Solution

$4(3+b)$	$=4\times 3+4\times b$
	$=12+4b$

Algebraic Techniques, SM-Bank 131

A number $\large y$ is halved and 3 is added. The result is then multiplied by 4.

Write an algebraic expression and expand it to represent this information. (2 marks)

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$2y+12$

Show Worked Solution

$4\bigg(\dfrac{y}{2}+3\bigg)$	$=4\times \dfrac{y}{2}+4\times 3$
	$=\dfrac{4y}{2}+12$
	$=2y+12$

Algebraic Techniques, SM-Bank 130

A number $\large x$ is doubled and 1 is added. The result is then multiplied by 5.

Write an algebraic expression and expand it to represent this information. (2 marks)

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$10x+5$

Show Worked Solution

$5(2x+1)$	$=5\times 2x+5\times 1$
	$=10x+5$

Algebraic Techniques, SM-Bank 129

A number $b$ is tripled and $5$ is subtracted. The result is then doubled.

Write an algebraic expression and expand it to represent this information. (2 marks)

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$6b-10$

Show Worked Solution

$2(3b-5)$	$=2\times 3b+2\times (-5)$
	$=6b-10$

Algebraic Techniques, SM-Bank 128

A number $x$ is added to $4$ and the result is doubled.

Write an algebraic expression and expand it to represent this information. (2 marks)

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$2x+8$

Show Worked Solution

$2(x+4)$	$=2\times x+2\times 4$
	$=2x+8$

Algebraic Techniques, SM-Bank 127

Expand and simplify the expression $2(5-4x)+3(6x-1)$. (2 marks)

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$7+10x$

Show Worked Solution

$2(5-4x)+3(6x-1)$	$=2\times 5+2\times (-4x)+3\times 6x+3\times (-1)$
	$=10-8x+18x-3$
	$=7+10x$

Algebraic Techniques, SM-Bank 126

Expand and simplify the expression $3(x-1)+4(2x+3)$. (2 marks)

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$11x+9$

Show Worked Solution

$3(x-1)+4(2x+3)$	$=3\times x-3\times 1+4\times 2x+4\times 3$
	$=3x-3+8x+12$
	$=11x+9$

Algebraic Techniques, SM-Bank 125

Expand and simplify the expression $3xy-2x(y-5)$. (2 marks)

Show Answers Only

$xy+10x$

Show Worked Solution

$3xy-2x(y-5)$	$=3xy-2x\times y-2x\times (-5)$
	$=3xy-2xy+10x$
	$=xy+10x$

Algebraic Techniques, SM-Bank 124

Expand and simplify the expression $2(m+2)+7m$. (2 marks)

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$9m+4$

Show Worked Solution

$2(m+2)+7m$	$=2\times m+2\times 2+7m$
	$=2m+4+7m$
	$=9m+4$

Algebraic Techniques, SM-Bank 123

Use the distributive law to expand the expression $9x(x-2y)$. (2 marks)

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$9x^2-18xy$

Show Worked Solution

$9x(x-2y)$	$=9x\times x-9x\times 2y$
	$=9x^2-18xy$

Algebraic Techniques, SM-Bank 122

Use the distributive law to expand the expression $b(3-b)$. (2 marks)

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$3b-b^2$

Show Worked Solution

$b(3-b)$	$=b\times 3-b\times b$
	$=3b-b^2$

Algebraic Techniques, SM-Bank 121

Use the distributive law to expand the expression $x(x+2)$. (2 marks)

Show Answers Only

$x^2+2x$

Show Worked Solution

$x(x+2)$	$=x\times x+x\times 2$
	$=x^2+2x$

Algebraic Techniques, SM-Bank 120

Use the distributive law to expand the expression $-(a-8)$. (2 marks)

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$-a+8$

Show Worked Solution

$-(a-8)$	$=-1\times a-(-1)\times 8$
	$=-a+8$

Algebraic Techniques, SM-Bank 119

Use the distributive law to expand the expression $-10(y+4)$. (2 marks)

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$-10y-40$

Show Worked Solution

$-10(y+4)$	$=-10\times y+(-10)\times 4$
	$=-10y-40$

Algebraic Techniques, SM-Bank 118

Use the distributive law to expand the expression $4(m-3)$. (2 marks)

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$4m-12$

Show Worked Solution

$4(m-3)$	$=4\times m-4\times 3$
	$=4m-12$

Algebraic Techniques, SM-Bank 117

Use the distributive law to expand the expression $3(x+1)$. (2 marks)

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$3x+3$

Show Worked Solution

$3(x+1)$	$=3\times x+3\times 1$
	$=3x+3$

Algebraic Techniques, SM-Bank 116

Use the algebraic expression $4-x$ to complete the table. (3 marks)

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$4-x$

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$x$	$1$	$2$	$3$	$4$	$5$	$10$
$4-x$	$3$	$2$	$1$	$0$	$-1$	$-6$

Show Worked Solution

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$4-x$	$3$	$2$	$1$	$0$	$-1$	$-6$

Algebraic Techniques, SM-Bank 115

Use the algebraic expression $-x$ to complete the table. (3 marks)

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$-x$

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Show Answers Only

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$-x$	$-1$	$-2$	$-3$	$-4$	$-5$	$-10$

Show Worked Solution

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$-x$	$-1$	$-2$	$-3$	$-4$	$-5$	$-10$

Algebraic Techniques, SM-Bank 114

Use the algebraic expression $5x$ to complete the table. (3 marks)

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$5x$

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$x$	$1$	$2$	$3$	$4$	$5$	$10$
$5x$	$5$	$10$	$15$	$20$	$25$	$50$

Show Worked Solution

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$5x$	$5$	$10$	$15$	$20$	$25$	$50$

Algebraic Techniques, SM-Bank 113

Use the algebraic expression $x^2-3x$ to complete the table. (3 marks)

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$x^2-3x$

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Show Answers Only

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$x^2-3x$	$-2$	$-2$	$0$	$4$	$10$	$70$

Show Worked Solution

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$x^2-3x$	$-2$	$-2$	$0$	$4$	$10$	$70$

Algebraic Techniques, SM-Bank 112

Use the algebraic expression $x^2+1$ to complete the table. (3 marks)

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$x^2+1$

--- 0 WORK AREA LINES (style=lined) ---

Show Answers Only

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$x^2+1$	$2$	$5$	$10$	$17$	$26$	$101$

Show Worked Solution

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$x^2+1$	$2$	$5$	$10$	$17$	$26$	$101$

Algebraic Techniques, SM-Bank 111

Use the algebraic expression $2x+1$ to complete the table. (3 marks)

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$2x+1$

--- 0 WORK AREA LINES (style=lined) ---

Show Answers Only

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$2x+1$	$3$	$5$	$7$	$9$	$11$	$21$

Show Worked Solution

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$2x+1$	$3$	$5$	$7$	$9$	$11$	$21$

Algebraic Techniques, SM-Bank 110

Use the algebraic expression $x-5$ to complete the table. (3 marks)

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$x-5$

--- 0 WORK AREA LINES (style=lined) ---

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$x$	$1$	$2$	$3$	$4$	$5$	$10$
$x-5$	$-4$	$-3$	$-2$	$-1$	$0$	$5$

Show Worked Solution

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$x-5$	$-4$	$-3$	$-2$	$-1$	$0$	$5$

Algebraic Techniques, SM-Bank 109

Use the algebraic expression $x+3$ to complete the table. (3 marks)

$x$	1	2	3	4	5	10
$x+3$

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Show Answers Only

$x$	1	2	3	4	5	10
$x+3$	4	5	6	7	8	13

Show Worked Solution

$x$	1	2	3	4	5	10
$x+3$	4	5	6	7	8	13

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)

Show Answers Only

$\dfrac{3x}{5}$

Show Worked Solution

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

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)

--- 2 WORK AREA LINES (style=lined) ---
$\dfrac{9a}{7}\ ÷\ \dfrac{c}{b}$ (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
$\dfrac{r}{3}\ ÷\ \dfrac{5}{4s}$ (2 marks)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

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)

--- 4 WORK AREA LINES (style=lined) ---
$\dfrac{14x}{3}\ ÷\ \dfrac{7y}{6}$ (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

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)

--- 3 WORK AREA LINES (style=lined) ---
$\dfrac{8x}{5}\times \dfrac{25y}{12}$ (2 marks)

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Show Answers Only

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

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

Show Worked Solution

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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$\dfrac{11a}{12}$

Show Worked Solution

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

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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$\dfrac{5a}{4}$

Show Worked Solution

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

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

Show Worked Solution

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

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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$\dfrac{r}{4}$

Show Worked Solution

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

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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$\dfrac{5m}{14}$

Show Worked Solution

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

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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$\dfrac{a}{20}$

Show Worked Solution

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

Algebraic Techniques, SM-Bank 095

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

Show Answers Only

$\dfrac{5x}{12}$

Show Worked Solution

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

a.	\(4x^2-2x\)	\(=2x\times 2x+2x\times (-1)\ \ \ \ (\text{HCF}=2x)\)
		\(=2x(2x-1)\)

b.	\(-6ab+3a\)	\(=-3a\times 2b-3a\times(-1)\ \ \ \ (\text{HCF}=-3a)\)
		\(=-3a(2b-1)\)

c.	\(-5q-10q^2\)	\(=-5q\times 1-5q\times 2q\ \ \ \ (\text{HCF}=-5q)\)
		\(=-5q(1+2q)\)

\(\text{Length of large rectangle}\)	\(=a+b\)
\(\text{Width of large rectangle}\)	\(=a+b\)

\(\text{Area of Triangle}\)	\(=\dfrac{1}{2}\times \text{base} \times \text{height}\)
	\(=\dfrac{1}{2}\times \Big(2+6m\Big)\times m\)
	\(=\dfrac{m}{2}\Big(2+6m\Big)\)
	\(=\dfrac{m}{2}\times 2 +\dfrac{m}{2}\times 6m\)
	\(=m+3m^2\)

\(2pq^2+4pq^2\)	\(=2pq\times q+2pq\times 2p\ \ \ \ (\text{HCF}=2pq)\)
	\(=2pq(q+2p)\)

\(9a^2-6a\)	\(=3a\times 3a+3a\times (-2)\ \ \ \ (\text{HCF}=3a)\)
	\(=3a(3a-2)\)

\(9a^2-6a\)	\(=3a\times 3a+3a\times (-2)\ \ \ \ (\text{HCF}=3a)\)
	\(=3a(3a-2)\)

\(7y+14x\)	\(=7\times y+7\times 2x\ \ \ \ (\text{HCF}=7)\)
	\(=7(y+2x)\)