Simplifying Algebraic Expressions

Algebraic Techniques, SM-Bank 086

Simplify the following algebraic expression, giving your answer in simplest form. (2 marks)

$\dfrac{5xy\times 3xy}{10\times xy}$

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$\dfrac{3xy}{2}$

Show Worked Solution

$\dfrac{5xy\times 3xy}{10\times xy}$	$=\dfrac{15\times xy\times xy}{10\times xy}$
	$=\dfrac{15xy}{10}$
	$=\dfrac{3xy}{2}$

Algebraic Techniques, SM-Bank 085

Simplify the following algebraic expression, giving your answer in simplest form. (2 marks)

$\dfrac{3\times ab\times 4a}{a\times 2b}$

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$6a$

Show Worked Solution

$\dfrac{3\times ab\times 4a}{a\times 2b}$	$=\dfrac{12\times a\times ab}{2\times ab}$
	$=6a$

Algebraic Techniques, SM-Bank 084

Simplify the following algebraic expression, giving your answer in simplest form. (2 marks)

$\dfrac{4m\times 3mn\times 5n}{30m\times 2mn}$

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$n$

Show Worked Solution

$\dfrac{4m\times 3mn\times 5n}{30m\times 2mn}$	$=\dfrac{60\times m\times mn\times n}{60\times m\times mn}$
	$=n$

Algebraic Techniques, SM-Bank 083

A rectangular prism is known to have a length of $5$ cm, a width of $3$ cm and a height of $x+2$ cm. Write an expression for the volume of the prism, giving your answer in simplest form. (3 marks)

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$(15x+30)\ \text{cm}^3$

Show Worked Solution

$\text{Volume}$	$=\text{length}\times\text{width}\times\text{height}$
	$=5\times 3\times (x+2)$
	$=15\times (x+2)$
	$=15\times x +15\times 2$
	$=(15x+30)\ \text{cm}^3$

Algebraic Techniques, SM-Bank 082

Write a simplified expression for the volume of the rectangular prism below. (3 marks)

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$24abc$

Show Worked Solution

$\text{Volume}$	$=\text{length}\times\text{width}\times\text{height}$
	$=4c\times 3a\times 2b$
	$=24abc$

Algebraic Techniques, SM-Bank 081

Write a simplified expression for the volume of the cube below. (3 marks)

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$8x^3$

Show Worked Solution

$\text{Volume}$	$=\text{(side)}^3$
	$=(2x)^3$
	$=2x\times 2x\times 2x$
	$=8x^3$

Algebraic Techniques, SM-Bank 080

Write a simplified expression for the area of the shape below. (3 marks)

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$48ab$

Show Worked Solution

$\text{Area}$	$=\text{Area big rectangle}-\text{Area small rectangle}$
	$=6a\times 10b-2a\times 6b$
	$=60ab-12ab$
	$=48ab$

Algebraic Techniques, SM-Bank 079

Write a simplified expression for the area of the rectangle below. (2 marks)

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$28b^2$

Show Worked Solution

$\text{Area}$	$=\text{length}\times \text{width}$
	$=7b\times 4b$
	$=28b^2$

Algebraic Techniques, SM-Bank 078

Write a simplified expression for the area of the rectangle below. (2 marks)

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$10mn$

Show Worked Solution

$\text{Area}$	$=\text{length}\times \text{width}$
	$=5m\times 2n$
	$=10mn$

Algebraic Techniques, SM-Bank 077

Write a simplified expression for the area of the triangle below. (2 marks)

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$3x$

Show Worked Solution

$\text{Area}$	$=\dfrac{1}{2}\times\text{base}\times \text{height}$
	$=\dfrac{1}{2}\times 2x\times 3$
	$=\dfrac{1}{2}\times 6x$
	$=3x$

Algebraic Techniques, SM-Bank 076

The side length of a regular hexagon is $2x+5$. Write an expression for the perimeter of the hexagon. (2 marks)

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$6\times(2x+5)\ \ \text{or }\ 6(2x+5)\ \ \text{or }\ 12x+30$

Show Worked Solution

$\text{A regular hexagon has all sides equal}$

$\therefore\ \text{Perimeter}$	$=6\times\text{side length}$
	$=6\times(2x+5)$
	$=6\times 2x+6\times 5$
	$=12x+30$

Algebraic Techniques, SM-Bank 075

The side length of a square is $x-3$. Write an expression for the perimeter of the square. (2 marks)

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

Show Worked Solution

$\text{A square has all sides equal}$

$\therefore\ \text{Perimeter}$	$=4\times\text{side length}$
	$=4\times(x-3)$
	$=4\times x-4\times 3$
	$=4x-12$

Algebraic Techniques, SM-Bank 074

The side length of an equilateral triangle is $x+2$. Write an expression for the perimeter of the triangle. (2 marks)

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$3\times(x+2)\ \ \text{or }\ 3(x+2)\ \ \text{or }\ 3x+6$

Show Worked Solution

$\text{An equilateral triangle has all sides equal}$

$\therefore\ \text{Perimeter}$	$=3\times\text{side length}$
	$=3\times(x+2)$
	$=3\times x+3\times 2$
	$=3x+6$

Algebraic Techniques, SM-Bank 073

The perimeter of a regular hexagon is equal to $4x+3y$. Write an expression for the side length of the hexagon. (2 marks)

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$\dfrac{4x+3y}{6}$

Show Worked Solution

$\text{A regular hexagon has all sides equal}$

$\therefore\ \text{Side length}$	$=\dfrac{\text{Perimeter}}{6}$
	$=\dfrac{4x+3y}{6}$

Algebraic Techniques, SM-Bank 072

The perimeter of a square is equal to $5q-3$. Write an expression for the side length of the square. (2 marks)

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$\dfrac{5q-3}{4}$

Show Worked Solution

$\text{A square has all sides equal}$

$\therefore\ \text{Side length}$	$=\dfrac{\text{Perimeter}}{4}$
	$=\dfrac{5q-3}{4}$

Algebraic Techniques, SM-Bank 071

The perimeter of an equilateral triangle is equal to $4m+5$. Write an expression for the side length of the triangle. (2 marks)

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$\dfrac{4m+5}{3}$

Show Worked Solution

$\text{An equilateral triangle has all sides equal}$

$\therefore\ \text{Side length}$	$=\dfrac{\text{Perimeter}}{3}$
	$=\dfrac{4m+5}{3}$

Algebraic Techniques, SM-Bank 070 MC

Which of the following expressions is equivalent to $(m\times 4)÷(3\times n)$?

$\dfrac{3m}{4n}$
$\dfrac{3n}{4m}$
$\dfrac{3mn}{4}$
$\dfrac{4m}{3n}$

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$D$

Show Worked Solution

$(m\times 4)÷(3\times n)$

$=\dfrac{m\times 4}{3\times n}$

$=\dfrac{4m}{3n}$

$\Rightarrow D$

Algebraic Techniques, SM-Bank 069 MC

Which of the following expressions is equivalent to $m÷(3\times n)$?

$\dfrac{m}{3n}$
$\dfrac{3}{mn}$
$\dfrac{3m}{n}$
$\dfrac{mn}{3}$

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$A$

Show Worked Solution

$m÷(3\times n)$

$=\dfrac{m}{3\times n}$

$=\dfrac{m}{3n}$

$\Rightarrow A$

Algebraic Techniques, SM-Bank 068 MC

Which of the following expressions is equivalent to $x÷2\times y$?

$\dfrac{2x}{y}$
$\dfrac{xy}{2}$
$\dfrac{2y}{x}$
$\dfrac{x}{2y}$

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$B$

Show Worked Solution

$x÷2\times y$

$=\dfrac{x}{2}\times y$

$=\dfrac{xy}{2}$

$\Rightarrow B$

Algebraic Techniques, SM-Bank 067 MC

Which of the following expressions is equivalent to $a÷b\times c$?

$\dfrac{ac}{b}$
$\dfrac{a}{bc}$
$\dfrac{ab}{c}$
$\dfrac{bc}{a}$

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$A$

Show Worked Solution

$a÷b\times c$

$=\dfrac{a}{b}\times c$

$=\dfrac{ac}{b}$

$\Rightarrow A$

Algebraic Techniques, SM-Bank 066 MC

Which of the following expressions is equivalent to $a÷(b\times c)$?

$\dfrac{ac}{b}$
$\dfrac{a}{bc}$
$\dfrac{ab}{c}$
$\dfrac{bc}{a}$

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$B$

Show Worked Solution

$a÷(b\times c)$

$=\dfrac{a}{b\times c}$

$=\dfrac{a}{bc}$

$\Rightarrow B$

Algebraic Techniques, SM-Bank 065

Simplify the following algebraic expressions, giving your answer in simplest form.

$\dfrac{9mn}{3}$ (1 mark)

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$\dfrac{4y}{12x}$ (1 mark)

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$\dfrac{16a^2}{-4ab}$ (2 marks)

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$\dfrac{-18b^2}{-24b}$ (2 marks)

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a. $3mn$

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

c. $\dfrac{-4a}{b}$

d. $\dfrac{3b}{4}$

Show Worked Solution

a.	$\dfrac{9mn}{3}$	$=\dfrac{3\times 3\times m\times n}{3}$
		$=3mn$

b.	$\dfrac{4y}{12x}$	$=\dfrac{4\times y}{4\times 3\times x} $
		$=\dfrac{y}{3x}$

c.	$\dfrac{16a^2}{-4ab}$	$=\dfrac{4\times 4\times a\times a}{-4\times a\times b}$
		$=\dfrac{-4a}{b}$

d.	$\dfrac{-18b^2}{-24b}$	$=\dfrac{-6\times 3\times b\times b}{-6\times 4\times b}$
		$=\dfrac{3b}{4}$

Algebraic Techniques, SM-Bank 064 MC

Which of the following algebraic expressions is equivalent to $\dfrac{20xy}{5yz}$?

$5xy^2z$
$4xy^2z$
$\dfrac{4}{z}$
$\dfrac{4x}{z}$

Show Answers Only

$D$

Show Worked Solution

$\dfrac{20xy}{5yz}$

$=\dfrac{5\times 4\times x\times y}{5\times y\times z}$

$=\dfrac{4x}{z}$

$\Rightarrow D$

Algebraic Techniques, SM-Bank 063 MC

Which of the following algebraic expressions is equivalent to $x\times x\times y\times x\times y$?

$6xy$
$x^2y^3$
$x^3y^2$
$3x+2y$

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$C$

Show Worked Solution

$x\times x\times y\times x\times y$

$=x\times x\times x\times y\times y$

$=x^3y^2$

$\Rightarrow C$

Algebraic Techniques, SM-Bank 062 MC

Which of the following algebraic expressions is equivalent to $3a\times 2b\times ab\times 2$?

$7ab$
$12ab$
$6a^2b^2$
$12a^2b^2$

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$D$

Show Worked Solution

$3a\times 2b\times ab\times 2$

$=3\times 2\times 2\times a\times a\times b\times b$

$=12a^2b^2$

$\Rightarrow D$

Algebraic Techniques, SM-Bank 061 MC

Which of the following algebraic expressions is equivalent to $4\times x\times y\times 2$?

$42xy$
$8xy$
$4xy^2$
$16xy$

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$B$

Show Worked Solution

$4\times x\times y\times 2$

$=4\times 2\times x\times y$

$=8xy$

$\Rightarrow B$

Algebraic Techniques, SM-Bank 060

Simplify the following algebraic expressions, giving your answer in simplest form.

$-3mn\times 2m$ (1 mark)

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$4a^2b\times (-3b)$ (1 mark)

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$-8cd\times (-2d)$ (1 mark)

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$4a^2b\times 3abc$ (2 marks)

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a. $-6m^2n$

b. $-12a^2b^2$

c. $16cd^2$

d. $12a^3b^2c$

Show Worked Solution

a.	$-3mn\times 2m$	$=-3\times 2\times m\times n\times m $
		$=-6m^2n$

b.	$4a^2b\times (-3b)$	$=4\times -3\times a\times a\times b\times b $
		$=-12a^2b^2$

c.	$-8cd\times (-2d)$	$=-8\times -2\times c\times d\times d $
		$=16cd^2$

d.	$4a^2b\times 3abc$	$=4\times 3\times a\times a\times b\times a\times b\times c $
		$=12a^3b^2c$

Algebraic Techniques, SM-Bank 059

Simplify the following algebraic expressions, giving your answer in simplest form.

$4pq\times 6r$ (1 mark)

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$3m\times m$ (1 mark)

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$7x\times 3x$ (1 mark)

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$2ab\times 3bc\times c$ (1 mark)

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a. $24pqr$

b. $3m^2$

c. $21x^2$

d. $6ab^2c^2$

Show Worked Solution

a.	$4pq\times 6r$	$=4\times 6\times p\times q\times r $
		$=24pqr$

b.	$3m\times m$	$=3\times m\times m $
		$=3m^2$

c.	$7x\times 3x$	$=7\times 3\times x\times x $
		$=21x^2$

d.	$2ab\times 3bc\times c$	$=2\times 3\times a\times b\times b\times c\times c $
		$=6ab^2c^2$

Algebraic Techniques, SM-Bank 058

Simplify the following algebraic expressions, giving your answer in simplest form.

$5\times 4y$ (1 mark)

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$4m\times 3$ (1 mark)

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$2a\times 3b\times 4$ (1 mark)

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$4c\times 2b\times 5d$ (1 mark)

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a. $20y$

b. $12m$

c. $24ab$

d. $40bcd$

Show Worked Solution

a.	$5\times 4y$	$=5\times 4\times y $
		$=20y$

b.	$4m\times 3$	$=4\times 3\times m $
		$=12m$

c.	$2a\times 3b\times 4$	$=2\times 3\times 4\times a\times b $
		$=24ab$

d.	$4c\times 2b\times 5d$	$=4\times 2\times 5\times c\times b\times d $
		$=40bcd$

Algebraic Techniques, SM-Bank 057

State the coefficients of $\large x$ and $\large y$ that make the expressions below equivalent? (2 marks)

$9x+3y-$

$x+$

$y=6x+8y$

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$3 , 5$

Show Worked Solution

$9x+3y-$

$x+$

$y=6x+8y$

$\text{Equating the }x\ \text{values}\longrightarrow$	$9x-$		$x=6x$
	$\therefore\ $		$=3$

$\text{Equating the }y\ \text{values}\longrightarrow$	$3y+$		$y=8y$
	$\therefore\ $		$=5$

Algebraic Techniques, SM-Bank 056

Write an expression for the perimeter of the shape below. Give your answer in simplest form. (2 marks)

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Show Answers Only

$8x+6y+10$

Show Worked Solution

$\text{Perimeter}$	$=3+3y+4x+2+3+4x+2+3y$
	$=4x+4x+3y+3y+3+2+3+2$
	$=8x+6y+10$

Algebraic Techniques, SM-Bank 055

Write an expression for the perimeter of the rectangle below. Give your answer in simplest form. (2 marks)

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Show Answers Only

$20x+2$

Show Worked Solution

$\text{Method 1}$

$\text{Perimeter}$	$=7x+1+3x+7x+1+3x$
	$=7x+3x+7x+3x+1+1$
	$=20x+2$

$\text{Method 2 (Advanced)}$

$\text{Perimeter}$	$=2(7x+1)+2\times 3x$
	$=14x+2+6x$
	$=20x+2$

Algebraic Techniques, SM-Bank 054

Write an expression for the perimeter of the triangle below. Give your answer in simplest form. (2 marks)

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$5m+n$

Show Worked Solution

$\text{Perimeter}$	$=2m+(m+2n)+(2m-n)$
	$=2m+m+2m+2n-n$
	$=5m+n$

Algebraic Techniques, SM-Bank 053 MC

Which of the following expressions is equivalent to $16ab$?

$16+a+b$
$4a+3b+7a+2b$
$5ab-20ab+ab$
$2ab-3ab+4ab+13ab$

Show Answers Only

$D$

Show Worked Solution

$\text{Option D:}$

$2ab-3ab+4ab+13ab$

$=16ab$

$\Rightarrow D$

Algebraic Techniques, SM-Bank 052 MC

Which of the following expressions is equivalent to $15m-4n$?

$11mn$
$8m-5n+7m+n$
$5m-6n+10m-2n$
$9m-2n-6m-2n$

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$B$

Show Worked Solution

$\text{Option B:}$

$8m-5n+7m+n$	$=8m+7m-5n+n$
	$=15m-4n$

$\Rightarrow B$

Algebraic Techniques, SM-Bank 051 MC

Which of the following expressions is equivalent to $4a-3b$?

$7ab$
$2a-3b-6a$
$b+5a-4b-a$
$(2a)^2-3b$

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$C$

Show Worked Solution

$\text{Option C:}$

$b+5a-4b-a$	$=5a-a+b-4b$
	$=4a-3b$

$\Rightarrow C$

Algebraic Techniques, SM-Bank 050

Simplify the expression $2m-6m+4m-3m$. (1 mark)

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$-3m$

Show Worked Solution

$2m-6m+4m-3m$

$=-3m$

Algebraic Techniques, SM-Bank 049

Simplify the expression $5x-3y-4x-2y$ by collecting like terms. (2 marks)

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$x-5y$

Show Worked Solution

$5x-3y-4x-2y$	$=5x-4x-3y-2y$
	$=x-5y$

Algebraic Techniques, SM-Bank 048

Simplify the expression $2mn+3m+5mn-n$ by collecting like terms. (2 marks)

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$7mn+3m-n$

Show Worked Solution

$2mn+3m+5mn-n$	$=2mn+5mn+3m-n$
	$=7mn+3m-n$

Algebraic Techniques, SM-Bank 047

Simplify the expression $4a+3b-5a+6b$ by collecting like terms. (2 marks)

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$-a+9b\ \text{ or }\ 9b-a$

Show Worked Solution

$4a+3b-5a+6b$	$=4a-5a+3b+6b$
	$=-a+9b\ \text{ or }\ 9b-a$

Algebraic Techniques, SM-Bank 046

Simplify the expression $3x+2x+3y+y$ by collecting like terms. (2 marks)

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$5x+4y$

Show Worked Solution

$3x+2x+3y+y$

$=5x +4y$

Algebraic Techniques, SM-Bank 045 MC

Which of the following is a like term to $8m$?

$3m^2$
$8mn$
$2m$
$4+m$

Show Answers Only

$C$

Show Worked Solution

$\text{For like terms algebraic parts are identical:}$

$\text{Option A:}\longrightarrow m^2\neq m$

$\text{Option B:}\longrightarrow mn\neq m$

$\text{Option C:}\longrightarrow m = m\ \checkmark$

$\text{Option D:}\longrightarrow 4+m\neq m$

$\therefore\ 8m\text{ and}\ 2m\text{ are like terms.}$

$\Rightarrow C$

Algebraic Techniques, SM-Bank 030 MC

If $\large x$ is a negative number, which of the following is true?

$x+2$ is always negative.
$x\times -1$ is always negative.
$2-x$ is always negative.
$x-2$ is always negative.

Show Answers Only

$D$

Show Worked Solution

$\text{Considering the options}$

$\text{Option 1: If}\ x\ge -2\ \text{then}\ x+2\ge 0, \therefore \text{not always negative}$

$\text{Option 2:}\ x\ \text{is negative }\therefore x\times -1\ \text{is always positive}$

$\text{Option 3:}\ x\ \text{is negative }\therefore 2-x\ \text{is always positive}$

$\text{Option 4:}\ x\ \text{is negative }\therefore x-2\ \text{is always negative}$

$\Rightarrow D$

Algebraic Techniques, SM-Bank 029

A triangle has a base of length $3$ metres and a perpendicular height of $h$ metres.

Write an expression for the area of the triangle. (2 marks)

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$\dfrac{3h}{2}\ \text{m}^2$

Show Worked Solution

$\text{Area}$

$=\dfrac{1}{2}\times \text{base}\times\text{perpendicular height}$

$=\dfrac{1}{2}\times 3\times h$

$=\dfrac{1}{2}\times 3h$

$=\dfrac{3h}{2}\ \text{m}^2$

Algebraic Techniques, SM-Bank 028

A rectangle has sides of $b$ metres and $c$ metres in length.

Write an expression for the perimeter of the rectangle. (1 mark)

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Write an expression for the area of the rectangle. (2 marks)

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a. $(2b+2c)\ \text{m}$

b. $bc\ \text{m}^2$

Show Worked Solution

a. $\text{Perimeter}$

$=b+c+b+c$

$=(2b+2c)\ \text{m}$

b. $\text{Area}$

$=\text{length}\times\text{width}$

$=b\times c$

$=bc\ \text{m}^2$

Algebraic Techniques, SM-Bank 027

A square has sides $x$ centimetres in length.

Write an expression for the perimeter of the square. (1 mark)

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Write an expression for the area of the square. (2 marks)

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a. $4x\ \text{cm}$

b. $x^2\ \text{cm}^2$

Show Worked Solution

a. $\text{Perimeter}$

$=4\times x$

$=4x\ \text{cm}$

b. $\text{Area}$

$=\text{side}^2$

$=x^2\ \text{cm}^2$

Algebraic Techniques, SM-Bank 026

At the local markets mangoes cost $$x$ and oranges cost $$y$.

Benji bought 6 mangoes and 5 oranges. Write an expression for the total cost of Benji's mangoes and oranges. (2 marks)

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$6x+5y$

Show Worked Solution

$\text{Total cost}$

$=6\times x +5\times y$

$=6x+5y$

Algebraic Techniques, SM-Bank 025

Juan bought 8 identical Christmas presents online for his relatives.

Write an expression for the total price in dollars, if the cost of each present is $$y$. (1 mark)

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When he goes to the checkout Juan discovers that all the presents were reduced in price by $10 each.

i. Write an expression for the reduced price of one present. (1 mark)

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ii. Write an expression for the total reduced price of all 8 presents. (2 marks)

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a. $8y$

b. i. $y-10$

ii. $8(y-10)$

Show Worked Solution

a. $\text{Price}=8\times y=8y$

b. i. $\text{Reduced Price per present}=y-10$

ii. $\text{Total Reduced Price}=8\times (y-10)=8(y-10)$

Algebraic Techniques, SM-Bank 024

Rachel works as a waiter and is paid $27 per hour.

Write an expression for her wages in a week where she works $x$ hours. (1 mark)

Show Answers Only

$$27x$

Show Worked Solution

$\text{Wages} = 27\times x =$27x$

Algebraic Techniques, SM-Bank 023

Describe the expression $\dfrac{5}{3q}$ in words. (2 marks)

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$\text{The quotient of 5 and }3q$

Show Worked Solution

$\text{The quotient of 5 and }3q$

Algebraic Techniques, SM-Bank 022

Describe the expression $(x-y)\times 2$ in words. (2 marks)

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$\text{Twice the difference between }x\ \text{and }y$

Show Worked Solution

$\text{Twice the difference between }x\ \text{and }y$

Write an expression for the product of $4\large x$ and $\large y$. (1 mark)

Show Answers Only

$4\large xy$

Show Worked Solution

$\text{Product means multiply}$

$\therefore\ \text{product of } 4x$ $\text{and }y=4x\times y=4xy$

\(\dfrac{5xy\times 3xy}{10\times xy}\)	\(=\dfrac{15\times xy\times xy}{10\times xy}\)
	\(=\dfrac{15xy}{10}\)
	\(=\dfrac{3xy}{2}\)

\(\text{Volume}\)	\(=\text{length}\times\text{width}\times\text{height}\)
	\(=5\times 3\times (x+2)\)
	\(=15\times (x+2)\)
	\(=15\times x +15\times 2\)
	\(=(15x+30)\ \text{cm}^3\)

\(\text{Volume}\)	\(=\text{length}\times\text{width}\times\text{height}\)
	\(=4c\times 3a\times 2b\)
	\(=24abc\)

\(\text{Volume}\)	\(=\text{(side)}^3\)
	\(=(2x)^3\)
	\(=2x\times 2x\times 2x\)
	\(=8x^3\)

\(\text{Area}\)	\(=\text{Area big rectangle}-\text{Area small rectangle}\)
	\(=6a\times 10b-2a\times 6b\)
	\(=60ab-12ab\)
	\(=48ab\)

\(\dfrac{3\times ab\times 4a}{a\times 2b}\)	\(=\dfrac{12\times a\times ab}{2\times ab}\)
	\(=6a\)

\(\text{Area}\)	\(=\text{length}\times \text{width}\)
	\(=7b\times 4b\)
	\(=28b^2\)

\(\text{Area}\)	\(=\text{length}\times \text{width}\)
	\(=5m\times 2n\)
	\(=10mn\)

\(\text{Area}\)	\(=\dfrac{1}{2}\times\text{base}\times \text{height}\)
	\(=\dfrac{1}{2}\times 2x\times 3\)
	\(=\dfrac{1}{2}\times 6x\)
	\(=3x\)