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Grade 9

3.04 Simplifying expressions

Worksheet
Simplify algebraic expressions
1

Simplify the following:

a

\left( 5 u - 2 u\right) \times 3

b

\left( 13 a + 15 a\right) \div 7

c

9 p \div \left( 6 p - 3 p\right)

d

80 m n \div \left( 4 m\right) \div \left( 5 n\right).

e

\left( - 7 n \right) \times 4 - \left( 2 n + 5 n\right)

f

\left( 6 u \times 12 v w\right) \div \left( 6 v \times 32 u\right)

g

\left( 19 v - 5 v\right) \div 2 \times 3.

h

30 r \div \left( 3 r + 2 r\right) + 5 r.

i

5 v - 5 \left( 6 v - 2 v\right)

j

6 v + \left( 15 v - \left( 9 v - 2 v\right)\right).

k

- 8 t - \left( 8 t - 6 t \div 2\right)

l

\left( \left( - 8 r \right) \times 16 s t\right) \div \left( 24 s \times \left( - 9 r \right)\right)

m

\left( 7 m - 5 m\right) \times 4 m

n

\left(-4n\right)\times 8-\left(7n+4n\right)

2

Simplify the following:

a

\dfrac{16 x}{3 x + 5 x}

b

\dfrac{20 s}{4 s} \times 6 t

c

\dfrac{21 m}{3} + \dfrac{10 m}{2}

d

9 v \times 2 - \dfrac{25 v}{5}

e

15 m + \dfrac{51 m - m}{5 m}.

f

\dfrac{58 r + 5 r}{7 r \times 3}

g

\dfrac{7 r + 5 r}{7 r - 5 r}

h

\dfrac{7 r s + 23 r s}{5 r \times 3 s}

i

5 n^{2} + \dfrac{20 n^{2}}{5}

j

3 m - \dfrac{12 m^{2}}{3 m}.

k

5 n \times 4 n - \dfrac{8 n^{2}}{4}

l

\dfrac{7st+53st}{4s\times 5t}

3

Simplify the following:

a

2 t + 4 \times 8 t

b

2 c + 7 c \times 3 + 4 c

c

61 a - 6 \times 9 a - 2 a

d

8 \times 3 a + 6 \times 2 a

e

3 n^{2} + 8 n^{2} - 3 n^{2}

f

9 n^{2} + 3 n \times 6 n

g

5 n^{2} + 5 n^{2} \times 2 + 7 n^{2}

h

5 m n \times 3 n - 2 m n^{2}

i

20 w - 10 w \div 2 - 3 w

j

6 p \times 8 q \div 2

k

60 n^{2} \div 5 n \div 3

l

8 j k \times 15 k \div 10 j

m

20 p q \div 5 q \times 2 p

n

12 s - 8 s \div 4

o

12 u \div 3 - 2 u

p

8jk\times 15k\div 20j

Equivalent expressions
4

State whether the following expressions are equivalent to 8x+6y-9.

a
4x+6y-4x+6+3
b
10x-2x+5y+y-9
c
8x+3(2y-3)
d
2(4x-3y)-9
5

Write three algebraic expressions that are equivalent to each of the following expressions:

a
8(2x-5)+12
b
\dfrac{12x-20}{6}
c
6(2x+8)+4(6x-7)
d
\dfrac{16x}{5x-x^2}
6

Show that the expression \, \dfrac{6x^2-10x}{-2x}+2(x+7) \, is equivalent to the expression -5 \left(\dfrac{x}{5}-4\right)-1.

Algebraic fractions
7

Consider \dfrac{w}{3} + \dfrac{w}{9}.

a

Find the lowest common denominator of \dfrac{w}{3} and \dfrac{w}{9}.

b

Hence, write \dfrac{w}{3} + \dfrac{w}{9} as a single fraction.

8

Consider \dfrac{m}{4} - \dfrac{m}{12}.

a

Find the lowest common denominator of \dfrac{m}{4} and \dfrac{m}{12}.

b

Hence, write \dfrac{m}{4} - \dfrac{m}{12} as a single fraction in simplest form.

9

Simplify the following expressions:

a
\dfrac{8 x}{12} + \dfrac{10 x}{12}
b
\dfrac{y}{11} - \dfrac{9 y}{11}
c
\dfrac{17 y}{20} + \dfrac{14 y}{20}
d
\dfrac{7 x}{3} - \dfrac{5 x}{3}
e
\dfrac{3 x}{4} + \dfrac{3 x + 4}{4}
f
\dfrac{p}{2} - \dfrac{p}{4}
g
\dfrac{3 p}{7} + \dfrac{2 p}{35}
h
\dfrac{p}{3} + \dfrac{6 p}{15}
i
\dfrac{x}{3} + \dfrac{x + 2}{12}
j
\dfrac{4 x + 3}{3} - \dfrac{x - 3}{6}
Applications
10

Write an algebraic expression for the perimeter of the following shapes:

a
b
c
d
e
f
g
h
i
11

Write an algebraic expression for the area of the following shapes:

a
b
c
d
e
f
12

Consider the following rectangular prism:

a

Write an expression for its volume.

b

Write an expression for its surface area.

13

Xavier, Yasmin and Zachary are travelling from Thunder Bay to Windsor at x, y and z km/h respectively.

Let S_1 represent the average speed of Xavier and Yasmin's vehicles, and S_2 represent the average speed of Xavier and Zachary's vehicles.

Write a simplified expression for S_1 + S_2 in terms of x, y and z.

14

Frank wants to determine the profit he makes on each eraser he sells. Frank can sell 3 erasers for x dollars, and it costs y dollars to make 9 erasers.

a

Determine a simplified expression for the profit Frank makes on each eraser. Leave your answer as a single fraction.

b

Frank realised he could sell 3 erasers for \$3 more than they previously were, and reduce the cost of production of 9 erasers by \$2. Find the new profit per eraser.

15

Glen lived in a rectangular bedroom. His mother asked him to find out the area of his room, and since Glen did not have a measuring tape or ruler, he counted how many of his steps each wall was.

If Glen's foot length is x cm, and two adjacent walls were 12 and 10 footsteps long, write an expression for the area of his room.

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Outcomes

9.C1.3

Compare algebraic expressions using concrete, numerical, graphical, and algebraic methods to identify those that are equivalent, and justify their choices.

9.C1.4

Simplify algebraic expressions by applying properties of operations of numbers, using various representations and tools, in different contexts.

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