Middle Years

# 2.03 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

Algebraic fractions
4

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.

5

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.

6

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
7

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

a
b
c
d
e
f
g
h
i
8

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

a
b
c
d
e
f
9

Consider the following rectangular prism:

a

Write an expression for its volume.

b

Write an expression for its surface area.

10

Dylan, Jimmy and Valentina are travelling from Adelaide to Sydney at x, y and z km/h respectively.

Let S_1 represent the average speed of Dylan and Jimmy's vehicles, and S_2 represent the average speed of Dylan and Valentina's vehicles.

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

11

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.

12

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.