D & D2c) 4(n + 2n + 1) – 3n = d) x(x – 8) – 2(x – 8) = e) ax(x + y) + ax(x – y) = 3. Give an expression for the area of this shape. 4. Give an expression for the volume of

– 1 –

– © D & D Resources 2013 – Preparation for CAT –

ContentsAlgebraic Notation 2

Order of Operations 3

Powers in Algebra 4

Expanding Single Brackets 5

Expanding Quadratics 1 6

Expanding Quadratics 2 7

Factorising Common Factors 8

Factorising Quadratics 1 9

Factorising Quadratics 2 10

Rational Expressions 11

Algebraic Fractions – Addition and Subtraction 12

Algebraic Fractions – Multiplication and Division 13

Substitution into Formulae 14

Rearranging Formulae 1 15

Rearranging Formulae 2 16

Algebra Skills Review 1 17

Algebra Skills Review 2 18

Higher Level Thinking 1 19

Linear Equations 20

Linear Equations and Inequations 21

Applications of Linear Equations 22

Simultaneous Equations 1 23

Simultaneous Equations 2 24

Applications of Simultaneous Equations 25

Quadratic Equations 1 26

Quadratic Equations 2 27

Applications of Quadratic Equations 28

Exponential Equations 29

Equations Review 30

Applications of Equations Review 31




Answers 35

©D & DResourcesWriters and publishers of mathematical resources

– 2 –


Alg

ebra

ic N

otat

ion

1. Write in algebraic notation. a) 4 more than k

b) double y

c) 2 less than 6 times m

d) v divided by 9

e) w times itself

f) p less than q

2. Write algebraic expressions for the following. a) The number of months in m years

b) Samantha’s age in 10 years’ time if she is now y years old

c) The price of w articles at a cost of c cents each

d) Tui is paid d dollars for h hours work. What is her hourly rate?

e) A piece of string, which is s metres long, has 3 lengths of k cm cut from it. What is the length of the remaining piece?

3. Write an algebraic expression for. a) The sum of two numbers

b) The difference between two numbers

c) The product of two numbers

d) The sum of the squares of two numbers

e) The square of the sum of two numbers

A square of length x cm is cut from each of the four corners of the card. a) Give an expression for the length of the card.

b) Give an expression for the width of the card.

5. The sum of the squares of two numbers is equal to double its product. a) Choose two different numbers and prove or disprove this statement.

b) Write an algebraic expression for this statement.

c) Show algebraically that the numbers must be equal to each other in order for this statement to be true.

4. A rectangular piece of card is 20 cm long and 12 cm wide.

f) The square root of the product of two numbers

g) The difference between one number and the reciprocal of another number

xx x

x

x

x x

x

12 cm

20 cm

Answer the following questions.Answer the following questions.


– 5 –


1. Expand the following. a) 3(a + 5) =

b) 5(2x – 3) =

c) m(p – q) =

d) 2v(3v + 4w) =

e) –6(x – 2) =

f) 4a(a + 3b + c) =

g) xy(xy – z) =

h) –7c(4c + 3d) =

2. Expand and simplify. a) 7y + 2(y – 4) =

b) 2(5x + 2) – 3(x + 1) =

c) 4(n2 + 2n + 1) – 3n =

d) x(x – 8) – 2(x – 8) =

e) ax(x + y) + ax(x – y) =

3. Give an expression for the area of this shape.

4. Give an expression for the volume of this shape in expanded form.

5. Choose two consecutive numbers. Multiply them together and then subtract the bigger number from your answer. What do you notice?

x + 1

x + 1

x

2x

x + 2

x

10

Prove this result algebraically.

Expanding Single Brackets

Answer the following questions. Answer the following questions.


– 8 –


Fact

oris

ing

with

Com

mon

Fac

tors

1. Factorise the following.

a) 7x + 21 =

b) 3p – 15q =

c) w2 – 10w =

d) 8xy – 12x =

e) 9p2 + 15p =

f) a2c – abc =

g) 2x2 + 4xy – 6xz =

h) 5p – 10p2q =

i) 20m2n + 30mn2 =

j) 4y3 – 6y2 =

k) –v3 + 6vw =

l) –12x3 – 8x2y =

m) a(x + 2) + b(x + 2) =

n) a2

x + abx2 =

2. One factor of 2m3n – mn2 is mn. What is the other factor?

3. A diagram of a rectangular table top with two extension pieces is shown below.

Give an algebraic expression for the area and then factorise it.

4. The formula for the surface area of a cylinder is S = 2πr2 + 2πrh. Factorise this expression.

5. The volume of a cone with height h sitting inside a sphere of radius r is given by the formula V = 2

3πrh2 – 1

3πh3. Factorise this

expression.

6. The diagram shows a circle inscribed within a square. Find an expression for the shaded area and factorise it.

r

7. An isosceles triangle has length y and height 3x as shown in the diagram. A rectangle with height x has been drawn inside the triangle. The smaller triangles on each side are also isosceles.

Give an expression for the shaded area, simplify this expression and factorise it.

x x

y

3x


a

ab

20 cm

20 cm


– 13 –


2. Simplify x + y1x+ 1y

3. Simplify by factorising each expression first.

a2 + aba2 – b2

x ab – b2

a2b2

4. A large L shaped block and a smaller rectangular block with dimensions are shown below.

a) Give an expression for the volume of the large L shaped block.

b) Give an expression for the volume of the rectangular block.

c) Form an algebraic fraction and simplify it to give an expression for how many small blocks would be needed to have the same volume as the large L shaped block.

x3

x

2x

3y

y

1. Simplify the following.

a) 8pp2 x 8pp2

=

b) m5m4

x m5m4

=

c) 2k15

5k2 x 2k15

5k2

=

d) 4w2

9z3z2

8w x 4w2

9z3z2

8w =

e) x62x3

÷ x62x3

=

f) 3a2

2a6b

÷ 3a2

2a6b

=

g) 3mn2

8p9np2

÷ 3mn2

8p9np2 =

h) xyyzzx

x xyyzzx x xyyzzx

=

i) 6a2

bb2

ac3b2c

x 6a2

bb2

ac3b2c

÷ 6a2

bb2

ac3b2c

=

x2

x3

x

Answer the following questions.

Algebraic Fractions – M

ultiplication and Division


– 18 –


1. Factorise fully.

a) 2pq – 14p2

b) 6x2 + x3

c) –8wz – 12w2y

d) x2 + 14x + 33

e) x2 – 4x – 21

f) x2 – 100

g) x2 – 12x + 35

h) 3x2 – 27

i) 2x2 – 14x + 12

j) 2x2 + x – 6


a) x2 – 42x + 4

b) x2 – 3xx – 3


a) 4x5–y2

b) x6+ x –13

4. Simplify fully.

a) 5m2p2

6pq25mn x 5m2p2

6pq25mn

b) 3ab4c2

9bc8a2 ÷ 3ab4c2

9bc8a2

5. The magic number in a magic square can be

found by using the formula M = n(n2 +1)2

, where

M is the magic number and n is the size of the square n x n. Find the magic number in a 7 x 7 square.

6. Rearrange the following formulae to make x the subject.

a) y = mx + c

b) w = W = 4(x + y)3

c) x2 + y2 = 25

d) xk

= y

e) z = yx

Alg

ebra

Ski

lls R

evie

w 2 ©

D & DResourcesWriters and publishers of mathematical resources

– 22 –


App

licat

ions

of L

inea

r Equ

atio

ns

Answer the following questions. Answer the following questions.Answer the following questions. Answer the following questions.

b) Give an expression for the income received from the booklets.

1. A group of 20 adults and children attend a concert as a fundraiser. The adults pay $5 each and the children $2 each. Altogether $82 is raised. a) In the equation 2c + 5(20 – c) = 82, explain what c and (20 – c) refer to.

4. The width of a rectangle is 34

of its length.

a) If x is the length of the rectangle give an expression for the width of the rectangle.

b) Solve the equation to work out the number of adults who attended.

2. Tyrone and Tyler are collecting pinecones. At present they have 40 between them. a) If Tyrone has p pinecones, how many does Tyler have?

b) If Tyrone gives Tyler 3 of his pinecones, the boys will have the same number. Form an equation and solve it to work out how many each boy has at present.

3. Amy bought two seedlings for her garden. One variety was 8 cm tall when she bought it and grew at a rate of 2.5 cm per week. The other variety was 5 cm tall and grew at a rate of 3 cm each week. After a certain number of weeks the two seedlings were the same height. a) Form an equation to work out when the two seedlings were the same height.

b) Solve this equation.

b) Form an equation to find the length and width of the rectangle if the perimeter is 35 cm.

5. Patrick has a maximum budget of $60 to buy stamps. He will purchase $3 bird stamps and $1 flower stamps. The inequation can be written as 3B + F ≤ 60. a) Patrick wants to buy twice as many flower stamps as bird stamps. Form a new inequation in terms of B.

b) Solve the inequation to find out the maximum number of flower stamps that Patrick can buy.

6. A printing firm produces booklets for students. Each booklet costs $3 to produce and sells for $5. There are also other costs which amount to $900. a) If b represents the number of booklets, give an expression for the total cost.

c) Form an inequation and solve it to work out the number of booklets that would need to be produced to make a profit.


– 27 –


Quadratic Equations 2


1. Solve the following equations. a) x2 + x = 56

b) x2 = 28 – 3x

c) (x + 4)(x + 1) = 40

d) (x – 1)2 = 9

e) x(x + 7) = 60

f) (2x + 1)2 = 3x2 + 13

g) x + 83

= 3x

h) 5x2 = 6 – 13x

i) 2x −1x +1

= x – 1

2. Form a quadratic equation and solve it. a) I think of a positive number, square it, subtract 7 times the original number and add 4. The answer is 34. What numbers am I thinking of?

b) Two numbers have a difference of 5. The sum of their squares is 157. What are the numbers?

d) Two sisters differ in age by 10 years. If their ages are multiplied together and then divided by the sum of their ages, the answer is 12. What are the ages of the two sisters?

e) Subtracting 9 from a positive number will give the same result as multiplying the reciprocal of the number by 36. What is the positive number?

f) If I multiply two consecutive odd numbers together and add 7 to the result, the answer will be 10 times the smaller number. What are the possible odd numbers?


– 30 –


1. Solve the following equations to find x: a) 4(x – 3) = –8

b) 3x – 6 = 6x + 9

c) x4 = 16

d) (2x – 7)(x + 3) = 0

e) x2 – 5x – 50 = 0

f) −3x2

+ 1 = 7

g) 4x – 2(x + 5) = 12

h) x3 = –125

i) 6x2 – 9x = 0

j) 2x2 + 7x + 5 = 0

k) 5x4

= x + 32

l) 6x = 216

Equa

tions

Rev

iew


m) x2 = 20x – 100

n) (x – 6)2 = 16

o) (x3)2 = 64

p) 3x – 3 = 81

q) x2 – 8 = 17

2. Solve the following inequations: a) 3 – 7x ≤ –18

b) 2x5

≥ x +12

3. Solve the following simultaneous equations: a) x + 3y = 2 4x – 2y = 22

b) y = 2x – 1 x – y = –6

c) 4x + y = 4 x = y + 6


– 33 –



1. From a cliff at sea, a seagull ascends briefly then swoops down to land on the beach below. Its flight can be modelled by the equation h = 24 + 6t – 3t2, where h is its height in metres above the ground at any time t in seconds. a) What is the height of the cliff?

3. On her recent trip to France, Kiera bought two canvas paintings of places she had visited and is now having them framed. One painting has a rectangular frame in which the length is 10 cm longer than the width. The other painting has a square frame. The frames have a total perimeter of 300 cm. The rectangular framing will cost $0.50 per cm and the square framing will cost $1 per cm. The total cost to frame both paintings is $210. Form equations and solve them to find out the dimensions of the two paintings and the total cost of framing them.

d) Show that the rule for finding the difference between any two consecutive terms in this sequence can be simplified to 8(n + 1).

c) Nicole also looked at the sequence with the equation T = 4n2 + 4n + 1. She noticed this time that all the numbers were odd square numbers. Explain why this would be true.

b) After how many seconds does the seagull land on the beach?

c) What is the greatest height that the seagull reaches on its ascent?

b) Form an equation and solve it to work out which term in the sequence is equal to 331.

2. a) Nicole was looking at the following sequence of numbers. 1, 7, 19, 37, 61, 91, 127, 169 ……… The rule for the sequence is given as T = 3n2 – 3n + 1. She noticed that all the numbers in the sequence were odd. Explain why this would be true.

Higher Level Thinking 3


– 40 –


Page 17

Page 18

Page 19

1. a) 6x

b) p+ q2

c) 3.2d

2. a) 6w – 5 b) 4k2 – 7k c) –8qr2

d) 1 + m2

e) a2b – ab f) 2k

3. a) 15p10

b) 4a3

c) 64b8

d) 9g2

e) 21f4g6

f) 32m5n15

g) 4k6

h) 24d10

6d8= 4d2

4. a) 30x – 40 b) 10mp – 2m2

c) –3a3 – a3b + a2c

5. a) 6n2 – 12n + 2n – 2 = 6n2 – 10n – 2 b) x3 – 4x2 – 4 + x2

= x3 – 3x2 – 4

6. a) x2 + 6x – 16 b) x2 – 81 c) x2 – 12x + 36 d) 3x2 + 19x + 20 e) 6x2 – 13x + 6

f) 16x2 + 8x + 1 g) 49x2 – 1

1. a) 210

b) Any two consecutive numbers add to give a square number.

c) n(n +1)2

+ (n +1)(n + 2)2

= 2n2 + 4n + 22

= n2 + 2n + 1 = (n + 1)2

2. a) 3, 5, 7. 32 + 72 – 2 x 52 = 8 The answer is 8.

b) Let the numbers be x, x + 2, x + 4. x2 + (x + 4)2 – 2(x + 2)2

x2 + x2 + 8x + 16 – 2x2 – 8x –8 = 8

3. πr2 = x2

2

x2

r2 = 2π

xr= 2π

4. a) A = x2 and P = 4x

Since x = P4

then A = P4

⎛⎝⎜

⎞⎠⎟2

= P2

16

b) When P =12, A = 144 ÷ 16 = 9 cm2

c) P = 2x + 2y and A = xy

Since y = P – 2x2

then

A = x P – 2x

2⎛⎝⎜

⎞⎠⎟= xP2– x2

5. ax–xa

a + x=

a2 – x2

axa + x

(a – x)(a + x)ax

x1a + x

a – xax

= 1x–1a

1. a) 2p(q – 7p) b) x2(6 + x) c) –4w(2z + 3wy) d) (x + 11)(x + 3) e) (x – 7)(x + 3) f) (x – 10)(x + 10) g) (x – 7)(x – 5) h) 3(x2 – 9) = 3(x – 3)(x + 3) i) 2(x2 – 7x + 6) 2(x – 6)(x – 1) j) (2x – 3)(x + 2)

2. a) (x – 2)(x + 2)2(x + 2)

= x – 22

b) x(x – 3)(x – 3)

= x

3. a) 8x – 5y10

b) x + 2(x –1)6

= 3x – 26

4. a) 30mpq50mnp2

= 3q5np

b) 24a3b36bc3

= 2a3

3c3

5. M = 7(72 +1)2

= 175

6. a) x = y – cm

b) x = 3w4– y

c) x = 25 – y2

d) x = (ky)2

e) x = yz2

Hig

her L

evel

Thi

nkin

g 1

Alg

ebra

Ski

lls R

evie

w 2

Alg

ebra

Ski

lls R

evie

w 1


Documents

D & D2c) 4(n + 2n + 1) – 3n = d) x(x – 8) – 2(x – 8) = e) ax(x + y) + ax(x – y) = 3. Give an expression for the area of this shape. 4. Give an expression for the volume of