1. The document provides steps to solve equations, factorize expressions, calculate areas, volumes, and other mathematical problems. 2. It also includes word problems involving time, money, and other real-world contexts. 3. The mathematical content covers topics such as standard form, percentages, prime factors, and algebraic manipulation.

1. Simplify Fully
• m4 x m2
• 5m2t3 x 2m4t
• h9÷ h3
• 35e7x2
5e5x
• (m2)3
• x0
• 16 ½
• (4x6)1/2
m6
10m6t4
h6
7e2x1
m6
1
4 (power of ½ means square root)
2x3

2. • 3x + 2 = 17
3x = 15
x = 5
• 4x - 2 = 10
4x = 12
x = 3
Solve
• 3x + 5 = x + 10
2x + 5 = 10
2x = 5
x = 2.5
• 3x – 3 = 8x + 17
-3 = 5x + 17
- 20 = 5x
-4 = x

3. Factorise Fully
• 8x2 + 4x
• 6x2 + 12x
• 3xy2 + 9xy
• 2x + x2
4x(2x + 1)
6x(x + 2)
3xy(y + 3)
x(2 + x)

4. Factorise
• x2 + 7x + 10
• x2 - 6x + 8
• x2 + 3x -4
(x + 2)(x + 5)
(x - 4)(x -2)
(x + 4)(x -1)

5. Factorise
• x2 – 36
• y2 - 64
• x2 - 81
x2 – 62 = (x – 6) (x + 6)
y2 - 82 = (x – 8) (x + 8)
x2 - 92 = (x – 9) (x + 9)

6. Expand and simplify
• 3(4x + y)
• 3y(y – 3)
• (y + 8)(y – 3)
• (2x – 3)2
12x + 3y
3y2 – 9y
y2 -3y +8y -24 = y2 + 5y -24
(2x -3)(2x-3) = 4x2 -6x -6x + 9
= 4x2 -12x+ 9

7. Inequalities
X is an integer
State the possible values
1. -1 ≤ x < 3
2. 4 < x ≤ 7
3. -2 < x < 4
-1, 0 , 1, 2
5, 6, 7
-1, 0, 1, 2, 3

8. Solve the inequality
3x < 9
x < 3
4y > 12
y > 3
2x + 2 < + 12
2x < 10
x< 5
3y + 6 < 15
3y < 9
y < 3
1. 3x + 2 < 11
2. 4y – 2 > 10
3. 6x + 2 < 4x + 12
4. 3(y + 2) < 15

9. -3 < x ≤ 4

10. A box of grass seed covers 20m2
How many boxes are needed to cover the lawn
Area = 9 x 12 = 108 m2
Area = 9 x 6 = 54 ÷ 2 = 27 m2
Total area = 108 + 27= 135m2
Seven boxes required

11. 2m
6m
5m
2m
Mrs Philips is going to cover the floor with floor
boards
One pack covers 2.5 m2
How many packs does she need?
Area = 5 x 2 = 10 m2 Area = 4 x 2 = 8 m2
Total area = 10 + 8= 18 m2
Eight boxes required

12. Calculate the Volume of the Prism
Calculate CSA
Area = 4 x7 = 28 cm2
Area = 5 x 2 = 10 c m2
CSA = 10 + 28 = 38cm2
Volume = 38 x 10 = 380cm3

13. Calculate the Volume of the Prism
4 cm
5 cm
3 cm
7 cm
Diagram
accurately drawn
NOT
Calculate CSA
4 x 3 = 12 ÷ 2 = 6 cm2
Volume = 6 x 7 = 42 cm3

14. 1. What time did Jasmine leave home?
2. For how long did Jasmine stop to talk to her friend
on the way to the park
3. Jasmine stayed at the park for ½ hour, then
walked home at a steady rate of 7.5 km/hr
Jasmine's journey to the park
9.10am
10 mins

15. The perimeter of the shape is 38 cm
Calculate the area
2x – 1
2x
3x -2
+ 3x + 1
10x -2
10x -2 = 38
10x = 40
x = 4 calculate lengths
8
7
13
Area = 8 x7 =56m2
Area = 8 x 6 = 48 ÷ 2
= 24cm2
Area = 56 + 24 = 80c m2

16. Find the value of x
3x - 15
2x
2x
+ 2x + 24
9x +9
Angles in a quadrilateral = 360o
So 9x + 9 = 360
9x = 351
x = 39

17. Calculate the value of x – give reasons
53 – Corresponding angle
180 – 53 = 127
Angles on straight line
28
Vertically opposite
180 – 127 – 28 = 25
Angles in a triangle
add to 180

18. Calculate angle x
Exterior
angle of
Hexagon
= 360 ÷ 6
= 60 Exterior
angle of
Octagon
= 360 ÷ 8
= 45
x = 60 + 45 = 105

19. nth term
• 3 6 9 12.........
• 1 7 13 19........
• 5 12 19 26........
3n
6n - 5
7n - 2

20. State two things wrong with this question
1)
2)
Design a better question
No timescale
Response boxes overlap
How much money do you spend using your
mobile phone in a week?
None £1-£5 £6-£10 £11-£15 > £15

21. Here is part of Gary's bill
Work out how much Gary has to pay for the units of electricity used
Units used
7155
-7095
60
Cost
60 x 15 = 900
which = £9

22. Standard form
• Write as an ordinary number
• 8.5 x 10 4
• 6.7 × 10–3
• Write in standard form
• 867000
• 0.00045
85000
0.0067
8.67 x 105
4.5 x 10 - 4

23. Compare the distributions
Median girls = 29 boys = 25.5 / Girls have highest median
IQR girls = 8 boys = 9 / Boys have highest IQR

24. + 12
12
27
45
+ 27
+ 45
+ 57
57
60
CHECK

26. estimate
• 31 x 9.87
0.509
• 84 x 10
23
• 26 x 33
0.235
• 30 x 10 = 300 = 600
0.5 0.5
• 80 x 10 = 800 = 40
20 20
• 30 x 30 = 900 = 4500
0.2 0.2

27. 16 gingerbread 8 gingerbread 24 gingerbread
180 flour 90 flour 270 flour
40 ginger 20 ginger 60 ginger
110 butter 55 butter 165 butter
30 sugar 15 sugar 45 sugar

28. 1 + 3 + 5 = 9
180 = 20
9
Need
Cement 1 x 20 = 20 kg
Sand = 3 x 20 = 60 kg
Gravel = 5 x 20 = 100 kg
He does not have enough
cement, as he needs 20 kg
and he has 15 kg

29. 2 9
3 1 6 5 3 9
4 3 9 3 4 6 2 8
5 2 4 5
2 9
3 1 3 5 6 9
4 2 3 3 4 6 8 9
5 2 4 5
KEY
2 9 = 29

30. • £1.82 x 15 =
• 3.28 x 13 =
• 4.56 x 1.8=
Remove decimals
Multiple – box method
Put decimals back in

32. Express 120 as a product of its prime factors.
Find the highest common factor (HCF) of 90 and 120
602
302
2 15
3 5
120 =2 x 2 x 2 x 3 x 5
120 =2 x 2 x 2 x 3 x 5
90 = 2 x 3 x 3 x 5
HCF = 2 x 3 x 5 =30

33. • Janice asks 100 students if they like biology or chemistry or
physics best.
• 38 of the students are girls.
• 21 of these girls like biology best.
• 18 boys like physics best.
• 7 out of the 23 students who like chemistry best are girls.
• Work out the number of students who like biology best.
Biology Physics Chemistry Total
Girls 21 10 7 38
Boys 28 18 16 62
Total 49 28 23 100

34. The area of the triangle is equal to the area of
the square
What is the perimeter of the square
Area triangle = 9 x 8 = 72÷ 2 = 36 cm2
Area square = 36 cm2
Length of each side of square = 6 cm
6 + 6 + 6+ 6 = 24 cm

35. V = 3b + 2b2
Find the value of V when b = –4
s = 3d + 8
Write d in terms of s
V = 3 x (-4) + 2 x (-4)2
V = - 12 + 2 x 16
V = - 12 + 32
V = 20
d → x 3 → +8 →
sd ← ÷3 ← -8 ← s
S – 8 = d
3

36. Express the area in terms of
7cm
7cm
10cm
Area = r2
= x 7 x 7 cm 2
= 49 cm 2
Volume = 4 9 x 10
= 490 cm3