Answers
Sequences
1.1 Arithmetic sequences
❶ a
20, 24, 28
b 27, 30, 33
c 28, 33, 38
❷ a 13
b 20
c 39
d 56
❸ a 12, 13, 14
b 55, 66, 77
c 20, 10, 0
d 16, 19, 22
e 1, −1, −3
f 40, 47, 54
❹ a 17
b 11
c 14
d 22
e −1
f 1; 22
❺ a 3n
b 10n
c 9n
d 7n
❻ a 2n + 1
b 9n + 1
d 1n + 1 or n + 1
c 3n + 1
❼ a 2n − 1
b 3n − 1
d 11n − 1
c 4n − 1
❽ a 2n
b 2n + 5
c 2n − 3
❾ a 5n
b 5n − 4
c 5n + 2
❿ a 7n
b 7n + 3
c 7n − 8
⓫ a 3n + 5
b 4n − 3
d 9n − 7
c 6n + 2
e Arithmetic sequences
⓬ It is 4n + 3. The difference between terms is +4,
and each term is three more than the four times
table.
⓭ a −2n
b −5n
c −3n + 2
d −n + 6
⓮ − n + 8 = 7, 6, 5, 4, … ; 2n + 3 = 5, 7, 9, 11, … ;
8n = 8, 16, 24, 32, … ; 3n + 2 = 5, 8, 11, 14, … ;
n + 8 = 9, 10, 11, 12, …
⓯ a 6, 7, 8, 9, 10
b 4, 8, 12, 16, 20
c 2, 7, 12, 17, 22
⓰ a 17, 27, 37, 47, 57
b 5, 9, 13, 17, 21
c 5, 11, 17, 23, 29
d −2, 1, 4, 7, 10
e 7, 4, 1, −2, −5
⓱ a
c
e
⓲ a
c
add 5
b 5n + 3
503
d 9th term
Arithmetic sequence
4, 7, 10, 13
b 8th term
add 3
d 3n + 1
Answers
1
1.2 Other sequences
Arithmetic −8
b Geometric ÷2
c Geometric ×8
d Arithmetic +8
❷ a 16, 32, 64
b 300, 3000, 30 000
c 50, 25, 12.5
d 125, 625, 3125
❸ a 8, 13, 21
b 11, 18, 29
c 16, 26, 42
d 13, 20, 33
e 22, 34, 56
❹ a 6
b 12
c 5; 9
d 7; 23
e 5; 15
f 2; 7
❺ Yes, each term is the sum of the previous three
terms.
❻ It could also be ×2 (hard to tell with only two
terms).
❼ a square numbers
b n²
❽ a
❶ a
Pattern 4
b
c
❾ a
c
e
❿ a
c
⓫ a
26 dots
10th pattern
18, 24, 31
42, 48, 53
−4, −9, −15
11; 20
1; 10
b
d
21, 31, 43
26, 37, 50
b
d
28; 20
18; 30
Pattern 4
b
c
15 dots
1, 3, 6, 10, 15, 21, 28, 36; triangular
numbers
1
2 Graphs
b
2.1 Straight line graphs
❶ A (4,3)
B(2,0)
F(5,−5)
C(−1,4)
G(−3.5,2)
b 8
d 14
Key Stage 3 Mastering Mathematics: Develop and Secure Practice Book 2
b
c
y
0
3
6
9
12
x
0
1
2
3
4
y
1
4
7
10
13
x
0
1
2
3
4
y
−2
1
4
7
10
❽ a
1
2
3
y
−1
2
5
8
7
6
5
4
3
2
1
0
c
1
2
3
5
7
y
7
6
5
4
3
2
1
0
d
4
5
6
x
2
4
6
y
2
3
4
5
5
4
4 x
3
2
It is a straight line
y = x +2
1
0
x
0
1
2
3
y
2
3
4
5
y
8
7
6
5
4
3
2
1
5
3
0
2
4
2
x
6
3
1
1
y = x +2
2
4
2
x
3
7
1
6
1
6
0
5
0
y
3
4
y
8
2
3
x
8
1
2
y = 2x + 1
10
0
1
8
y
−1
2
0
8
11
b
❾ a
x
y
D(−2,−3)
H(3.5,−2.5)
E(0,−4)
❷ a 5
c 11
e 17
❸ a y=4
b x=2
c y = −3
d x = −1
❹ a x=3
b y=6
c y = −1
d x = −2
❺ a horizontal line crossing y-axis at (0, 1)
b vertical line crossing at (3, 0)
c horizontal line crossing at (0, −5)
d vertical line crossing at (−9, 0)
❻ a y=7
b y=9
c y = 10
d y = 13
❼ a x
0
1
2
3
4
y = 3x − 1
6
x
e
1
2
3
4
5
6
x
y = 6− x
x
0
1
2
3
y
6
5
4
3
y
y
⓭ a
8
4
7
2
6
0
−3 −2 −1
−2
5
4
1
2
3
4
5
6
b
x
0
1
2
3
y
4
3
2
1
i
ii
i
ii
c
y
No
Yes
Yes
No
8
2.2 Real life graphs
7
❶ a
6
c
❷ a
c
❸ a
b
c
d
e
❹ a
c
e
❺ a
c
❻ a
4
3
2
1
1
2
3
4
5
6
x
❿ The points don’t form a straight line. (3, 6) is
wrong.
⓫ a i y=3
ii y = 7
iii y = −3
b i x=1
ii x = −2
iii x = −1.5
⓬ a x
y
−3
−2
−1
0
1
2
3
−6
−4
−2
0
2
4
6
1
b 2
0.5
iv
b i
iii
d ii
300 m
5 minutes
700 m
15 minutes
14 mins & 27 mins
5 feet
b 14 feet
8 years
d 3rd and 7th years
8th year
at 4 mins
b ignore it
dotted line
Number of customers
5
y
6
5
65
60
55
50
45
40
35
30
25
20
0
Mon
4
3
1
0
−3 −2 −1
−1
−2
−3
−4
−5
−6
y=3
x = −0.5
Yes
1
2
3
x
b
c
❼ a
Tues
Wed Thurs Fri
Day of the week
Sat
Sun
Sun
Tues
25
Temperature °C
2
c
d
e
Answers
−10
x
y = 4− x
0
x
−8
1
b
3
−6
2
f
2
−4
3
0
1
20
15
10
5
0
6 a.m.
9 a.m. 12 noon 3 p.m.
Time
6 p.m.
9 p.m.
3
Key Stage 3 Mastering Mathematics: Develop and Secure Practice Book 2
b
c
d
❽ a
c
9am − 12 noon
6pm − 9pm
2.5 hours
iii
b
ii
❷ a
i
3 Angles
3.1 Parallel lines
❶ a 180°
b 360°
❷ angle b
❸ a = 50°, b = 130°, c = 130°, d = 102°, e = 78°,
f = 78°, g = 65°, h = 115°, i = 115°
❹ a and d are parallel
❺ a a = 138°
b b = 51°
c c = 36°
❻ a p = 135°
b q = 38°
c r = 152°
❼ a f = 131°
b g = 144°
c h = 38°
❽ n = 80° (e.g. angles on a straight line add up
to 180°)
o = 100° (e.g. vertically opposite angles are
equal)
p = 100° (e.g. alternate angles are equal)
q = 80° (e.g. angles on a straight line add up
to 180°)
r = 43° (e.g. vertically opposite angles are
equal)
s = 137° (e.g. angles on a straight line add up
to 180°)
t = 137° (e.g. co-interior angles add up to 180°)
u = 43° (e.g. corresponding angles are equal)
❾ a = 86°, b = 94°, c = 86°, d = 94°, e = 86°, f =
94°, g = 86°
❿
4 Constructions
4.1 Bearings
❶ a
⓫ r = 124° because allied angles add up to 180°.
s = 124° because it is alternate to angle r and
alternate angles are equal.
3.2 Polygons
❶ a and d. A polygon is an enclosed 2D shape with
straight edges.
4
triangle
b quadrilateral
c pentagon
d hexagon
e octagon
f decagon
❸ a 360°
b 360°
❹ a a = 129°
b b = 98°
c c = 50°
❺ a x = 96°
b not regular as the angles are different sizes.
❻ a 12
b 12
❼ a 72°
b 108°
❽ a 120°
b 60°
❾ a 40°
b 140°
c 9
d 9
❿ They don’t add up to 180°.
⓫ 36 sides
⓬ A hexagon
⓭ a 540°
b 1080°
c 900°
d 2160°
⓮ a hexagon
b y = 44°
⓯ p = 106°
⓰ a 108°
b 120°
c 135°
d 144°
⓱ He should divide the total by 9 to find the size of
each angle (140°).
a 360/9 = 40 b 180 − 40 = 140
322°
b 239°
c 264°
d 50°
e 70°
f 290°
❷ a = 54°
b = 327°
c = 110°
d = 231°
❸ Correct angles drawn
❹ a 270°
b 045°
c 225°
d 315°
❺ a South
b South-East
c East
d North
❻ No, bearings must have three digits, so it would
have to be 030°.
❼ a 120°
b 055°
c 260°
293°
b 070°
c 310°
❾ a 031°
b 154°
c 326°
d 259°
e 098°
f 270°
❿ a Durton
b Egwell
⓫ a i x = 105°, y = 255°
ii 075°
iii 255°
b i x = 60°, y = 300°
ii 120°
iii 300°
c i x = 80°, y = 280°
ii 100°
iii 280°
⓬ The bearing of B from A is 050°. The correct
bearing of A from B is 230°.
⓭ Correct diagram
⓮
N
Lighthouse
❻ a
80 cm
b 3 cm
c 2m
d 50 cm
❼ Yes: 1 cm = 25 cm, so 4 cm = 100 cm and
4 cm = 1 m
❽ a 1:5
b 50 cm
c 11 mm
❾ 1 cm : 2 m is 1 : 200
1
❿ a
10 cm : 2.5 m is 10 : 250 = 1 : 25 =
25
b 12.5 m
c 300 cm
⓫ a 125 m
b 8 cm
⓬ a 3 km
b 800 000 cm
c 16 cm
4.3 Constructions
❶ Correct angles drawn
❷ 60°
❸ a
Hut
101º
3 cm
4.7 cm
50º
190°
050°
29º
6 cm
b
8.7 cm
30º
5 cm
120º
4.2 Scale drawing
1 cm = 10 mm
b 1 m = 100 cm
c 1 km = 1000 m
❷ a 1 : 10 = 8 : 80
b 2 : 7 = 6 : 21
c 4 : 5 = 24 : 30
d 3 : 20 = 15 : 100
e 1 : 4 = 5 : 20
f 3 : 8 = 12 : 32
g 1 : 50 = 4 : 200
h 1 : 25 = 3 : 75
❸ a
1 cm on the model boat = 50 cm on the real
boat
b 400 cm
c 9 cm
❹ a 280 cm
b 0.5 m
❺ a 50 cm
b 4 cm
Answers
❽ a
c
❶ a
30º
5 cm
7 cm
4 cm
8.1 cm
60º
d
❹ a
30º
scalene, isosceles, right angled
80º
4.4 cm
40º
3.3 cm
60º
5 cm
b
8.6 cm
150º
20º
3 cm
c
55º
10º
5.9 cm
4.6 cm
55º
4 cm
70º
4 cm
d scalene, scalene, isosceles
❺ Yes − the angles are in the same position relative
to the 6 cm line.
5
❻ The angle isn’t between the 5 cm and 7 cm sides. ⓬ a
❼ No, the angles are 53°, 87° and 40°.
c
❽ a
⓭ a
65º
Key Stage 3 Mastering Mathematics: Develop and Secure Practice Book 2
4.5 m
3.8 m
50º
65º
4.5 m
b
❾ a
b
c
isosceles triangle
Correct diagram
BC = 3.2 km
68°
5 Calculations
5.1 Calculations review
❶ a
50
b 160
c 80
d 270
❷ a 3
b 20
c 100
d 123
❸ a 350
b 2400
c 2400
d 268
❹ a 128
b 1200
c 40
d 900
❺ a 98
b 482
c 656
d 704
❻ a 13.9
b 40.2
c 85.4
d 42.678
❼ a
She hasn’t lined up the digits correctly using
place value.
b £435.75
❽ a 56
b 245
c 364
d 387
❾ a 24.17
b 92.52
c 23.12
d 2.67
❿ a
4 − 7 is negative so you need to exchange
a ten for 10 units in order to do the
subtraction.
b 367
⓫ a
1022
446
576
187
148
259
39
b
97
54.86
18.16
9.52
4.14
6
317
220
36.7
8.64
5.38
28.06
3.26
24.8
c
⓮ a
b
⓯ a
c
⓰ a
b
⓱ a
c
⓲ a
c
⓳ a
b
c
d
131 miles
b 301 miles
115 miles
0.3 m
b 0.55 m
0.125 m
d 75.4 cm
435 + 158 = 593
682 − 157 = 525
486
b 1533
8944
d 81 026
Forgotten to multiply by 20 not 2.
2025
197
b 14.9
0.535
d 4.725
3.05
b 16.48
1.075
d 4.465
41 remainder 4
41, with 4 left
42 boxes
41 lollipops
5.2 Multiplying decimals
❶ a
2.8
b 7
c 11.2
❷ a 42
b 121
c 459.2
d 38.01
❸ a 49 × 7 ÷ 10
b 3 × 85 ÷ 10
c 19 × 26 ÷ 100
d 235 × 74 ÷ 1000
❹ a 170.1
b 170.1
c 17.01
d 1.701
e 17.01
f 17.01
g 1.701
h 0.1701
❺ a 3.5
b 3.2
c 0.54
d 0.2
❻ a 37.8
b 20.8
c 9.73
d 5.76
❼ a 13.95
b 5.2
c 0.657
d 205.58
❽ Yes, 6.8 × 0.35 = 68 × 35 ÷ 1000, and
2380 ÷ 1000 = 2.380 which is 2.38
❾ a £5.16
b £3.43
c £11.37
d £13.14
❿ $71.28
⓫ a 3.9 m²
b 68.89 cm²
c 3.375 m²
❶ a
c
❷ a
c
1.2
2.9
118
59
❸ a
3.8 38
=
0.2
2
b
15 150
=
0.3
3
c
0.01 1
=
0.04 4
d
4.2
420
=
0.05
5
❹ a
b
3.25
b
318
27 9
=
6
2
b
126 63
=
8
4
c
15 5
=
9
3
7.2 3.6
=
4
2
❺ a 70
b 90
c 60
d 140
❻ a 163
b 1.2
c 4.3
d 0.18
❼ a 26 800
b 124
c 436
d 0.4875
❽ She has calculated 4 ÷ 5 not 5 ÷ 4.
❾ a
Bigger 53 when you divide by a number
between 0 and 1 the answer is
bigger
b 66.25
❿ a 8.5 m
b 1.3 cm
c 2.45 mm
⓫ a 50
b 64
c 1520
d 8.5, so 8 boxes
d
6 Negative numbers
6.1 Negative numbers
❶ a
b
c
0, 2, 4, 6, 8, 9
0, 7, 9, 11, 14, 25
0, 8, 14, 17, 29, 31
❷ a
c
❸
6>1
24 < 42
−10
−5
–8
–6
0<9
513 < 531
b
d
0
–1
5
2
10
6 7
5 °C
b −3 °C
c 13 °C
d −2 °C
❺ a 1
b −3
c −5
d −8
❻ a 8
b 5
c 0
d −3
❼ No, −8 is lower on a vertical number line than 5
so it is smaller.
❽ a 5 > −7
b −2 < 2
c −5 < 1
d −3 > −6
❾ a 42
b −82
❿ a
A number line drawn with values marked
correctly.
b 4 and −4
⓫ a −7, −5, −1, 0, 3, 7
b −6, −4, 0, 3, 4, 6
c −7, −5, −1, 0, 1, 3
⓬ a 0, 1, 2 or 3
b −6, −5, −4 or −3
c 1, 0, −1 or −2
❹ a
Answers
5.3 Dividing decimals
6.2 Adding and subtracting
negative numbers
❶ a
c
e
g
❷ a
c
e
g
❸ a
c
e
g
❹ a
c
e
g
❺ a
c
e
g
6
9
6
2
7
6
−2
5
2
4
−4
−4
−1
−3
−4
−6
4
−7
3
−10
b
d
f
h
b
d
f
h
b
d
f
h
b
d
f
h
b
d
f
h
8
10
3
0
7
0
−4
4
4
0
−5
0
−2
−4
−5
−3
5
−12
2
−9
7
❻ a
Key Stage 3 Mastering Mathematics: Develop and Secure Practice Book 2
11
b 7
c −2
d 5
e 6
f 8
g 2
h −3
❼ a 3
b 6
c 5
d 12
e −4
f 0
g −10
h −11
❽ No, that is only true when subtracting a negative
number. −5 −7 = −12
❾ a 22
b −30
c 11
d −12
❿ −5 °C
⓫ a
3
−1
b
12
6
−2
7
8
−9
17
3 °C
b
c 6 °C
d
e 21 °C
f
⓭ 11 °C
⓮ a 9−6=3
b 9 + −6 = 3
c −3 + 7 = 4
d −2 − 4 = −6
e 5 + −12 = −7
f 11 − 3 = 8
g −9 − 1 = −10
h −14 + 5 = −9
i −6 + 2 = −4
j −4 + 5 = 1
k −10 + 2 = −8
l −9 + 4 = −5
⓬ a
9 °C
13 °C
37 °C
6.3 Multiplying and dividing
negative numbers
❶ a
c
e
❷ a
c
e
8
❼
4
−8
7
−6
b 3 × −2
❹ a −12
b −40
c 14
d 90
e −12
f 64
g −36
h 22
❺ a −5
b −2
c 9
d −9
e 12
f −3
g 3
h −3
❻ No, 5 × 3 = 15, −5 × −3 = 15. When the signs are
the same, the result is positive.
❸ a
35
24
24
3
2
7
b
d
f
b
d
f
24
18
60
7
8
6
×
−5
6
0
−7
−3
−9
45
−54
0
63
27
−4
20
−24
0
28
12
10
−50
60
0
−70
−30
2
−10
12
0
−14
−6
−8
40
−48
0
56
24
−4 × 7 = −28
b 32 ÷−8 = −4
c −5 × 6 = −30
d −14 ÷ 7 = −2
e −10 ÷ −5 = 2
f 6 × −9 = −54
g 7 × −7 = −49
h −24 ÷ 8 = −3
❾ a
Any answer from 1 × −14, −1 × 14, −2 × 7
and 2 × −7
b Four pairs in total: 1 × −14, −1 × 14, −2 × 7
and 2 × −7
❿ a −2 × 5 < −3 × −4
b −7 × −5 > −10 ÷ 2
c 28 ÷ −4 > −9 × 3
⓫ a −24
b 60
c 27
d −1
⓬ a 10
b −10
c 10 and −10
⓭ a 16
b 25
c 36
d Always positive
⓮ a −1
b −1000
c −125
d Always negative
⓯ a 225
b −729
❽ a
❻ a
7.1 Fractions review
❶ a
c
3, 6, 9, 12, 15
4, 8, 12, 16, 20
12, 24, 36
12
2
b
9
6
7
c
4
5
d
5
8
❼
e
5
6
f
3
5
❽ a
g
2
3
h
6
11
c
❸ a
1
4
=
3 12
b
1 7
=
5 35
❾ a
c
3 15
=
4 20
d
2 18
=
3 27
❿
2 16
=
3 24
f
b
c
d
❷ a
e
g
❹ a
b
c
6
1
=
4 24
h
e
5
9
2
8
g
=
=
25
45
18
72
9
3
6
=
=
12 4
8
15 5
25
=
=
21 7
35
c
❺
33
55
=
3
5
2 10 8
=
=
3 15 12
=
1 4
7
=
=
5 20 35
⓬
7
15
⓭ a
10
odd one out
14
h
1
3
b
5
7
2
3
d
5
6
£12
21 m
b
d
5 kg
€44
b
1
× 8 =1
8
1
× 3 =1
3
1
1
c
2×
=1
d
7×
e
3 4
×
=1
4 3
f
5 4
×
=1
4 5
⓯ a
4 8 20
=
=
7 14 35
f
⓫ £6
c
24
40
d
1
3
1
12
2
9
27
40
1
1
of 10 = of 20
2
4
e
d
b
5 7 10 7
3 1
− = − =
=
6 12 12 12 12 4
⓮ a
8
4 24
=
=
18 9 54
4
7
7
10
17
20
1
24
Answers
7 Fractions
c
⓰ a
c
e
⓱ a
c
e
2
1
5
4
5
2
12
12
2/5
17/20
7/1000
0.3
0.55
0.375
7
=1
b
d
b
d
1
10
9
3
7
12
30
27/100
3/50
b
d
0.17
0.36
b
d
f
9
⓲ a
Key Stage 3 Mastering Mathematics: Develop and Secure Practice Book 2
c
e
⓳ a
c
e
⓴ a
a
c
e
a
b
3/10
b 1/4
1/5
d 1/25
9/25
23%
b 70%
45%
d 40%
52%
3/5
b 2/5
4 : 16
b 8 : 12
14 : 6
d £18 : £30
20 m : 15 m
f $50 : $70
F = 7/11 S = 4/11
28 sweets
7.2 Multiplying fractions
❶ a
b
d
12
c 18
❷ 1/3 not shaded
8
20
⓬ 4/5 × 1/3 = 4/15
9 10 1
because 9 and 36 have a common
×
=
35 36 14
factor of 9, and 10 and 35 have a common factor
of 5
⓮ a 21/160
b 248/425
c 407/437
d 455/3828
⓭
⓯ a
b
2 4
8
×
=
5 7 35
3
5
×
3 9
=
4 20
c
7 5 5
×
=
8 7 8
d
3 4 3
×
=
4 7 7
7.3 Dividing fractions
❸ a
b
❹ a
❶ a
1 4 4
× =
3 5 15
2 3 6
× =
3 4 12
×
=
c
❷ a
c
❸ a
c
❹ a
c
b
3/20
❺
❻ Correct diagram; 2/12 = 1/6
❼ a 1/6
b 5/24
c 2/21
d 20/63
❽ a 1/14
b 4/33
c 4/15
d 1/6
❾ a 1/15
b 5/18
c 1/9
d 5/12
e 9/28
f 2/9
❿ a 1/6 m²
b 8/35 m²
c 3/16 m²
⓫ You can only cancel when numerators and
denominators share common factors (5 and 10
have a common factor of 5, but they are both on
the denominator so cannot be cancelled).
10
1/15
5/28
1/3
3/2
2/1
7/1
1 5
×
3 2
3 1
×
4 5
b
d
b
d
b
b
d
2/5
4/5
4
7/4
3/1
2 8
×
7 1
5
×1
6
❺ a
8
b 15
c 12
d 9
❻ a 1/6
b 1/6
c 1/20
d 2/21
❼ a 3/5
b 2/7
c 3/16
d 5/18
e 3/5
f 8/27
g 20/33
h 10/9
❽ No, he has taken the reciprocal of the first
fraction, not the second.
❾ a
c
❿ a
c
3/4
4/9
77/95
−243/325
b
d
b
d
7/10
4/7
234/391
248/483
b
2/3 m
c
2/5 m
5/4 m
×
1
2
3
7
7
9
3/5
1
5
3/4
1
3
1
2
1/4
3/14
7/18
3/10
1/10
3/8
1/6
3
4
3/8
2
5
1/5
9/28
7/12
9/20
3/20
9/16
6/35
14/45
6/25
2/25
3/10
7a + 6b
5a + b
5/16
15/56 35/72
6/7
3/7
18/49
5
16
1/3
3/8
1/8
15/32
5/24
18/35
6/35
9/14
2/7
5/32 15/112 35/144
3
16
1
16
15/64
5/48
1/6
1/5
1
15
1
4
1
9
1/7
2
3
7/27
21
36
−4
−2
12
7
22
10
D = 200
D = 12
A = 12
b
d
b
d
b
d
b
d
b
b
4
22
−16
6
40
32
10
5
D = 20
A = 300
A=
a + 5b
a+b
4b
20c + 3d
8c + 7d
3c + 5d
5
8
3a + 10b
2a + 5b
2a + b
3a
2/15
3
4
❼ a v = 11
b v = 12
c v = 11
❽ Not used BIDMAS; F = 22
❾ a a = adult, c = child
b £45
❿ a 50 + 40h
b T = 50 + 40h, where T is total cost in
pounds, and h is number of hours worked
c £170
⓫ a 4m
b 7a + 8b
c 4x + 6y
d 3w − 4
⓬ a 5a²
b 3c² + 4c
c 8g² − g³
d xy + 2xz
c
10a + 16b
b
8.1 Working with letter symbols
review
c
❷ a
c
❸ a
c
❹ a
c
❺ a
c
❻ a
negative sign in front of the 4g).
⓮ a
1/4
8 Expressions and formulae
❶ a
⓭ No, it should be 3g + 4h (Will has ignored the
3c − 3d
12c − 4d
5c + 2d
8d
Answers
⓫ a
⓬
7c − 6d
5c − 6d
2c
ab
b 3cd
c 14ef
d 40ghi
⓰ a x²
b 12y²
c 18m³
d 12vw³
⓱ a 26j
b 36j²
⓲ a 22p m
b 30p² m
⓳ a a=5
b b = 15
c c=4
d d = 18
⓴ a x=4
b x =2
c x = 20
d x=7
a w = −1
b x = 5.5
c y = 0.3
d z =0
4 × 5 + 3 = 23 not 17 so Yotam is wrong.
⓯ a
8.2 Expanding brackets
❶ a
❷ a
20
b
18
b 18
❸ a 4a
b
c c²
d
e −7e
f
g 27g
h
❹ a 9m + 5
b
c 8c + 2
d
❺ 7x + 6y − 2x − 4y
❻ a 2a + 10
b
c 3c + 12
d
e −2e − 10
❼ a 14a + 10
b
c 12c + 66
d
e −20e + 5
❽ a 2a + 2b + 18
b 8c − 4d + 12e
c 3f ² + 7gh
d −12i − 3j³ + 15k
20
15b
12d²
−8f
10h²
2y − 1
w−9
2b − 10
5d − 5f
14b − 10
6d − 9f
11
❾ 2 ( x + 7) = 2x + 14
4 (3 + 2x ) = 8 x + 12
3 ( 4 x − 5) = 12x − 15
2 ( 4 x − 1) = 8 x − 2
−2 (8 − x ) = 2x − 16
Key Stage 3 Mastering Mathematics: Develop and Secure Practice Book 2
❿ Nihal has forgotten to multiply the final term by
c
⓭ a
c
⓮ a
b
c
d
⓯ a
b
c
d
⓰ a
c
e
⓱ a
c
10p + 13
b 8k − 7
29r + 15
d 68a − 16
6w + 2
b 13x + 7
31y −10z
4(2p + 5)
8p + 20
8p + 20 = 44
p=3
5(y + 4) + 6 + 2(3y + 1)
11y + 28
11y + 28 = 50
y=2
m² + 4m
b 2n² − n
6p − 5p²
d 6q² + 10qr
−6s² + 14st
2x² + 5x
b 5x² + 3x
13x²
b
c
d
e
❷ a
c
e
❸ a
c
e
1, 2, 4, 8
1, 2, 4, 5, 10, 20
1, 3, 5, 9, 15, 45
1, 3, 9
1, 11
4
b
5
d
1
5x
b
24
d
5b
❹ a
4x + 8 = 4 x + 2
b
12
(
(
6 w + 15 x − 9 y = 3 (2w + 5 x − 3 y )
5a + 20 = 5 a + 4
2(a + 4)
b 5(b − 9)
c 7(1 + 3c)
d 11(3d − 1)
e 9(e + 9)
❼ a 4(3m + 2) b 5(5 + 6m)
c 9(m − 4)
d 7(4m + 3n)
e 12(2m − n)
❽ She hasn’t factorised it fully. It should be 6(4x +
5).
❾ The terms don’t share any common factors other
than 1.
❿ a x(y + 5)
b d(c − 4)
c m(m + 3)
d p(p − 2)
⓫ a 2a(4b + 3) b 5h(2g − 9)
c 3x(4x + 1)
d 3w(5 − 4w)
⓬ Not fully factorised − it should be 9p(3p − 2)
❻ a
8.3 Factorising expressions
❶ a
d
( )
b 12a + 9 = 3 ( 4 a + 3 )
c 16a − 8 = 8 ( 2a − 1 )
d 16a − 6b = 2 ( 8a − 3b )
⓫ a 2 f + 7 = 2 f + 14
⓬ a
14 x − 12 y = 2 (7x − 6 y )
❺ a
2, it should be 10m + 8.
( )
b 3 (2g − 8 ) = 6 g − 24
c 4 ( 5h + 9) = 20h + 36
c
2 (3 x + 4 ) = 6 x + 8
⓭
3x ( 4 x − 2) = 12x2 − 6 x
4 x (3 + 2 y ) = 12x + 8 xy
3 ( 4 x − 3) = 12x − 9
2 y (3 y − 2x ) = 6 y2 − 4 xy
8.4 Rearranging formulae
−5
c +9
❷ T = 28
❸ a x=2
c x = 16
❹ x=3
❺ a y
c p
❻ a y=x−6
c y=x+z
g
❼ a h=
4
g
c h=
f
❶ a
3
12
−3y
2
)
35 x − 10 y = 5 7x − 2 y
)
b
d
÷4
×3
b
d
x=5
x = 39
b
d
b
d
A
S
y=x+3
y = x − aw
b
h = 3g
d
h = 3gi
9 Equations
9.1 Solving equations
−2
b ÷8
c +5
d ×4
❷ a a=3
b b =7
c c =8
d d=6
e e=3
f f =5
g g = 20
g h = 12
❸ a 7
b 4
c 20
d 16
e 2.5
f 3
❹ a 6
b 10
c −16
d 4
e −10
f 16
g −4
h −6
❺ a x = −3
b x = −4
c x = −10
d x = 5.5
e x=6
❻ a y = −6
b y = 11
c y = −21
d y = 10
e y = −4.5
❼ − 9 − 2 = −11 so Paulo is incorrect
❽ a p = −4
b p=1
c p = −14
d p = −13
❶ a
❾ Sam; 4a = 3 so a = 3
4
❿ a 2m = 11, m = 5.5
b 10m = 5, m = 0.5
1
c 7m = 1, m =
7
5
d 9m = −5, m = −
9
m
e
= 3.6, m = 36
10
m
= 4.5, m = 9
f
2
g m − 1.5 = 6, m = 7.5
h m + 1 = −3, m = −4
⓫ a y+3
b y + y + 3 = 33, 2y + 3 = 33
c y = 15, Amy is 15 years old
⓬ a x + 2x = 24, 3x = 24
b x = 8, 2x = 16 so Tim has £8 and Uleeti has £16
⓭ a 2x + 1 + x + 3x = 6x + 1
b 6x + 1 = 31
c x=5
d 15 cm
⓮ a x is less than or equal to 6
b x is more than 0.5
c x is less than −8
d x is more than or equal to 1/3
⓯ a x is more than 3
b x is less than −7
c x is more than or equal to 0.1
⓰ a m5
b p < −4
5
c y > 0.7
d w
9
⓱ n + 2 10 n plus 2 is less than or equal to 10
n>2
n is more than 2
n 2
n is less than or equal to 2
n 2
n is more than or equal to 2
2n > 10
twice n is more than 10
n − 2 > 10 2 less than n is more than 10
n + 2 10 2 more than n is more than or equal
to 10
n
< 10
half n is less than 10
2
n
> 10
half n is more than 10
2
n<2
n is less than 2
Answers
4
a
( w − 3)
❾ a x=
2
( w + 1)
b x=
5
( w − y)
c x=
3
❿ a c = y − mx
( y − c)
b m=
x
( y − c)
c x=
m
⓫ a r = 2(q − 7)
b r = 3(q + 5)
c r = 3(q − ms)
⓬ Bron is correct − without the brackets, only
s is being multiplied by 5, not the whole
expression.
⓭ Kelli should multiply by 2 first: c = 2P − d
❽ c=
9.2 Solving equations with an
­unknown on both sides
❶ a
c
a=4
c=7
b
b =5
13
3x + 5 = x + 11
2x + 5 = 11
2x = 6
x=3
b 5x − 3 = 2x + 24
3x − 3 = 24
3x = 27
x=9
c 4x + 3 = 7x − 9
3 = 3x − 9
12 = 3x
4= x
❸ a x=5
b x=3
c x=8
d x=7
❹ a x=1
b x=5
c x=4
d x = −2
❺ 4 × 3 − 1 = 11, 2 × 3 + 7 = 13. Suleiman is
incorrect.
❻ a m = 3.5
b w = −2
c p = 2/3
d g=0
❼ It doesn’t matter but you would usually subtract
2y from both sides first to deal with the variables.
❽ a 2k + 9 = 5k − 6
b k=5
❾ a 8y − 24 = 5y + 3
b y=9
c Angle ABC = 84°
Key Stage 3 Mastering Mathematics: Develop and Secure Practice Book 2
❷ a
9.3 Solving equations with
­brackets
2y + 14
b 3y − 15
c 12y − 42
d 20y + 4
e 45y − 10
f 20 + 70y
g 40 − 32y
h 6y + 4
❷ a i 2x + 10
ii x = 4
b i 12x − 3
ii x = 1
c i 25 + 10x
ii x = 2.5
❸ a m=4
b h = 10
c p=3
d f = 1.5
❹ a w=2
b c=6
c g=8
d z=3
❺ 2(3 × 2 − 4) = 4 not 8. He hasn’t expanded the
bracket properly.
❶ a
14
❻ a
b
❼ a
b
❽ a
c
❾ a
b
c
d
❿ a
b
c
⓫ a
b
i 5(x + 9)
ii 5x + 45
5x + 45 = 55, x = 2
4(5w + 3) = 72
20w + 12 = 72, w = 3
a = −8
b b = −2
c = 1.5
d d = 2/9
w = 7; 2(3 × 7 − 5) = 32
w = −4; 4(2 + 3 × −4) = −40
w = 0.5; 5(2 × 0.5 − 1) = 0
w = −0.5; 4(2 × −0.5 + 3) = 8
x; 2x; 2x + 3; 4(2x + 3); 4(2x + 3) = 60
x=6
(6 × 2 + 3) × 4 = 60
6(4y − 3) = 54
y=3
10 Working with 2D shapes
10.1 Types of quadrilateral
❶ a
c
❷
1
2
b
d
1
4
reflex
acute
right angle
reflex
obtuse
acute
straight line
a = 30
b b = 190
c c = 160
❹ a
A rectangle has 2 pairs of equal sides.
The diagonals do not cross at right angles.
It has 2 lines of symmetry. The opposite
sides are parallel. All the angles are right
angles.
❸ a
b
A kite has 2 pairs of equal sides. The
diagonals cross at right angles. It has 1 line
of symmetry. It has no parallel sides. Two of
the angles are the same.
An arrowhead has 2 pairs of equal sides. It
has 1 line of symmetry. None of the sides
are parallel. Two angles are the same. It
contains one reflex angle.
❺ a
i 4
ii 4
b i 0
ii 2
c i 1
ii 1
d i 0
ii 1
❻ No, a rectangle also has four right angles.
❼ A trapezium has one pair of parallel sides.
❽ a d = 55°, e = 55°, f = g = 125°
b b = 135°, c = 45°
c a = 20°, b = 100°, c = 120°
❾ No, because all the angles are not the same size.
A square is the only regular quadrilateral.
❿ a (5, 4)
b (5, 4)
c (1,1), (3,1), (4,1), (5,1) or (6,1)
⓫ a
parallelogram, rhombus, trapezium, kite,
arrowhead
b square, rectangle, rhombus, parallelogram,
kite, arrowhead, trapezium (all of them!)
10.2 Area
perimeter = 32 mm, area = 55mm2
b perimeter = 24 m, area = 36m2
c perimeter = 32 cm, area = 56cm2
❷ a 24 cm²
b 21 m²
c 160 mm²
❸ a 16 cm²
b 84 m²
c 90 km²
❹ It should be base × perpendicular height,
so 8 × 5 = 40 cm²
❺ a 30 cm
b 45 cm²
❻ AD = 7 m
❼ b = 8 cm so perimeter = 30 cm
❽ a 30 cm²
b 6 m²
c 24 cm²
❾ a 20 m²
b 80 cm²
c 28 km²
❶ a
❿ a
36 cm
60 cm²
72 cm²
36 cm²
7 cm & 12 cm
4.2 m & 3.5 m
10 mm & 16 mm
70 cm²
b 22 m²
60 mm²
60 × 2 = £120
210 ÷ 30 = 7 packets
b
⓫ a
b
⓬ a
b
c
⓭ a
c
⓮ a
b
Answers
c
11 Properties of 3D shapes
11.1 Properties of 3D shapes
❶ edge
vertex
face
❷ A vertex is a corner.
An edge joins two vertices.
A face is a surface.
❸ a Yes
b A rectangle is 2D and a cuboid is 3D
c Circle
❹ a Triangle
b Cuboid
c Octagon
❺ Yes − if you cut it along the length the crosssection is always an H.
❻ The circle cross-section isn’t always the same
size (and a circle isn’t a polygon).
❼ Hexagonal prism. The cross-sectional face has
six edges.
❽ No, the cross-sectional shape doesn’t have to be
regular.
❾ Prisms: a, d, g; Pyramids: b, c, f, h; Neither: e
❿ Nicki − it has the same cross-section all along
the length.
⓫ Hexagonal-based pyramid. The base has six
edges and the faces meet at a point.
⓬ a 6
b 12
c 8
d 6
e 12
f 8 They are the same for cube and cuboid.
15
Key Stage 3 Mastering Mathematics: Develop and Secure Practice Book 2
⓭
❽ Multiple solutions e.g.
Name
Triangular
prism
Square
based
pyramid
Pentagonal
based pyramid
Hexagonal
prism
(irregular)
Faces
5
5
6
8
Edges
9
8
10
18
Vertices 6
5
6
12
⓮ a
c
e
b
d
True
False
True
False
False
3
1
5
4
❾
5 cm
3 cm
6 cm
11.3 Surface area and volume of
a cuboid
❶ a
11.2 Nets
c
❶ a and g
6 cm³
16 cm³
b
d
8 cm³
18 cm³
❷
b and e
c and h
d and f
❷
Number of
Cuboid cubes in
one layer
Number
of layers
Total
number
of cubes
Volume
A
4
2
8
8 cm³
B
8
3
24
24 cm³
C
5
4
20
20 cm³
D
9
9
81
81 cm³
❸ a
❸ b
❹ d
❺
5 cm
3 cm
4 cm
2 cm
5 cm
8 cm
2 cm
❻
5 cm
7 cm
❼ The 4 × 3 rectangles need to be split up with a
4 × 2 rectangle between them so they form the
base, front, top, back in order.
16
6
2
36 cm³
b 120 cm³
c 125 cm³
❹ Cuboid A
❺ x = 4m
❻ a 50
b 10
c 500
❼ a i 20 cm²
ii 10 cm²
iii 8 cm²
iv 76 cm²
b i 18 cm²
ii 18 cm²
iii 9 cm²
iv 90 cm²
❽ He hasn’t doubled the areas to take into account
front and back etc.
❾ a
Net A = 78 cm²; Net B = 124 cm²;
Smallest = A
b Net A = 45 cm³; Net B = 48 cm³;
Largest = B
❿ a SA = 96 cm²; V = 64 cm³
b SA = 158 mm²; V = 120 mm³
c SA = 52 m²; V = 24 m³
⓫ a 25 cm²
b 150 cm²
c 125 cm³
b
c
⓫ b
9 cm²
3 cm
27 cm³
12 Percentages
i 58%
ii 7%
iii 125%
iv 110%
v 204%
i 75%
ii 30%
iii 126%
iv 105%
v 212%
i 0.28
ii 0.06
iii 1.59
iv 1.02
v 2.8
b
12.1 Working with percentages
❶
Percentage 100% 50% 25% 75% 10% 20% 1% 5%
1
1 1
3
1
1
1
1
Fraction
10 5
4
2
4
100 20
Decimal
❷ a
1
0.47 =
0.5
0.25 0.75 0.1
0.09 =
9
= 9%
100
c
0.31 =
31
= 31 %
100
d
0.99 =
99
= 99%
100
e
0.82 =
82
41
=
= 82 %
50
100
f
0.45 =
45
9
=
= 45 %
20
100
62%
c 32%
e 75%
❹ 65%
❺ a 60%
b 40%
❻ a 35%
b 65%
❼ a 37.5%
b 62.5%
❽ a 40%
c 156
❾ a 5 g
c 125 g
❿ a Yes
c Yes
e No
0.01 0.05
47
= 47 %
100
b
❸ a
0.2
c
b
d
f
90%
35%
40%
44
b
85 g
b
d
No
No
59
100
7
ii
100
41
iii 1
100
2
iv 1
25
3
v 2
20
d
i
⓬ All of them − they are equivalent fractions.
⓭ a More
b
⓮
160%
36 stickers
12.2 Percentage increase and
decrease
❶
b
Answers
⓬ a
Number
50%
10%
1%
400
200
40
4
2600
1300
260
26
80
40
8
0.8
12
6
1.2
0.12
❷ a
c
e
g
❸ a
6
4.5
7.5
3.2
b
d
f
h
Percentage
100%
10%
5%
15%
Amount
£60
£6
£3
£9
360
64
8.4
51
17
Key Stage 3 Mastering Mathematics: Develop and Secure Practice Book 2
b
c
❹ a
b
❺ a
£69
£51
33 is more than what she started with.
She added 3; she should have subtracted 3
100%
10%
5%
20%
300
30
15
60
70
7
3.5
14
1420
142
71
284
48
4.8
2.4
9.6
6
0.6
0.3
1.2
b
i 330
ii 84
iii 1349
iv 40.8
v 7.8
vi 195
❻ a 99
b 20
c 75
d 64
e 2525
f 437
g 92
h 70
i 125
❼ £2880
❽ £7200
❾ 200 + 10% = 220 220 − 10% = 198;
198 is less than the original amount of 200
❿ a £72
b £2472
⓫ a £200b £600
c £8600
12.3 Percentage change
❶ a
89%
b 62%
c 30%
d 52%
e 55%
f 60%
g 117%
h 110%
i 166%
j 125%
❷ a Increase of £5
b Decrease of 2 m
c Decrease of 22p
d Increase of 25 g
❸ a 5g
b 10% decrease
❹ a £1
b 20% increase
❺ The increase (3) should be divided by the
original width (12) so 3/12 = ¼ = 25%
18
❻ a
£5
b 25% profit
❼ a £9
b 15% loss
❽ a £30
b 150% profit
❾ 26% profit
❿ 33% loss
⓫ a 7% profit
b 72% profit
c 140% profit
d 167% profit
⓬ a 28% loss
c 9% loss
⓭ a 0.1 cm
c 0.5 ml
⓮ a 2% error
b 4% error
c 0.5% error
d 0.9% error
e 2.2% error
f 1.2% error
b
d
b
19% loss
77% loss
2g
13 Multiplicative reasoning
13.1 Ratio and proportion
review
❶ a
b
❷ a
c
e
g
❸ a
c
e
g
i
❹ a
c
e
❺ a
b
❻ a
b
2/5
3:2
1 : 10
3:2
2:1
9:7:4
8
28
21
60, 90
16, 10
5:1
3:8
7:2
5:9
2:5
18
35
b
d
f
h
b
d
f
h
2:3
1:2
1 : 4 : 12
5:1:2
20
27
15
4, 28
b
d
f
4:1
8:3
1:5
3.5
3
2.5
2
1.5
1
0.5
Ingredients
For 15
biscuits
5 biscuits
20 biscuits
Flour
180 g
60 g
240 g
Butter
120 g
40 g
160 g
Sugar
60 g
20 g
80 g
0
c
d
❻ a
⓭ 2.5 bananas, 150 g raspberries, 100 ml yogurt,
125 ml milk
⓮ 264 g self-raising flour, 180 ml milk, 30 g sugar,
72 g butter
❶ A (0, 0)
B(4, 2)
E(4, −3) F(−2, −2)
❷ a
C(0, 3)
G(−4, 0)
y
–1 0
–1
ii 6.6 gallons
ii 81 litres
y
Fahrenheit
80
60
40
0
b
❼ a
b
1
–2
20 x
15
120
5
2
–3
10
Litres
20
3
–4
5
2.2 gallons
13.5 litres
D(−3, 4)
H(1, −4)
4
–5
i
i
0
100
13.2 Conversion graphs
0
10
20
Celsius
i 50 °F
(16, 8)
40 x
30
ii 28 °C
y
16
1
2
3
4
5
x
14
12
–2
Miles
10
–3
–4
L (or an irregular hexagon)
220 g
b 3.6 ounces
1400 g or 1.4 kg
d 27 ounces
3.4 l
3.5 pints
26.4 pints
i 0 litres = 0 gallons
ii 18 litres = 4 gallons
iii 4.5 litres = 1 gallon
8
6
–5
b
❸ a
c
❹ a
b
c
❺ a
y
4
Answers
b
❽ a
c
❾ a
b
c
❿ a
b
⓫ £70
⓬ e.g.
b
75 g
30 g peanut butter, 90 g strawberries
9 : 36
b 15 : 25
15 : 21
d 12 : 54
7/10
21 students
240 students
£2500
£500
Gallons
❼ a
4
2
0
c
d
e
0
4
8
12
16
Kilometres
20
24 x
(16, 10)
7.5 miles
19 kilometres
19
❽ a
Key Stage 3 Mastering Mathematics: Develop and Secure Practice Book 2
c
❾ a
c
e
❿ a
b
d
b
d
48 lira
3800 lira
£10 = €11
£7.30
£360
14 Working with data
£8.40
£630
£1 = €1.10
€550
14.1 Frequency tables
British
pounds
0
1
5
10
Thai baht
0
40
200
400
y
b
400
Thai baht
350
300
250
200
150
100
50
0
c
d
i
ii
i
ii
0
2 4 6 8 10 x
British pounds
120 baht
£7.50
£3.00
16 000 baht
13.3 Best buys
❶ a
11p
b 24p
c £6
d £0.65
❷ a
4-pack unit price = 50p; 9-pack unit
price = 40p
b 9-pack
❸ a
200 g jar unit price = £1.55; 500 g jar unit
price = £1.50
b 500 g jar
❹ a
2 pint unit price = 40p; 4 pint unit price =
26p; 6 pint unit price = 25p
b 6 pint bottle
❺ a 53.3p
b 48.6p
c The 720 g box
❻ a Tray of 6 pears
b 9 pack
c 1 kg bag
❼ a £5.20
b £3.25
❽ a £5.46
b £13.75
20
5×8
b 9×2
c 3×5
d 4×1+6×3
e 5×4+2×5+7×6
❷ a 2
b 6
c 13
d 150
❸ a mode = 9, median = 8
b mode = 15, median = 14.5
c mode = None, median = 4
❹ a mean = 5, range = 4
b mean = 2, range = 5
c mean = 21, range = 12
❺ a mode = 4, median = 4.5, mean = 5,
range = 6
b mode = 9, median = 6.5, mean = 6, range = 10
❻ a
0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2,
2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5
b 6 people had no pets
c 5
d 7 people
e 1 pet
f 1 pet
g 32
❼ a Score
Frequency
❶ a
b
c
d
❽
1
5
2
7
3
4
4
6
5
3
6
5
Total
30
5
30
2
Number of
people
Frequency
n × Frequency
1
2
1×2=2
2
2
2×2=4
3
7
3 × 7 = 21
4
6
4 × 6 = 24
5
9
5 × 9 = 45
6
4
6 × 4 = 24
Total
30
120
❾
Frequency
n × Frequency
0
2
0×2=0
1
4
1×4=4
2
9
2 × 9 = 18
3
6
3 × 6 = 18
4
1
4×1=4
Total
22
44
2 goals
22 games
44 goals
b
d
f
4
See table
44/22 = 2
❹ a
1
4
c
i 8
ii 4
iii 2
iv 2
Avonford Town
❺ a
b
1
2
c
i
Score
Frequency
n × Frequency
7
1
7×1=7
8
3
8 × 3 = 24
9
6
9 × 6 = 54
10
5
10 × 5 = 50
Total
15
135
Drink
Every test result was higher than 6
135/15 = 9
8th piece
9
Score
Frequency
n × Frequency
1
5
1×5=5
2
4
2×4=8
3
2
3×2=6
4
14
4 × 14 =56
Total
25
75
a
c
e
25 times
4
75/25 = 3
b
d
4
75
14.2 Pie charts
❶ a
1
4
ii 90°
8
5/8 b
75
10°
18°
10°
225°
b 6°
d 5°
Lemonade Orangeade Cherryade
Cola
Frequency
13
9
4
10
Angle of
sector
13 × 10
= 130°
9 × 10
= 90°
4 × 10
= 40°
10 × 10
= 100°
❾ a
4°
b
❿ a
Sport
Frequency
Angle of sector
Football
30
30 × 4 = 120°
Hockey
16
16 × 4 = 64°
Rugby
13
13 × 4 = 52°
Tennis
23
23 × 4 = 92°
Netball
8
8 × 4 = 32°
360/20 = 18°
b
Music
Pop
Rock
Jazz
Classical
Other
Frequency
8
6
3
1
2
Angle of
sector
8 × 18
= 144°
6 × 18
= 108°
3 × 18
= 54°
1 × 18
= 18°
2 × 18
= 36°
c
Other
l
a
sic
as
Cl
60°
b 90°
c 135°
d 35°
e 40°
❷ Correct angles drawn
❸ a 10
b
c 6
d
e 120
f
g 222
h
British
b
d
❻ a
c
❼ a
c
❽ a
b
a
b
c
d
⓫
5
See table
120 ÷ 30 = 4
Number of
goals
a
c
e
❿
b
d
f
5 people
30 houses
120 people
Answers
a
c
e
4
18
124
270
Pop
Jazz
Rock
21
⓫ a
Key Stage 3 Mastering Mathematics: Develop and Secure Practice Book 2
b
❷ a
6°
Language
Frequency
Angle of
sector
French
13
13 × 6 = 78°
German
9
9 × 6 = 54°
Spanish
26
26 × 6 = 156°
c
❸ a
c
❹ a
c
Mandarin
8
8 × 6 = 48°
❺
Other
4
4 × 6 = 24
b
d
b
d
b
8.3
7.3
r=5
r = 0.4
r=5
r = 0.7
26.8
3.0
r = 3.5
r = 32.1
r =2
radius
c
Other
French
centre
Mandarin
diameter
circumference
German
❻
Spanish
⓬ a
b
5°
Colour
Frequency
Angle of sector
Red
7
7 × 5 = 35°
Black
19
19 × 5 = 95°
Silver
14
14 × 5 = 70°
Blue
15
15 × 5 = 75°
White
5
5 × 5 = 25°
Other
12
12 × 5 = 60°
c
Other
Red
White
Black
Blue
Silver
15 Circles
Diameter
a
5m
10 m
b
6 mm
12 mm
c
20 cm
40 cm
d
4.5 m
9m
❼ a
3.1
b 25.1
c 7.9
d 43.7
❽ a 31.4 cm
b 22.0 mm
c 14.1 cm
d 2.8 m
❾ a 12.6 m
b 69.1 mm
c 15.7 cm
d 3.8 m
❿ a 18.8 cm
b 25.1 m
c 44.0 mm
d 47.1 km
e 7.5 m
f 5.7 cm
⓫ She has used a radius of 4.5 cm not a diameter.
It should be 28.3 cm.
⓬ a 500 cm
b 1571 cm
c 8 rolls
d £36
⓭ a 145 cm
b 7 full turns needed
⓮ a 6.4 cm
b 3.2 cm
⓯
Radius
Diameter
Circumference
15.1 Circumference
5 cm
10 cm
31.4 cm
4.5 m
9m
28.3 m
❶ a
4.0 mm
8.0 mm
25 mm
1.3 cm
2.5 cm
8 cm
c
e
22
Radius
3 cm
200 cm
8 cm
b
d
0.9 cm
70 cm
⓰ 60 cm
❶ a
32 cm²
b 35 m²
c 120 mm²
d 56 km²
❷ a 37.7 cm
b 44.0 m
❸ a 50.3
b 314.2
c 36.3
d 4.8
❹ a 78.5 cm²
b 314.2 mm²
c 18.1 m²
d 1.5 cm²
❺ a 254.5 m²
b 12.6 cm²
c 38.5 mm²
d 18.9 km²
❻ a 28.3 cm²
b 50.3 m²
c 153.9 mm²
d 176.7 km²
e 4.5 m²
f 2.5 cm²
❼ a
She hasn’t used the radius, and she hasn’t
used BIDMAS.
b It should be A = π × 32 = 9π = 28.3 cm².
❽ 32.9 m²
❾ 15.7 m²
❿ a C = 31.4 m, A = 78.5 m²
b C = 44.0 cm, A = 153.9 cm²
⓫ a 4.4 cm
b 5.6 m
c 1.7 mm
⓬ a 2.4 m
b 4.8 m
c 15.1 m
16 Pythagoras’ theorem
16.1 Investigating triangles
❶ a
c
❷ a
c
e
g
i
k
❸ a
c
❹ a
b
❺ a
b
❻ a
b
AB, BC, AC
b PQ, QR, PR
XY, YZ, XZ
25
b 144
289
d 0.81
20.25
f 13.69
15
h 18
26
j 6.5
0.4
l 7.3
5 cm
b 13 m
7.7 mm
36 cm2, 64 cm2, 100 cm2
36 + 64 = 100
100 m2 and 576 m2
i 676 m²
ii 26 m
49 mm2 and 625 mm2
i 576 mm²
ii 24 mm
i 32 + 42 = 25
ii 25 cm²
iii 5 cm
b i 3.62 + 4.82 = 36
ii 36 m²
iii 6 m
❽ a i 64cm2
ii 8 cm
b i 36m2
ii 6 m
❾ 5² + 8² = 25 + 64 = 89; 9² = 81; 89 ≠ 81 so the
triangle is not right-angled.
❼ a
Answers
15.2 Area of a circle
16.2 Using Pythagoras’ theorem
❶ a
c
❷ a
c
❸ a
c
25
149
3
8
4.9
12.5
❹
a2 + b2 = c2
b
52
b
5
b
d
9.3
8.0
82 + 152 = y2
64 + 225 = y2
289 = y2
289 = y
y = 17m
❺ a 8.1 cm
b 15.6 m
c 9.8 km
❻ a 51 cm
b 100 m
❼ 9.5 cm
❽ 120.9 m
❾ a 3.9 cm
b 7.1 m
c 13.3 mm
❿ x is not the hypotenuse; the hypotenuse is 5 m
a2 + b2 = c2
32 + x2 = 52
9 + x2 = 25
16 = x2
16 = x
x = 4m
⓫ 30 cm
⓬ 15.36 cm²
23