Answers Cambridge Checkpoint Mathematics Practicebook 9
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Answers to Practice Practice Book exercises exercises 1 Integers, powers and roots
F Exercise 1.1
Directed numbers
1 a −3.3
b −8.7
c 13.3
d −13.3
2 a 12 3 a 3.7
b 12.3 b −20.5
c −1.9 c 20.5
d 1.9 d −3.7
5 a N = = −7
b N = = −8.5
c N = = −10.8
6 a −6.8
b 1.2
c −27.6
d −3.5
c 14.8
d −3.7
4 −4.4 °C
7
×
−1.2
3
−1.1
1.32
−3.3
−0.5
0.6
−1.5
8 a 7.4
b 9.4
e −2
9 A and and B are are 6 and −6 so A − − B is is either 12 or −12.
Exercise 1.2
F 1 a
Square roots and cube roots
7
b 12
c 19
2 a 92 = 81 < 95 and 10 2 = 100 > 95 so 9 < b 43 = 64 < 95 and 5 3 = 125 > 95 so 4 < 3 a 19 <
385 <
4 a 12 <
N <
20
15
5 a 26 6
3
200 <
95 <
3
95 <
10
5
b 7 < 3 500 < 8
c 8 <
b 10 <
c 0 <
M <
20
b 25.5 6 because 63 = 216.
200 >
d 7
698 . < 3
R <
9
d 3 < 3 555 . 7500
8 a 5.5
b 21
c 29
d 7.4
e 13.2
9 a 2.45
b 7.75
c 24.49
d 6.53
e 1.56
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Cambridge Checkpoint Mathematics 9
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Unit 1
Answers to Practice Book exercises
F Exercise 1.3
Indices 1 36
b 243
c
2 a 0.125
b 0.25
c 0.25
d 0.333…
e 0.001
d
1 8
1 a 625
e 1
3 112, 121, 62, 26 and 43 (the same), 34 4 3−3, 2−4 and 4−2 (the same), 5−1, 1−5 5 a 42
b 44
c 40
d 4−1
e 4−3
6 a 24
b 28
c 20
d 2−2
e 2−6
7 −4 8 a
1 2
or 2−1
F Exercise 1.4
b 21 5
16
Working with indices
1 a 85
b 74
c 26
d r 6
e s 6
2 a 43
b 6
c c 4
d 24
e e
3 a 0 × a 5 is equal to a 5. All the rest are equal to a 6. 4 a 729
b 81
2
−1
5 a a 6 a 1
b 6 b 1
c 8 c 1
d 1 d 1
7 a a 4
b 55
c f 2
d 104
8 a 1
b 0
c 4
9
2
−2
k
100
Cambridge Checkpoint Mathematics 9
e e
2
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Answers to Practice Practice Book exercises exercises 2 Sequences and functions
F Exercise 2.1 1
Generating sequences
a Linear, term-to term-to-term -term rule is ‘add 4’ 4’.. b c d e f g h i
Linear, term-to-term term-to -term rule is ‘add 1’.. 1, add 2, add 3, ...’ . 1’ Non-linear, term-to-term rule is ‘add Linear, term-to term-to-term -term rule is ‘subtr ‘subtract act 7’. Non-linear,, term-to-term rule is ‘subtract 4, subtract 5, subtract 6, ...’ Non-linear ...’ . Linear, term-to term-to-term -term rule is ‘subtr ‘subtract act 3’. Linear, term-to-term rule is ‘add 1 1 ’. 2 Linear, term-to term-to-term -term rule is ‘subtr ‘subtract act 1.1’ 1.1’.. Non-linear, term-to term-to-term -term rule is ‘add 5, add 4, add 3, ...’ .
2 a 9, 5, 1, −3 d 10, 9, 6, 1
b
1
1 2
, 3,
4
1 2
,6
e 64, 32, 16, 8
c −3, −2, 0, 3 f −64, −32, −16, −8
3 20. Check students’ methods. 4
1 , 3
1, 3, 9. Check students’ methods.
5 a 6, 7, 8, 9
b −6, −5, −4, −3
c 3, 5, 7, 9
d 2, 5, 10, 17
e 4, 7, 12, 19 6 a i 5 c i 5
f −1, 2, 7, 14 ii 7 ii 20
g 2, 8, 18, 32 iii 23 iii 500
h 4, 10, 20, 34 b i 0 d i −99
ii 5 ii −96
iii 45 iii 0
7 Term = 5 × position number + 4 8 Term = position number 2 + 3
F Exercise 2.2
Finding the nth term
1 a 5, 10, 15, 50 d −6, −2, 2, 30
b 5, 6, 7, 14 e 9, 8, 7, 0
c 10, 12, 14, 28 f −8, −18, −28, −98
2 A: i, B: iv, C: ii, D: v, E: iii 3 a 2n + + 18 f 10 − 3n
b 2n + + 2 g 14 − 7n
c 8n − − 5 h −20 + 5n
d 4n − − 12 i −n − −1
e 8 − n
4 a 58 f −50
b 42 g −126
c 155 h 80
d 68 i −21
e −12
5 The sequence increases by 2 each time, so should include a 2n term, term, not a 5 n term. term. 6 Yes. The number of squares increases by 4 each time (term-to-term rule is ‘add 4’), so the n th th term will start with 4n . The number of squares in the patterns is: i s: 1, 5, 9, 13 and 4 × 1 − 3 = 1, 4 × 2 − 3 = 5, 4 × 3 − 3 = 9, 4 × 4 − 3 = 13. 7 Mia. Each pattern increases by three dots (term-to-term rule is ‘add 3’), so the n th th term will start with 3 n . The numbers of dots in the t he patterns are: 6, 9, 12, 15 and 3 × 1 + 3 = 6, 3 × 2 + 3 = 9, 3 × 3 + 3 = 12, 3 × 4 + 3 = 15.
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Cambridge Checkpoint Mathematics 9
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Unit 2
Answers to Practice Book exercises
F Exercise 2.3
Finding the inverse of a function
1 a
b
2 a 3 a
y = = x + +
8
x → x − − y = =
b
7
x + 4
3
b
x − 5 4 a x → 2 5 a i x → 5 – x
iii x → 100 – x
y = = x − −
x → x + + y = =
c y = =
8
c
7
− 3
x
x
8
d
x → x 7
d
c y = = 3(x − − 4)
4
d
y = =
8x
x → 7x y = =
4x − − 3
x + 2
b ii
x →
5
x → x − 3
or 3 − x −3 3 x − 4 4 iv x → or − x −7 7
c
x → 2(x +
( − ) or1
5)
d
x → 2x −
5
1 x 3
b i and iii
2
11
6 a
x → 5(x + +
1)
b
7 a
x → x − 3
b 4 × 2.25 2.25 + 3 = 12 12
4
5
− 1 = 1.2
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Answers to Practice Practice Book exercises exercises 3 Place value, ordering and rounding
F Exercise 3.1
Multiplying and dividing decimals mentally
1 a 1.2
b 2.6
c 3.6
d 8.1
e 3.3
f 0.24 2 a 20 f 250
g 0.28 b 40 g 300
h 0.45 c 30 h 3000
i 1.4 d 40 i 200
j 5.55 j e 200 j 400 j
3 A, C, E, I (0.024); D, G, J, L (0.24); B, F, H, K (2.4) 4 a B
b B
c C
d B
5 a 0.12 f 30
b 1.35 g 9
c 0.072 h 5
d 0.15 i 7
e 0.055 j 40 j
6 Top: 2.5 × 0.2 = 0.5, not 5. Bottom: 5 × 0.1 = 0.5, not 50. Answer = 1. 7 a 20
b 30
c 500
d 0.2
8 a i 1.1 b Larger
ii 2.2 ii
iii 3.3 iii
iv 4.4 iv
v 5.5 v
9 a i 80
ii 40 ii
iii 20 iii
iv 16 iv
v 10 v
vi 6.6 vi
b Larger
F Exercise 3.2
Multiplying and dividing by powers of 10
1 a 2800 e 280 000 i 0.028
b 28 000 f 0.2 j 0.28 j
c 280 g 28 k 0.028
d 2880 h 0.2 l 28.8
2 a 3.4 e 0.034 i 3400
b 3.4 f 0.034 j 30 400 j
c 0.034 g 34 k 30
d 0.034 h 3.4 l 340
3 Powers of ten – easy! 4
a
b 0.004 × 10
4 × 10
3
0
400
10
÷
=4
0.4 × 10
1
2
0.04
40
67
2
−
10
÷
1
2
3
3
670 × 10−
= 0.67
10
÷
1
6.7 × 10−
10
÷
670
10
÷
6.7
2
67 × 10−
5 a i 5000 b Larger
ii 500 ii
iii 50 iii
iv 5 iv
v 0.5 v
vi 0.05 vi
6 a i 0.099 b Smaller
ii 0.99 ii
iii 9.9 iii
iv 99 iv
v 990 v
vi 9900 vi
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10
÷
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Unit 3
Answers to Practice Book exercises
F Exercise 3.3
Rounding
1 a 21.7
b 18.55
c 0.847
d 0.99
e 9.5960
f 34.590
2 a 74.0
b 73.95
c 73.953
d 73.9530
e 73.953 02
f 73.953 017
3 a 2000
b 760
c 5.37
d 0.08
e 0.20
f 6.04
4 a D
b A
c C
d D
5 a 300 000
b 250 000
c 254 000
d 254 100
e 254 060
f 254 060
h 254 059.95
i 254 059.952
ii ii ii ii ii ii ii ii ii ii
b d f h
g 254 060.0 6 2700 km 7 0.0259 g 8 200 000 9 a c e g i
i i i i i
120 12 000 1 000 25 20
F Exercise 3.4
119 12 600 962 18.6 17.2
i i i i
400 80 3 4
ii ii ii ii ii ii ii ii
401 83.6 2.89 5.19
Order of operations
1 a 28 f 13 k 14
b 5 g 0 l 41
c 25 h 9 m 19
d 6 i 19 n 17
e 11 j 62 j o 9
2 a <
b =
c >
d >
e >
3 a
û,
12
b
ü
c
û,
−3
d
ü
e
û,
f = 6
4 a i Added before multiplying b i Should have have squared the the 3 before before subtracting the the result from 14 14 c i Should hav havee worked worked out the the numerator numerator and denominator denominator separately separately before dividing
f
û,
4
ii 30 ii ii 50 ii ii 2 ii
5 No. Harsha doesn’t understand that 3 x means 3 × x . Ahmad doesn’t understand the BIDMAS rules. Answer = 46 6 a 22
2
b 7
c 100
Cambridge Checkpoint Mathematics 9
d 90
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Answers to Practice Practice Book exercises 4 Length, mass, capacity and time
F Exercise 4.1 1 a 53.25 g
Solving problems involving mesuremen mesurements ts b 1.875 g
2 3 days 3 8 hours and 20 minutes 4 1165 miles 1864 km ≈
5 a 53.3 cm
b 9
6 a 63
b 3
F Exercise 4.2
Solving problems involving average speed
c $340
1 58 km/h 2 12 km 3 20 minutes and 50 seconds 4 13 00 or 1 p.m. 5 14 km/h 6 a 38 minutes
b 39 km/h
c 6 hours and 20 minutes
7 10.4 m/s 8 32 m/s 9 0.432 km/h 10 17 500 mph
F Exercise 4.3
Using compound measures
1 Aeroplane A. For aeroplane A, speed = 349 km/h; for aeroplane B, speed = 332 km/h. 2 34 – 20 = 14 km/h 3 a Monday = 21.25 km/h, Thursday = 22.5 km/h. 4 a 6 pack = 96.5 cents cents each, 20 pack = 98.5 cents cents each.
b Thursday b The 6 pack
5 a The can = 0.148 cents/ml, the bottle = 0.1345 cents/ml.
b The bottle
6 Neither. 375 g box = 0.44 cents/g, 650 g box = 0.44 cents/g. 7 175 ml tube. 50 ml tube = 1.58 cents/ml, 175 ml tube = 1.31 cents/ml. seconds/clue, e, 80 clues = 16.5 seconds/clue. 8 a 34 clues = 18 seconds/clu b The 80-clue crossword.
9 a i 21.6 km/h ii 13.5 km/h b To his grandmo grandmother’s ther’s house c 16.6 km/h
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Answers to Practice Practice Book exercises exercises 5 Shapes
F Exercise 5.1
Regular polygons
1 The exterior angle is 360° ÷ 8 = 45°. The interior angle is 180° − 45° = 135°. 2 a 150° 3
a
b 156°
Num umb ber of sid ides es
Ext xte eri rio or an angl gle e
Int nter eriior an angl gle e
5
72°
108°
10
36°
144°
20
18°
162°
40
9°
171°
b It is halved. 4 a 18
b 20
5 a 36
b 45
6
c 120
a The exterior angle is 24°. 360 ÷ 24 = 15. Yes, it has 15 sides. b The exterior angle is 48°. 360 360 ÷ 48 = 7.5 7.5 which is not a whole number. number. It is not possible.
7 9 sides 8 24 sides
F Exercise 5.2 1 a 1080°
More polygons
b 1620°
c 1800°
2 130° 3 If the shape is a polygon with 7 sides, the sum of the angles is 5 × 180° = 900°. If the shape has 6 sides, the sum is 720°. If the shape has 5 sides, the sum is 540°. 4
a 4, because because the sum of the angles of a quadrilateral is 360°. b (N − − 2) × 180 = 720 → N − − 2 = 4 → N = = 6. Six sides. c (N − − 2) × 180 = 1440 → N − − 2 = 8 → N = = 10. Ten sides.
5 a 70°
b The sum is (2 × 80°) 80°) + (3 × 30°) + 110° 110° = 360°. 360°.
6 a (N − − 2) × 180 = 1980 → N − − 2 = 11 → N = = 13. It could be the sum of the interior angles of a 13-sided polygon. b 2160 7 a 90°
b 30°
8 a 5 sides
b 8 sides
F Exercise 5.3
c 60°
Solving angle problems
1 a 40° + 30° = 70°, the exterior exterior angle of triangle PQR. PQR. b 30°, alternate angles c 40°, alternate angles 2 a Triangle OXZ is isosceles, so a = = (180 − 72) ÷ 2 = 54. b 72° is the the exterior angle of isosceles triangle OZY, OZY, so b = = 72 ÷ 2 = 36. c Angle OZY = b° = = 36°, as triangle OZY is isosceles. Angle XZY = a° + + OZY = 54° + 36° = 90°. 3 The angle opposite 104° is also 104° because the shape is a kite. d° = = 360° − (104° + 104° + 57°) = 95°
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Cambridge Checkpoint Mathematics 9
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Unit 5
Answers to Practice Book exercises
4 Extend the line segment DC. = 72, alternate angles; b = = 180 − 72 = 108. a = = 57, alternate angles; d = = 180 − 57 = 123. c =
A
B 57°
5 The angles of the squares and triangles at the point add up to 2 × 60° + 2 × 90° = 300°. The remaining angle is 360° − 300° = 60°, which is the angle of an equilateral triangle, so it will fit exactly. 6
72°
c ° d ° D
b °
a ° C
is the fourth angle of a quadrilateral so r is = 360 − (95 + 110 + 100) = 55. s makes makes a triangle with r = and 100° so s = = 180 − (55 + 100) = 25. r and makes a triangle with 95° and r so so t = = 180 − (55 + 95) = t makes
Exercise se 5.4 F Exerci 1
30.
Isometric drawings
Other views are possible. a
b
c
2 21 cm and 28 cm 3
4
or
2
Cambridge Checkpoint Mathematics 9
and
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Answers to Practice Book exercises 5
Unit 5
Other views are possible.
F Exercise 5.5
Plans and elevations
1 a
A
B
P
b P
c
A
B
P A
2 a
b
3 a 5
b
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B
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Unit 5
Answers to Practice Book exercises
Exercise se 5.6 F Exerci
Symmetry in three-dimens three-dimensional ional shapes
1 a
b
2
Two like this
Two like this
4
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Answers to Practice Book exercises
Unit 5
3
4 a
5 a Four Four
b
Cube
b
Two like this
Two like this
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Answers to Practice Practice Book exercises 6 Planning and collecting data
Exercise se 6.1 F Exerci
Identifying data
1 a More men than women like gardening gardening.. b c d e f
W omen hair look silly. Girls arethink bettermen thanwith boysspiky boys at texting quickly. quickly . Boys can can throw throw a ball more accurately than than girls can. can. Good cooks go to resaura resaurants nts more often than bad cooks do. Girls who drink lots of water have clearer skin skin than those who who don’t. don’t.
2 a For example: 1. Right-handed students students are better at writing their name using their left hand than left-handed left-handed students are at writing their name using their right hand. 2. ‘Are Are you right or left handed?’ handed? ’, ‘Please write wri te your name, using your left hand and then your right hand.’ hand.’ 3. People People’’s left- or right-handedness and their own names, names, written with both their left and right hands. 4. Ask people to to write on a piece piece of lined paper. paper. 5. About 75 students. 6. Only give people one one chance to write their name as neatly as possible. normal writing, how how many left-handed students students there there are. b Age, neatness of normal right-handers handers but not many left-handers, how to judge how much worse people’s handwriting is when c Lots of rightwriting with the wrong hand, some students may think it is a silly idea and refuse.
3 a For example: 1. There are usually more pictures in my dad’s dad’s newspaper than t han in my magazine. 2. How many many pictures are there there in the newspaper and and in the magazine. 3. Number of pictures in several copies of of the newspaper and in the same number of copies copies of the magazine. 4. Read through both the newspapers and magazines magazines and count count all pictures. 5. Five copies of the newspaper newspaper and of the magazine 6. Count every picture b Does the newspaper have have different different number of pictures depending depending on the the day? c Might not be able to get copies copies of of her dad’s newspapers? 4 a Need equal numbers of boys and and girls in the sample. Need to have a wide range of students, not just those on on her bus. have a range of ages, not just in his year group. Need to ask a range of of students, not just those those that b Need to have obviously like hockey.
F Exercise 6.2
Types of data
1 a b c d
Primar y. Easy to do a survey on your family. Primary. Secondary.. Only the Secondary the airport would be able to to get such a large amount of of information. Secondary. Secondar y. No one could collect that informati information on by themselves. Either: Secondary. Can’t collect this informati information on for the whole country/world. Or: Primary. Could survey women in my area. Either: Secondary. Can’t collect this informati information on for the whole country/world. e Or: Primary. Could do a survey of the motorcycles motorcycles passing my house or ask at local garages or motorcycle sale-rooms. f Either: Secondary. Can’t collect this informati information on for the whole country country/world. /world. Or: Primary. Could survey the people going to my local supermarket.
2 a Madrid is the capital of of Spain, what is sold in this area of Spain might might be the same as is sold in other areas. b Tourists probabl probablyy read different magazines when they are on holiday compared to when they are at work. Many tourists will not be Spanish, but most people living in Madrid are, so they might read different magazines. Copyright Cambridge University Press 2013
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Unit 6
Answers to Practice Book exercises
3 a Mexico is a neighbour of USA, and and so they might buy the same types of of laptop. b USA is the richest country in the world, world, so possibly possibly the people who buy laptops there there will spend more money than people in Mexico.
F Exercise 6.3 1
Number
Designing data-collectio data-collection n sheets Tally
Frequency
1 2 3 4 5 6
Total 2
Make of motorcycle
Tally
Frequency
BMW Ducati Harley Davidson Honda Moto Guzzi Other
Total 3
Number of brothers
Tally
Frequency
0 1 2 3 4
Total values . 4 a No option for 1, 2, 3 or 4 pairs of shoes; overlapping values. b
Number of pairs of shoes
Tally
Frequency
0 1–5 6–10 more than 10
Total
2
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Answers to Practice Book exercises
F Exercise 6.4 1 a
Unit 6
Collecting data
Number
Tally
Frequency
1
//// /
6
2
////
5
3
////
4
4
////
5
Total
20
b 1 is the most common number rolled. 3 is the least common number rolled. 2 a
Mass (grams)
Tally
Frequency
70–79
//
2
80–89
////
5
90–99
//// /
6
100–109
////
5
110–119
//
2
Total
20
b The most common mass for new-born kittens is 90–99 g. 3 a
Number of texts sent
Tally
Frequency
0–9
////
4
10–19
////
5
20–29
////
4
30–39
////
4
40–49
/
1
Total
18
b The most common number of of texts sent was 10–19.
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Answers to Practice Practice Book exercises exercises 7 Fractions
F Exercise 7.1
Writing a fractio fraction n in its simplest form
1 a
3 5
b
2 a
1 2
2
3 a
5
3
4
a
3
g
1
a
b
a
4
1 6
g
7
1 14
8
3
4
39 −
10
5 For example: 3
6
a
8
7
a
5 m 8
20
m
1
3 4
6
3
a
18
g
1 9
a
3
e
2
4
35
6 a
5 12
f
9 11
d
4
12
f
b
2
c
4
20
3
7
7
4 5
7
e
3
5
6 11
Adding and subtracting fractions 9
10
1
5
14
78 −
b
8
h
4
20
3 4
87 =
8 21
20
c
1
i
1
=
1
2
d
1 27
4 21
1
c
8
i
1
8
2
26
k
+1 4
21
87 20
d j
e
5
j 9
2 9
4
=
3
13 30
5
7 9
=
10
2 21 1 12
f
1
l
11
f
7
7 20
l
9
7 36
3 48
4 21
7 20
e
8
k
4
11 24 11 40
1 1 1 3 1 + 1 = 3, 1 + 1 = 3 2 2 3 4 12
b
2
1 10
m
c Check students’ methods.
b Check students’ methods.
b
b
Multiplying fractio fractions ns
9
c 15
1 3
7 15
h
1 5
3 20
b
5
16 25
f
22
1 16
e
5 6
5 For example:
3 7
c
b 12
2 a
d
2 9
F Exercise 7.3 1 a 9
3 5
b
165 =
f
d
6 7 18 25 25 1 , + = + = 3 7 21 21 21 21
33
h
20
2
5
b
e
c
Exercise se 7.2 F Exerci 1
4 5
d 33
1 3
c
3
c
35
i
7 11
c
4
g 10
1 4 1 2
48
e 30
1 5
d
13
d
19
9 14
j
24
1 9
d
11
h
2
e e k
11
f 33
1 5
21
f
f
6 11
l
40
1 3
21 9 13
6 25
2 3
2 21
5 10 1 1 1, 2 × = × = 2 2 4 3 7 21
b
1 15
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Unit 7
Answers to Practice Book exercises
F Exercise 7.4 1
a 35
b 24
g 25
h
15
2 a
15 16
b
4
g
10 11
h
3 a
25 26
b f
e 4 4 For example: 5 a
5 12
1 2
÷
b
F Exercise 7.5
c 22 3 4
1 6
i 88
c
1
11 116
i
13
3
1 8
c
3 5
3
1 9
g
6
d
÷
3 5
=
3 7
c
13 21
5 6
c
2 3
g
5 6
h
7 12
i
9 20
1 14
e
2
1 2
4 9
k 1
1 4
l
20
f
4
l
42
1 5 1
5 21
d
1
h
16 63
d
1
1
d j
4
1 5
b h
15
3 10
1 20
c i
1 18
d j
3 a
1 6
b
2 9
c
9 28
d
g
1 5
h
7 20
i
15 22
4 a
1 2
b
1 4
c
1 3
g
8 9
h
2
i
11 12
7 a
3 28
b
5 28
c
5 7
8 a
3 7
b
2 7
1 10
7 10
1
1 10
e
1
7 20
f
3 50
k
1
1 20
l
1
5
1
10
11 40
16
j 1
2 a g
6
k
Working with fractio fractions ns mentally
b
7 15
f 28
10 9
1 4
5
1 5
35 1
e 45
1
1 2 = 2, 4 3
4 5
j
19 26
4
d 27
1 a
1
2
Dividing fractio fractions ns
25
j d
Cambridge Checkpoint Mathematics 9
j
3
11 24
2
e k
7 12
f l
15
3 28 12 35
e
20 63
f
81 200
1 6
k
1 2
l
15 22
1 6
e 6
f
1
1 8
l
1
11 14
1
1 2
k
9
7 8
11 24
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Answers to Practice Practice Book exercises 8 Constructions, shapes and Pythagoras’ theorem
Exercise se 8.1 F Exerci
Constructing Con structing perpendicular lines
1 Check students’ drawings, all measurements ± 2 mm and ± 2°. 2 Check students’ drawings, all measurements ± 2 mm and ± 2°. 3 Check students’ drawings, all measurements ± 2 mm and ± 2°. 4 a Check students’ students’ drawings, all measurements ± 2 mm and ± 2°. b i 30° ± 2° ii 180° − 90° 90° − 60° = 30° 30° 5 Check students’ drawings, all measurements ± 2 mm and ± 2°. 6 Check students’ drawings, all measurements ± 2 mm and ± 2°. students’ drawings, drawings, all measurements measurements ± 2 mm and ± 2°. 7 a Check students’ 90° − 90° = 180° 180° b i 180° ± 2° ii 360° − 90°
F Exercise 8.2
Inscribing shapes in circles
1 a Check students’ students’ constructions of an inscribed equilateral equilateral triangle, including including construction construction lines. b Check studen students’ ts’ constructions of an inscribed regular octagon, octagon, including including construction construction lines. 2 a Check students’ constructi constructions, ons, includin includingg constructi construction on lines. b 7.1 cm ± 2 mm c Shaded area = 78.5 – [students’ x 2] = 25.2 to 30.9 cm 2 students’ constructions of the inscribed regular octagon octagon and the inscribed circle, including including construction construction 3 Check students’ lines. Inner circle: radius of 6.2 cm to 6.7 cm, area of 120.70 cm2 to 140.95 140 .95 cm2 Area of octagon = 137.28 cm2 to 147.41 147. 41 cm2 students’ constructions of inscribed square, including including construction construction lines. Measurement Measurement of of side length 4 a Check students’ of 11.1 cm to 11.5 cm. Area of square = 123.21 cm2 to 132.25 cm2. Praise, but do not allow alternat alternative ive method, not involving constructi construction. on. b Check students’ students’ explanations explanations involving involving knowledge that one one area must be a quarter (or four times) times) the area of the other when dimensions are doubled. c Check students’ students’ constructions of inscribed square, including including construction construction lines. Measurement Measurement of of side length of 5.5 cm to 5.9 cm. cm. Area of square = 30.25 cm2 to 34.81 cm2. Praise, but do not allow alternati alternative ve method not involving constructi construction. on.
F Exercise 8.3 2
1 a
a
2 a
c
Using Pythagoras Pythagoras’’ theorem 2
= 676, a = 26 cm cm
b
a
= 2500 – 1600 = 900, c = = 30 cm
b
c
2
= 62 + 2.52 = 42.5, a = = 6.5 cm cm
2
= 2.52 – 22 = 2.25, c = = 1.5 m
3 13 cm 4 14.14 cm 5 9.43 km 6 2.24 m or 224 cm 7 33 cm 8 502 cm2 9 78.5 cm2
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Answers to Practice Practice Book exercises exercises 9 Expressions and formulae
F Exercise 9.1 1 a a 7
Simplifying Simplif ying algebraic expressions
b b 10
4
c c 15
4
d d 10
5
e e 6
6
g g 2 a 4a 4 g 8g 8
h h b 16 16b b 8 h 3h 6
i i c 36 36c c 12 i 2x 3
3 a B
b A
c A
f f 4 7
j j j d 64 64d d 6 j 5x 8 j
k k e 10 10e e 11 k 5x 4
l l f 12 f 10 l 11 11x x
d D
4 a One group has x 6 terms and one group has x 9 terms. b 9x 12 ÷ x 9 = 9x 9x 3: this is the only one with power of 3; all others are to the power of 9 or 6.
F Exercise 9.2 1 a n + + 1 f
n
4
− 5
k 3( 3(n n + + 20)
Constructing algebraic expressions
b n − − 10
c 100 100n n
g 6n − − 7
h
n
8
+
9
d i
n
1000 1
− 1
n
e 2n + + 3 j j
10 2n
l 20( 20(n n − − 3)
2 a 6x
b 3x + + 10
c 12 12x x − − 2
d 13 13x x − − 4
3 a xy
b y 2
c 4xy
d 16 16x x 2
b i 2b + + 2 d i 12 12d d − − 2
ii 5b − ii − 20 ii 5d 2 − 5d ii 5d
4 a i 2a + ii 5a + + 16 ii + 15 2 c i 4c − ii c − 8c − 16 ii 8c
5 a i 2( 2(a a + + 3) + 2(3a 2(3a + 1) = 8a 8a + + 8, 4(2a 4(2 a + + 2) = 8a 8a + + 8 ii 3( ii 3(a a + + 3) + 3(3a 3(3a + 1) = 12a 12a + + 12, 6(2a 6(2a + + 2) = 12a 12a + + 12 iii iii 5( 5(a a + + 3) + 5(3a 5(3a + 1) = 20a 20a + + 20, 10(2a 10(2a + + 2) = 20a 20a + + 20 b n black black rods + n striped striped rods = 2n 2n white white rods (or similar explanation given in words) c i 4( 4(a a + + 3) + 2(2a 2(2a + 2) = 8a 8a + + 16 , 8(a 8(a + + 2) = 8a 8a + + 16 ii 6( ii 6(a a + + 3) + 3(2a 3(2a + 2) = 12a 12a + + 24, 12(a 12(a + + 2) = 12a 12a + + 24 iii iii 8( 8(a a + + 3) + 4(2a 4(2a + 2) = 16a 16a + + 32, 16(a 16(a + + 2) = 16a 16a + + 32 d 2n black black rods + n white white rods = 4n 4n grey grey rods (or similar explanation given in words)
F Exercise 9.3
Substituting Substitu ting into expressions
1 a −8 e −8 i −4 1
b −4 f 3 j 12 j
c −7 g 5 k −26
d −2 h 94 l −11
2 a 15 e 8 i 8
b 20 f −64 j 2 j
c −20 g 2 k −25
d 11 h −7 l 10
2
3 a For example: Let a = = 2, 10a 10a 2 = 10 × 22 = 40 and (10a (10 a )2 = (10 × 2)2 = 400, so 10x 10x 2 ≠ (10 (10x x )2 b For example: Let b = = 2, (2b (2b )3 = (2 × 2)3 = 64 and 2b 2 b 3 = 2 × 23 = 16, so (2b (2b )3 ≠ 2 2b b 3 c For example: Let c = = 2 and d = = 3, 3c 3c − − 3d 3d = = 3 × 2 − 3 × 3 = −3 and 3( 3(d d − − c ) = 3(3 − 2) = 3, so 3c 3c − − 3d 3d ≠ 3( 3(d d − − c )
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Unit 9
Answers to Practice Book exercises
Exercise se 9.4 F Exerci
Deriving and using formulae H
1 a H = = 24D 24D
b H = = 240
c
2 a D = = 150
b D = = 180
c S = = 20
d T = 5.5
3 a F = = 25 e e = = 5
b F = = 54 f a = = 7
c I = = 40
d I = = 21
+ 3 4 a d +
= 2d 2d + + 3 b T =
= 19 c T =
d
5 a 50%
b 8%
c 110%
6 a 450 m
b 1303 m
c 1078 m
D
=
24
d D = 20
−
d T 3 =
2
e 12
d 1615 m
7 Anders is correct. 20 °C = 68 °F and 68 °F > 65 °F.
F Exercise 9.5
Factorising
1 a 6(a 6(a + + 4) d g (7g (7g + + 1)
b 3(3c 3(3c – – 5) e 4(2 – 3 j )
c 4 f (e + + 4) f m(7m (7m – 4)
2 a 5(z 5(z + + 3) e 2(3 2(3v v + + 4) i 3(4 − 5w )
b 2( y – – 7) f 7(2 7(2u u − − 3) j 8(2 + 3x ) j
c 4(5x 4(5x + + 1) g 6(2 − u ) k 2(4 + 7 y )
d 3(3w 3(3w − − 1) h 7(2 + 3v ) l 7(2 − 5z )
3 a m(7 (7m m + 1) 1 + 4s 4s ) e 3s ((1 i 7e (2i (2i − − 1)
b 5a (a – – 3) 3 − 4 y ) f 4 y ((3 j 4a (3 j (3 + 2b 2b )
c t (t + + 9) 8(2e e − − i ) g 8(2 k 3g (7 (7 + 5h 5h )
d 4h (2 (2 − h ) 3(5e e + + 2i 2i ) h 3(5 l 15 15w w (2 (2 − t )
4 a 2(a 2(a + + 2h 2h + 4) d e (3e (3e + + 4 + f + f )
b 5(b 5(b – – 5 + j + j ) e k (7 (7 – k – – a )
c 4(3tu 4(3tu + + 4u 4u – – 5) f 3n (2n (2n – – 3 + m)
5 5(3x 5(3x − − 2) − 5(2 + x ) = 15x 15x − − 10 − 10 − 5x 5 x = = 10x 10x − − 20 = 10(x 10(x − − 2) Tanesha’s mistake was expanding −5(2 + x ) to give −10 + 5x 5x , which adds to 15x 15 x − − 10 to give 20x 20x − − 20.
Exercise se 9.6 F Exerci
Adding and subtracting algebraic fractions
2 x 3
b
g
y
h 1 4 y
2 a
x
1 a
2
c
+
y
b
4a + 5b 20
2x + y
h
c
21a + 4b 28
3
d
j j
18
6
x
i 1 5 y
9
2
g
3x 5
9x + y
d
4 x 7 2 y 9
i
18
− y
e
18
j j
3 a A, D, F each equal 1 x and and B, C each equal 1 x . 4
k
15x
12 10a + 15b
e
2
5a
− 7b
35
3x 4
f
17 y
l
24
7x
− 8 y
f
2 x 5
5 y 16 7x
12
k
15a
18
− 2b
24
− 15 y
l
12a
− 35b 42
b E, which equals 1 . 3
c You can ignore the letter, work out the fracti fractions, ons, then put the letter back in at the end.
2
Cambridge Checkpoint Mathematics 9
Copyright Cambridge University Press 2013
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Answers to Practice Book exercises
F Exercise 9.7
Expanding the product of two linear expressions
1 a x 2 + 7x 7x + + 10 d x 2 − 3x 3x − − 18 2 g x + 15 15x x + + 50
b x 2 + 7x 7x + + 6 e x 2 − 6x 6x + + 9 2 h x + 5 5x x − − 50
c x 2 + 2x 2x − − 8 f x 2 − 13x 13x + + 40 2 i x − 15 15x x + + 50
2 a B
b A
c C
2
2
2
46d + 3 a d a d 2 + − 4a 6a d + + + 49 4 a b c d
Unit 9
+ 10e 8b e 8b + + b e b e 2 − 10 +16 + 25
d C
22 f c + + + 11 cf fc 2 + − 2c +
i a 2 − 1 ii a 2 − 16 ii iii a 2 − 81 iii There is no term in x , and the number term is a square number. a 2 − 64 a 2 − b 2
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Answers to Practice Practice Book exercises exercises 10 Processing and presenting data
F Exercise 10.1
Calculating statistics
1 a 0
b 1
c 1.7 or 1.71
2 a 1
b 2
c 2.55
3 a 50–55 minutes 4
d i 1
ii 2 ii
iii 2.67 iii
b The median is in the the class 55–60.
a The mode is 5 and the the median is 4. b The mean is 156 ÷ 40 = 3.9 3.9 which is less than the median of 4.
5 10.125 kg 6
a 31–35 seconds
F Exercise 10.2
b 21–25 seconds
c
26–30 seconds
Using statistics
1 a The median is 41 and the mean is 40.3. Both show that that the average is above above 40. The mode is not a good good choice here. b There is no reason to complain if the average average is above above 40. 2 The median is 19 (< 20) and the mean is 21.1 (> 20). 3 a You can use the modal class or the median. median. The modal class for the the boys is 36–40 and for the the girls it is 41–45. The median for the boys is in the class 36–40 and for the girls it is in the class 41–45, so the girls have done better than the boys. b The range for the the boys is greater greater because there there are no girls in the lowest class. 4 a The mean.
b They had all played 14 games.
c 1 045 980
5 a The Book Book Club. Club. The medians are 35 and 47. 47. b The M Music usic Club. The ranges are are 24 years years and 15 years. 6 The men. The median for the the men is in the the class 40–44; for women women it is in the class 35–39.
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Answers to Practice Practice Book exercises exercises 11 Percentages
F Exercise 11.1 1 a $468
Using mental methods b $702
c $117
d $23.40
example: Find 25% (a quarter) and 10% (divide by by 10) and add them. 2 a For example: b i 15.4 kg ii 98 m iii $30.80 3 a 219.3
b 2646
c 57.6
d 320
4 a 204.12 kg
b $136.08
c 816.48 m
d
5
40.824 litres
Amount
164
328
82
16.4
32.8
65% of the amount
106.6
213.2
53.3
10.66
21.32
6 a C, 69% of 272 b The rest are all equal. This one one should be 69% of 282 to be the same as the others. 7 a 1024
b 1536
c 2112
d
3712
8 D C A B
Exercise 11.2
Comparing different quantit quantities ies
F 1 No. 76% for English and 82% for science.
p eople is greater. 2 9% of the young people and 27% of the older people wear glasses. The percentage of the older people 3 a Rovers 45%, United 62% 4 a 56% (or 55.6%) 5 a b c d
b United, becaus becausee the percentage is higher.
b 44% (or 44.1%)
i 36% ii 64% 68% were boys and 32% were girls. Yes. 23/75 = 31%. Yes. 62% of the girls chose tennis but only 31% of the boys.
F Exercise 11.3
Percentage changes
1 Carpet 14% (or 13.8%), table 22% (21.7%), chair 42% (41.7%) 2 7% (6.5%) 3 3% (3.125%) 4 a 6.6% increase
b 10.4% decrease
c 7.2% decrease
5 a 10%
b 9.1%
c 8.3%
6 a 130% (129.7%)
b About 67.5 million
7 a 12.5%
b 11.1%
8 a A: 50%, B: 33.3%, C: 25%
b A is the best.
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d 7.1%
c 75 km/h (74.9)
Cambridge Checkpoint Mathematics 9
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Unit 11
Answers to Practice Book exercises
F Exercise 11.4
Practical examples
1 Radio 30% profit; Television 11.1% profit; Computer 11.5% loss; Jewellery 25.5% profit. 2 $225 3 a $53 300
b $270
c 4%
4 12.4% profit 5 Hock Hockey ey stick $81.75; football boots $100.28; track suit $140.61. 6 $1287.50 7 18% 8 a $0
2
b $1350
c $5250
Cambridge Checkpoint Mathematics 9
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Answers to Practice Practice Book exercises exercises 12 Tessellations, transformations and loci
F Exercise 12.1
Tessellating shapes
1 Check students’ tessellations; each should show at least five of the shape being tessellated.
explainations, involving involving corners of square = 90° and 360 ÷ 90 = 4 (i.e. no remainder), 2 Check students’ explainations, with a suitable diagram. 3 Exterior angle = 36°, so interior angle = 144° and 360 ÷ 144 = 2.5 (i.e. not a whole number).
F Exercise 12.2 1
Solving transformation problems
y
3 2
c
1
a
0 –4 –3 –2 –1 0 A 1 –1 d –2 b –3
2
2
3
4
x
y
6 5 4 c
3
B
2
a
b
1 0 0
3
1
2
3
4
y
5
6
7
x
6
7
x
d
6 5
a
4
b C
3 2 1
c 0 0
4 a
1
2
3
4
5
b
y
y
6
6
5
5
4
4 a
3
3
2
2
1
1
0
b
0 0
1
2
3
4
5
6
7 x
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0
1
2
3
4
5
6
7 x
Cambridge Checkpoint Mathematics 9
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Unit 12 5
Answers to Practice Book exercises y
4 3
P
2 1 0 –4 –3 –2 –1 0 –1
1 R
2
3
4
x
–2 Q –3 –4
6 a A(2, 6), B(7, 6), C(6, 3) and D(0, 2). c A´(6, 2), B´(6, 7), C´(3, 6) and D´(2, 0). -coordinates have changed places. d The x- and y -coordinates
b
y
B′ y = x
7 6
C′
A
B
5 4 C
3 2
A′
D
1
D′
0 0
1
2
3
4
5
6
7
x
7 The shape is symmetrical about the line y = = 3, so when it is reflected in the line y = = 3 the shape stays in the same position. The shape has rotational symmetry of order 4 about the centre (3, 3), so when the shape is rotated 90° about (3, 3) it again stays in the same position. So the starting shape and finishing shape are exactly the same, in exactly the same position.
F Exercise 12.3 1
Transforming shapes y
6 5 d
4 c 3
c
d
2 1 b 0 –6 –5 –4 –3 –2 b–1 0 –1
1
2
3
–2
4 5
6
x
a
–3 A
–4
a
–5 –6
2 a b c d e f g
Reflection in the y -axis. -axis. Reflection in the line y = = 1. Reflection in the line y = = 2. Reflection in the line y = = −2. Reflection in the line x = = −2. Rotation 90° anticlockwise about (0, 0). Rotation 180° about (0, 1).
h Rotation 180° about (–2, –1).
2
Cambridge Checkpoint Mathematics 9
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Answers to Practice Book exercises 3 a i
Rotation 90° anticlockwise about (−1, 3).
ii Translation
Unit 12
4 –4
iii Reflection in the line x = = −1. iv Reflection in the line x = = −3.5. b i Check students’ combinations combinations of at least two transformations. two transformations. ii Check students’ combinations of at least two
F Exercise 12.4 1
Enlarging shapes
y 4 3 2 1 0 –4 –3 –2 –1 0 –1
1
2
3
4
x
–2
2 a Enlargement scale factor 3, centre (6, 2). b Enlargement scale factor factor 2, centre (3, 5). 3 Enlargement scale factor 3, centre (6, 1). 4 a y 4 3 2 1 0 0 –6 –5 –4 –3 –2 –1 –1
1
2
3
4
x
–2 –3
enlargement with centre centre of enlargement anywhere anywhere except except (0, 0) and words words to b Grid showing a square and its enlargement the effect that in this case multiplying the coordinates by 2 does not make the equal to the coordinates of the enlarged square
Exercise 12.5
F 1
Drawing a locus
A 4.5 cm
2
3
3 cm
3 cm
3 cm
3 cm
4 cm P
4 cm 6 cm 4 cm
4
Q 4 c m
Check students’ circles; they must have a radius of 3 cm. G 6 cm
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Cambridge Checkpoint Mathematics 9
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Unit 12
Answers to Practice Book exercises
5 C 6 cm
1.5 cm
8 cm
6 a
7
b
c
W
X
Z
Y
8
W
X 160 km
Check students’ circles; they must have radii of 5 cm and 4 cm.
4
Cambridge Checkpoint Mathematics 9
Copyright Cambridge University Press 2013
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Answers to Practice Practice Book exercises exercises 13 Equations and inequalities
F Exercise 13.1
Solving linear equatio equations ns
1 a g = = 12
b g = = −5
c g = = −10
d g = = 7
= 5.25 2 a p =
= 0 b p =
= 7 c p =
= 0.5 d p =
= 2 47 3 a y =
= 1116 b y =
= 3 18 c y =
= 5 79 d y =
4 a x = = −8
b x = = −3
c x = = −1213
d x = = −2
+ 15 = 10x 10 x − − 20 → x = = 7 5 a 5x +
+ 3 = 2x 2x − − 4 → x = = 7 b x +
− 32 + 20 − 4x 4x = = 0 → 4 4x x − − 12 = 0 → x = = 3 6 a 8x − 2(x 2( x − − 4) + 5 − x = = 0 → 2x 2 x − − 8 + 5 − x = = 0 → x − − 3 = 0 → x = = 3 b = 4 7 a x =
= −3 b x =
= 11 c x =
b x = = 4 72
8 a 5x + + 30 = 60 − 2x 2 x
4 x − − 8 = 40 − 2x 2x → 6 6x x = = 48 → x = = 8. 9 Multiplying out the brackets: 4x Dividing by 2: 2(x 2(x − − 2) = 20 − x → 2 2x x − − 4 = 20 − x → 3 3x x = = 24 → x = = 8. Both give x = = 8.
F Exercise 13.2
Solving problems
1 a n + + 2(n 2(n + + 3) = 90 → 3 3n n + + 6 = 90
b n = = 28
c 28 and 62
+ 50 and 2x 2 x + + 80 2 a x +
+ 80 = 144 b 2x +
= 32 c x =
+ 2s 2s + + 2s 2s + + 5 = 100 → 5 5s s + + 5 = 100 3 a s +
= 19 b s =
c 43 cm
4 a y + + 3 y + y + y − − 2 + 4( y − − 2) = 116
b y = = 14
c 48
5(x x − − 8) = 2(x 2(x + + 10) 5 a 5(
b 20
6 a 2a + + 6(a 6(a − − 2) = 44 or a + + 3(a 3(a − − 2) = 22 → 4 4a a − − 6 = 22
F Exercise 13.3
b 7 cm and 15 cm
Simultaneous Simultaneo us equati equations ons 1
1 x = = 6, y 6, y = = 18 = 6, y 6, y = = −3 2 x = 3 x = = 2, 2, y y = = 5
4 a x = = 6, y 6, y = = 24
b x = = 4, y 4, y = = 6
c x = = 1, y 1, y = = −3
5 (2 × 4) + (3 × 5) = 23 and (5 × 4) + (2 × 5) = 30 6 x = = 10, y 10, y = = 20 = 1.6, y 1.6, y = = 18.4 7 x = = 14, y 14, y = = −9 8 x =
9 x = = −2, y −2, y = = 4
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Cambridge Checkpoint Mathematics 9
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Unit 13
Answers to Practice Book exercises
F Exercise 13.4
Simultaneous Simultaneo us equatio equations ns 2
= 18, y 18, y = = 2 1 a x =
= 9.5, b = = 5.5 b a =
2 a x = = 9, y 9, y = = 3
b x = = 9, y 9, y = = 6
3 a x = = 2.5, y 2.5, y = = 10
b x = = 12, y 12, y = = 14
= 7.5, y 7.5, y = = 5.5 4 a x =
2x + y + y = = 20.5 and not 19. b Using the values in part a, 2x
5 a x = = 6, y 6, y = = 10
b x = = 3.5, y 3.5, y = = 3
F Exercise 13.5
= −4, q = = 8 c p =
c a = = 3, b = = −1
Trial and improvement
= 9 1 a x =
= 10 b x =
= 3 c x =
2 8.7 3 3.2 4 a 0.9
b 12.5
5 If x = = 4.8, x ² − 4x 4x = = 3.84. If x = = 4.9, x ² − 4x 4x = = 4.41. 6 x = = 2.3. Here are some possible values.
x
2
3
2.5
2.2
2.3
x ² + 3x 3x
10
18
13.75
11.44
12.19
7 x = = 2.7 8 x = = 1.6 and x = = 4.4
F Exercise 13.6
Inequalities Inequalit ies
> 2 1 a x >
b x ≥ −6
< 0 c x <
d x ≤ 10
2 a –3 –2 –1
0
1
b 0
3.5
c –3
0
d –10
0
10
20
3 a Could be true.
b Could be true.
c Must be true.
d Cannot be true.
4 a x ≥ 0.5
< 3 b x <
c x ≤ 13
< 6.5 d x <
5 a x ≤ 10
b x > > 4
c x ≥ 2
6 a A + + A + + 5 + 2(A 2(A + + 5) < 100 → 4 4A A + + 15 < 100 b A < < 21.25 c Because if A < < 21.25 then 2(A 2( A + + 5) < 52.5. 7 a x + + 2x 2x + + 3(x 3(x − − 10) < 360 → 6 6x x − − 30 < 360 b x < < 65 c Yes es.. 2x = = 3(x 3(x − − 10) → x = = 30 and this is in i n the solution set.
2
Cambridge Checkpoint Mathematics 9
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Answers to Practice Practice Book exercises 14 Ratio and proportion
F Exercise 14.1
Comparing and using ratios
1 a Banana yellow 1 1 : 0.6, Mellow yellow 1 : 0.71
b Mellow yellow
2 a Gavin 1 : 3.5, Matt 1 : 3.3
b Gavin
3 a 1 : 13.12
b 1 : 15.67
c Raine’s
4 a 1 : 1.41
b 1 : 1.34
c The Bounders
5 300 g cement and 5 kg lime 6 a 2.5 kg cement
7
b 27.5 kg
Activity
Child : staff ratios
Number of children
Number of staff
Horse-riding
4:1
22
6
Sailing
5:1
17
4
Rock-climbing
8:1
30
4
Canoeing
10 : 1
26
3 17
Total:
8 a $744
b $525
9 a $154
b Check students’ methods for checking.
F Exercise 14.2 1 a b c d e f
c $312
d €258.50
Solving problems
Yes, as the number number of bottles bought increases, so does the total total cost (the ratio stays the same). No, the ratio does not stay the same. Yes, as the number of stamps bought increases, so so does the total total cost (the (the ratio stays the same). Yes, as the distance increases, so does the time taken (the ratio, on average, stays the same). No, the ratio does not stay the same. No, the ratio does not stay the same.
2 a $50
b $75
c $187.50
3 a $33.30
b $20.35
4 a $1.18
b $1.15
c 120 tea bags
5 a The box of 50 pens
b The 750 750 g pack pack of of cereal
c The 450 450 ml pot pot of of yoghurt yoghurt
6 £208 7 a 8
S$517.50
€187 = $239.74, $254
b A$104 = €198.12. He should buy the phone in Paris.
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Answers to Practice Practice Book exercises 15 Area, perimeter and volume
F Exercise 15.1 1 a 70 000 cm2
Converting units of area and volume b 8000 cm2
2
d 5002mm g 9 m j 3 cm2 2 a d g j
2 000 000 cm3 8000 mm3 9 m3 7 cm3
3 a 70 ml d 7 litres g 8000 cm3
c 32 500 cm2
2
e 40 mm h 3.4 m2 k 2.8 cm2
b e h k
240 000 cm3 500 mm3 0.48 m3 0.23 cm3
b 348 ml e 8.4 litres h 3900 cm3
2
f 920 mm i 0.5 m2 l 0.8 cm2 c f i l
5 600 000 cm3 7200 mm3 82.2 m3 77.6 cm3
c 2500 ml f 0.92 litres i 880 cm3
4 a 12.2425 m2. Check that students check correctly, using estimation. correctly, using inverse operati operations. ons. b $312. Check that students check correctly, Wall 2: 8.28 m2 5 Wall 1: 10.08 m2 Wall 3: 9.72 m2 Wall 4: 10.44 m2 Total = 38.52 38. 52 m2 Check students’ own choice of method for checking the answer.
F Exercise 15.2
Using hectares
1 a 40 000 m2 e 8200 m2
b 52 000 m2 f 340 m2
c 9000 m2
d 452 000 m2
2 a 7 ha e 0.07 ha
b 3.2 ha f 237.5 ha
c 67 ha
d 0.88 ha
3 a 151200 m2
b 15.12 ha
4 a 39861 m2
b 3.9861 ha
5 a 28275 m2 b 2.8275 ha c $6220.50 students’ ts’ own methods for checking checking their answers answers by estimation. d Check studen 6 Area = 98 701 m2 Cost = $38 493.39 $38 493.39 < $40 000 so he can afford it. Check students’ own own methods for checking their answers by estimation.
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Unit 15
Answers to Practice Book exercises
F Exercise 15.3
Solving circle problems
1 a c
A = =
12.6 cm2, C = = 12.6 cm cm 2 A = = 254.5 cm cm , C = = 56.5 cm
b d
A = =
2 a c
A = =
113.5 cm cm2, P = = 43.7 cm 2 A = = 402.1 cm cm , P = = 82.3 cm
b d
A = =
= 8.37 cm 3 a d = = 7.07 cm 4 a r =
66.5 m2, C = = 28.9 m 2 A = = 21.2 m , C = = 16.3 m 904.8 mm2, P = = 123.4 mm 2 A = = 88.4 m , P = = 38.6 m
= 28.49 mm b d = = 3.82 m b r =
= 1.51 m c d = = 0.53 m c r =
= 11.30 cm d d = = 10.78 mm d r =
5 1.4 cm (14 mm) 6 7.59 m (759 cm) 7 27 cm2 8 a 168.18 cm2
b 120.82 cm2
Exercise se 15.4 F Exerci
Calculating Calculatin g with prisms and cylinders
1 a 150 cm3
b 129.6 cm3
2
Area of cross-section
Length of prism
Volume of prism
a
2
8.4 cm
20 c m
168 cm3
b c
24 cm2 58 m2
6.5 cm 5.7 m
156 cm3 330.6 m3
d
56.85 mm2
62 m m
3524.7 mm3
3 a V = 480 cm3, SA = 416 cm2 c V = 675 cm cm3, SA = 558 cm2
b V = 576 cm3, SA = 544 cm2
4 a V = 754.0 cm3, SA = 477.5 cm2 c V = 42 411.5 mm3, SA = 8482.3 mm 2
b V = 492.6 cm3, SA = 401.1 cm2
5
6 a
2
c 427.5 cm3
Radius of of ci circle
Area of of ci circle
Height of of cy cylinder
Volume of of cy cylinder
a
7 cm
153.94 cm
12 c m
1847.26 cm3
b
1.5 m
7.07 m2
2. 4 m
16.96 m3
c
9 cm
254.47 cm2
7.51 cm
1910 cm3
d
2.19 m
15 m2
3. 8 m
57 m 3
e
4.55 mm
65 mm2
22 mm
1430 mm3
x = = 4.3
2
b
x = =
3.3
Cambridge Checkpoint Mathematics 9
c
x = =
2.1
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Answers to Practice Practice Book exercises 16 Probability
F Exercise 16.1
Calculating probabiliti probabilities es
1 a 0.9
b 0.7
c 0.45
2 a 0.95
b 0.9
c 0.15
3 a 5%
b 80%
c 15%
4 a 0.15
b 0.85
c 0.2
5 a
1 16
7 16
b
6 a 0.17
b 0.31
Exercise se 16.2 F Exerci 1 a
2 a
T
+
+
H
+
+
H
T
2
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
2
+
+
+
+
+
+
1
+
+
+
+
+
+
1
2
3
4
5
6
3
5 a
b The probabilities are 1 and
+
4
4 a
Sample space diagrams
+
6 5
3 a
c 0.11
4
+
+
+
+
3
+
+
+
+
2
+
+
+
+
0
2
4
6
C
+
+
+
B
+
+
+
A
+
+
+
A
B
C
D
+
+
+
+
C
+
+
+
+
B
+
+
+
+
A
+
+
+
+
A
B
C
D
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1 4
. One is twice the other.
b i
11 36
ii
1 6
iii
2 9
b i
1 6
ii
5 6
iii
1 6
b i
1 3
ii
2 3
b i
1 4
ii
3 4
Cambridge Checkpoint Mathematics 9
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Unit 16 6
Answers to Practice Book exercises
1 and 4 5 5
7 a
3 16
8 a 0.1
b
13 16
b 0.18
F Exercise 16.3
c 0.16
Using relative frequency
1 a 0.46 b 0.67 c 0.21 2 a i A: 0.72, B: 0.77 ii A: 0.18, B: 0.16 iii A: 0.10, B: 0.08 student being normal weight weight is higher, higher, and the probabilities of a student student being b School B. The probability of a student underweight or overweight are lower than in school A. 3 a 0.64
b 0.8
4 a City 0.12, Mountain View 0.17 b City 0.57, Mountain View 0.33 probability of a good good grade, and and a lower lower probability of of a poor grade. c City. It has a higher probability 5 a i 0.27 ii 0.16 b Afternoon trains are more likely to be on time and are are less likely to to be early or late. late. 6 a i 0.58 ii 0.17 b It is not a good way. way. One reason is that it makes a difference whether a team is playing at home or away. away.
2
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Answers to Practice Practice Book exercises exercises 17 Bearings and scale drawing
F Exercise 17.1
Using bearings
1 a 065°
b 145°
2 a 057°
b 237°
3 a 110°
b 045°
c 200°
d 315°
c 155°
d 275°
e 330°
4 a Ai 036° Aii 216° b Answer to ii = answer to i + 180° c Ai 083° Aii 263°
Bi 124°
Bii 304°
Ci 073°
Cii 253°
Bi 137°
Bii 317°
Ci 022°
Cii 202°
5 a Ai 238° Aii 058° b Answer to ii = answer to i – 180° c Ai 232° Aii 052°
Bi 288°
Bii 108°
Ci 261°
Cii 081°
Bi 336°
Bii 156°
Ci 198°
Cii 018°
F Exercise 17.2
Making scale drawings
drawings. s. 1 a Check students’ scale drawing
b 178 km
c 286°
drawings. s. 2 a Check students’ scale drawing
b 229 km
c 090°
3 a Check students’ scale drawing drawings. s.
b 26 km
c 247°
4 7.4 km, 218° 5 Their paths cross, so they could collide. It depends on when they start moving and how fast they travel. Students’ scale drawings should show that the paths cross. 6 a 24 km
b 14 cm
7 a 256 km
b 5.75 cm
8 She ran 12.5 km, so she raised $450.
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Answers to Practice Practice Book exercises exercises 18 Graphs
F Exercise 18.1
Gradient of a graph 1 4
1 a 1
b 2
c
2 a −1
b −5
c − 13
3
y
4 b
3 2 1 a 0 –4 –3 –2 –1 –1
1
2
3
4
5
6
7
8
9
10
x
8
9
10
x
c
–2 –3 –4
4
y
8 7 6 5 d 4
b
3 c
2 1
0 –4 –3 –2 –1 –1
1
2
3
4
5
6
7
–2 –3 –4 –5 –6 –7 –8
5 a 2.5
b −1.5
6 a 0.1
b 0.05 or
7 a 25
b 0.1
c 0.5 1 20
Copyright Cambridge University Press 2013
d −5
c −0.1 c −1
d 2
Cambridge Checkpoint Mathematics 9
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Unit 18
Answers to Practice Book exercises
F Exercise 18.2 1 a
The graph of y = = mx + + c
y
b All have gradient 4.
8 7 6 5
ii
iii
i
4 3 2 1 0 –2 –1 –1
1
2
3
4
x
–2 –3 –4 –5 –6 –7 –8
2 a A and B
b −4 (C)
c A and D
3 a y = = −2x −2x
b y = = −4 − 2x 2x
c y = = 4 − 2x 2x
4 a If x = = 0, y 0, y = = 50 − 10 × 0 = 50; if x = = 5, y 5, y = = 50 − 10 × 5 = 0 5 a −25
b 25
c 50
b −10
d 75
6 A, C and D are parallel; B and E are parallel.
F Exercise 18.3 1
2
Drawing graphs
a i y = = −x −x + + 12 12
ii y = ii = −2x −2x + + 12
iii y = iii = − 12 x + + 6
b i −1
ii −2 ii
iii − 12 iii
= 1.5x 1.5x − − 3 a y =
b
y 8
c 1.5
6 4 2 0 –4 –2 –2
2
4
6
8
x
–4 –6 –8
3 a y = = 0.1x 0.1x + 1.4 is the equation of a straight str aight line. 4 a 1.5 1.5
b −0.4
c −1
b 0.1 d 5
top line. It passes through (0, 5). 5). 5 a The top b 5x + + 8 y = = 0 (through the origin) and 5x 5 x + + 8 y = = 20 (through (0, 2.5))
c y = = −10x −10x
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Answers to Practice Book exercises 6 a y = = 0.05x 0.05x + 5 or y or y = =
1 x + + 20
1 20
b 0.05 or
5
Unit 18
c
y 10
8
6
4
2
–140
–120
–100
–80
–60
–40
–20
0
20
40
60
80
1 00
1 20
–2
7 a −10
b −0.1
8 a – 2
b c = = 2 and d = = 3
3
c 50
d 0.02
Exerci Exercise se 18.4 Simultaneous equations
F 1 a x = = −4 and y and y = = −7
b x = = −1 and y and y = = 2
c x = = 2 and y and y = = −1
2 a x = = 3.2 and y and y = = 4.6
b x = = −0.8 and y and y = = 2.6
c x = = 1.4 and y and y = = −1.9
3 a i y = = −x −x + + 5
ii y = ii = x − − 3
b
y 6 4 2 0 –2 –2
2
4
6
8
x
–4 –6
= 4 and y and y = = 1 c x = 4 a, b y
+ 5 = x − − 3 d −x +
8 6 4 2 0 –6 –4 –2 –2 –4 –6 –8
2
4
6
8
10
x
→
→
8 = 2x 2x x = = 4 and then y then y = x − − 3 = 4 − 3 = 1 c x = = −3 and y and y = = 6
x 140
Cambridge Checkpoint Mathematics 9
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Unit 18 5
Answers to Practice Book exercises
a and b
y
c i x = = 2 and y and y = = 60
ii x = ii = 4 and y and y = = 20
100 80 60 40 20 0 –6 –4 –2 –20
2
4
6
8 10 x
–40
6 The graph should look like this. this. x = = 3.4 and y and y = = −0.9 y 2 1 0 –4 –3 –2 –1 –1
1
2
3
4
5
6
x
–2 –3 –4 –5 –6
F Exercise 18.5
Direct proporti proportion on
Other scales are possible for the graphs. 1 a y
b 1.64
c D = = 1.64E 1.64E
b y = = 7.35g 7.35g
c i $24.99
d i $484
ii ii
(100, 164)
150 ) $ ( s r a100 l l o D
50
0
2
a
50 100 Euros (€)
x
y
40
(5, 36.75)
30 s r a l l o 20 D
10
0
1
2
3 4 Grams
5
x
ii 2.72 grams ii
€179.88
3
Cambridge Checkpoint Mathematics 9
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Answers to Practice Book exercises 3
a 15 000 litres
b
Unit 18
y
15000 l e u f f o s e r t i L
(60, 15000)
10000
5000
0
c f = = 250m 250m 4
a
d i 41 250 litres
) y m c 4 ( r i a h 3 f o 2 h t g 1 n e L
a
5
15
10 Weeks
y
ii 400 minutes ii minutes or 6 hours and 40 minutes b 0.3
c l = = 0.3w 0.3w
d 333 weeks weeks or or about 6.4 years years
b 16.5
about 6 minutes and 4 seconds c Just over 6 minutes or about
x
) m40 c ( e30 c n a20 t s i D10
0
x
(12, 3.6)
0
5
10 20 30 40 50 60 Minutes
(2, 33)
1 2 Minutes
F Exercise 18.6
x
Practical graphs
Other scales are possible on the graphs. h
1
= 2 + 0.5d 0.5d a h =
b
7 s 6 e r t e 5 m n 4 i t h g 3 i e H2
1 0 0
1
2
3
4
5 6 Days
7
8
9 10 d
c i 4 metres
ii 9 days ii
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Unit 18 2
Answers to Practice Book exercises
a D = = 3w 3w + + 20
b
c i $32
D
ii 10 weeks ii
60 50 s r40 a l l o 30 D
20 10 0 0
1
2
3
4
5
6
7
8
9
10
w
Weeks
3
a n = = 14 000 − 500m 500m
b
n
c 8 minutes
14000 12000 e l p o e P
10000 8000 6000 4000 2000 0 0
1
2
3
4
5
6
7
8
9
10
m
Minutes
4
a t = 30 − 4d 4d
b
c On the 8th day ( 7 1 days)
t
2
32 28 24 s t 20 e l b a T16
12 8 4 0 0
5 a L = = 20 000 − 1500d 1500d
1
2
b
3
4 5 Days
6
7
8 d
L
20000 18000 16000 ) 14000 s e r t 12000 i l ( r e10000 t a W8000 6000 4000 2000 0 0
1
2
3
4
5
6
7
8
Days
c 12 500 litres
d 13 days (13 1 ) 3
9 10 11 11 12 13 13 14 d
5
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Answers to Practice Book exercises 6
a P = = 25 + 0.1 y
b
P
) s 30 n o i 25 l l i m20 ( n o15 i t a l u10 p o P 5 0
0
10
20
30 Years
40
50
y
c 30 years
Unit 18
Cambridge Checkpoint Mathematics 9
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Answers to Practice Practice Book exercises 19 Interpreting and discussing results
F Exercise 19.1 1
Interpreting and drawing frequenc frequency y diagrams
a 32 b
Time, t (minutes)
Frequency
Mid-point
10 ≤ t < < 12
4
11
12 ≤ t < 14
16
13
14 ≤ t < < 16
7
15
16 ≤ t < < 18
5
17
c
Time taken by 9C to complete cross-country run 18 16 14 12
y c n e u q e r F
10 8 6 4 2 0 10
d
12 14 16 Time (minutes)
18
5 8
2 a 50 b
Wednesday Height, h (c (cm)
Saturday
Frequency
Midpoint
Height, h (c (cm)
Frequency
Midpoint
120 ≤ h < < 140
4
130
120 ≤ h < < 140
25
130
140 ≤ h < 160
6
150
140 ≤ h < 160
16
150
160 ≤ h < < 180 180 ≤ h < 200
22 18
170 190
160 ≤ h < < 180 180 ≤ h < 200
7 2
170 190
c
Heights of people on roller coaster Wednesday
Saturday
35 30 25
y c n e u q e r F
20 15
10 5 0 120
140
160 180 Height (cm)
200
d For example: On Saturday Saturday there were were fewer taller people and more more shorter people. There There were only two two people with a height between 180 cm and 200 cm on Saturday compared with 18 on Wednesday. There were 25 people between 120 cm and 140 cm on Saturday compared with four on Wednesday.
7
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Unit 19
Answers to Practice Book exercises
3 a
Hours of training each week by athletes at two clubs Falcons Club
Harriers Club
30 y c n e u q e r F
20 10 0 0
5
10 15 20 Number of hours
25
b For example: The most popular training time for the Falcons Falcons Club was between 5 and 10 hours, hours, whereas for the Harriers Club it was between 15 and 20 hours. In the Falcons Club only 22 athletes trained for more than 15 hours a week compared with 42 athletes from the Harriers Club. c Falcons Club 68, Harriers Club 70 surveyed at each club was nearly nearly the same. d Yes, because the number of athletes surveyed
F Exercise 19.2
Interpreting and drawing line graphs
1 a
Average Ave rage monthly rainfall in Faro, Portugal
100 90 80 70
) m60 m ( l l 50 a f n i a 40 R
30 20 10 0 J
F
M
A
M
J
J
A
S
O
N
D
Month
b For example: The year year starts with just under 80 mm of rain in January, then there is less rain every month until until July. July is the driest month. After July it starts getting wetter each month for the rest of the year, with a large increase in rain in October. c February and March
2 a
Company profit 7
) s n o i l 6.5 l i m $ ( t i 6 f o r P
5.5 2002
2004
2006 2008 Year
2010
2012
b For example: example: The profit profit is increasing increasing by a roughly similar similar amount each year. year. c $6 million d Answer from $6.8 million million to to $6.9 million (inclusive) (inclusive)
1
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Answers to Practice Book exercises 3 a
Unit 19
Daily temperatures in Marrakech in July Maximum temperature (ºC) Minimum temperature (ºC)
) 40 C º ( 30 e r u t 20 a r e p10 m 0 e T
Mon
Tues
Wed
Thur Day
Fri
Sat
Sun
temperatu res increased from Monday to to Thursday, then decreased for the next b For example: The maximum temperatures two days, finally increasing again on Sunday. Sunday. The minimum temperatures stayed the same for the first two days then increased until Thursday, then decreased each day for the rest of the week. c Wednesday
4 a 42 million b 2002 to 2004 c 2008 to 2010 d Yes, the figures are increasing each year, but by a smaller amount each time. The increases between the years shown are 5 million, 4 million, 3 million and 2 million, so an estimate for the number of visitors in 2012 could be an extra 1 million added on to the 2010 figure, i.e. 50 million.
F Exercise 19.3 1
a
25
e r o c s t s e t g n i l l e p S
Interpreting and drawing scatter graphs
Time spent reading and spelling test score
20 15 10 5 0 0
5
10 15 Hours reading
20
25
Positive ve correlation. correlation. The more more hours reading reading a student does, does, the better better their spelling test test score. b Positi
2 a
Art and Science exam results
90 80 )
70
60 ( % t l u s 50 e r e c 40 n e 30 i c S 20 10 0 30
40
50 60 70 Art result (%)
80
90
b Negative correlation. The better the students’ result in art, the worse their science result.
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Unit 19 3 a
Answers to Practice Book exercises Numberr of packets of biscuits and crisps sold Numbe
30
s t e 25 k l c d a o20 p s f s o p s 15 r i e r c10 b f m o u 5 N
0 0
5 10 15 20 25 Number of packets of biscuits sold
30
correlation. The number of packets packets of biscuits sold has no relationship to the number number of packets b No correlation. of crisps sold. station, the less it is worth. 4 a Negative correlation. The further the house is from the railway station, The house that doesn doesn’t ’t fit the tren trend d is worth $146 000 and is 6 km from the train station. station. b
For example: example: It might not be in a very good state state of repair, repair, which which is why why it isn’t isn’t worth worth as much much as it should be.
F Exercise 19.4 1
a
Interpreting and drawing stem-and-l stem-and-leaf eaf diagrams June
6
6
6
5
3
August
3
9 1
2
8
0
2
7 0
3 4
6 0
7 3
8 7
5
0
2
4
6
2
0
5
6
8
8
Key: For June, 0 | 2 means 20 customers For August, 3 | 6 means 36 customers
b
i Mode
ii Median
iii Range
iv Mean
June
46
43
48
44
August
58
51
26
49
c In August the mode, median and mean are all greater than in June, showing that on average there are more customers. The range, however, is smaller in August than in June, showing that there is more variation in the numbers of customers riding in June. d Yes, because the mode, median and mean are all greater in August than in June.
2 a
i Mode
ii Median
iii Range
iv Mean
Girls’ times
27.3
26.05
2.6
26.1
Boys’ times
26.5
27.4
3.6
27.3
boys, showing showing that their times times are more varied. The girls have have a lower lower b For example: The range is larger for the boys, median and mean which shows that using these averages they were quicker at solving the puzzle. girls’,, which makes them appear faster. c The mode, as the boys’ mode is lower than the girls’ d The median or the mean, as the girls’ median and mean are lower than the boys’, so the girls were faster. e The girls, as their median and mean are lower, therefore they were faster than the boys.
3 a
Top shelf
9
8
7
6
5
Middle shelf 9
4
10
9
2
2
11
5
4
2
0
12 12
0
7
9
0
13 1 3 14
0 2
2 4
6 5
8 7
Key: For the top shelf, 4 | 10 means 104 boxes of cereal For the middle shelf, 11 | 5 means 115 boxes b oxes of cereal
9
9
3
Cambridge Checkpoint Mathematics 9
4
Copyright Cambridge University Press 2013
RM.DL.BOOKS GROUPS
Answers to Practice Book exercises b
Mode
Median
Range
Mean
Top shelf
112
123
26
120.5
Middle shelf
139
137
32
134.5
Unit 19
For example: example: The sales of cereal were were better on the middle shelf as on average more more boxes boxes were were sold (the (the mean, median and mode were all greater on the top shelf than the middle shelf). shelf ). The sales on the middle shelf were more varied, but included the largest number of boxes sold on one day. day. The smallest number of box boxes es sold on one day were on the top shelf.
F Exercise 19.5
Comparing distribut distributions ions and drawing conclusions
1 For example: On Saturday 20 more cars were parked for less than 2 hours than on Wednesday. On Saturday the most popular length of time in the car park was between 2 and 4 hours, whereas on Wednesday Wednesday it was between 6 and 8 hours. On Wednesday there were 38 cars parked for between 4 and 6 hours, compared with 16 on Saturday.
2 For example: The most popular mass of suitcase going to Spain was between 18 and 20 kg compared with 22 to 24 kg going to Sweden. There were 10 cases over 24 kg going to Sweden compared with 4 going to Spain. There were 16 cases less than 18 kg going to Spain compared with 6 going to Sweden.
3 a Yes, as the graph has a positive correlation. b No, she should get a mark between about 52% and 60%. 4 Team A Team B
Mode
Median
Range
Mean
18 28
19 27.5
16 7
21.25 27.25
For example: Steph Steph is correct in saying that on average team A are younger as their mode, median and mean are all less than team B. However, team A have a larger range which means that team B are more similar in age, so this part of her statement is incorrect.
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