1 a 2 b 10 c 4 d 12. e 7 f 11 g 12 h 30. i 8 j 7 k 5 l a 9 b 14 c 3 d 10. e 11 f 5 g 15 h 1. i 12 j 2 k 35 l 49.
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1 Cambridge Essentials Mathematics Core 7 A1.1 Answers A1.1 Answers 1 a 2 b 10 c 4 d 12 e 7 f 11 g 12 h 30 i 8 j 7 k 5 l 20 2 a 9 b 14 c 3 d 10 e 11 f 5 g 15 h 1 i 12 j 2 k 35 l 49 m 25 n 1 o 1 p 18 3 a 12 b 12 c 16 d 16 e 24 f 24 g 18 h 18 4 a 20 b 20 c 8 d 8 e 12 f 12 g 30 h ; 6 a 24 b 40 c 17 d 10 e 11 f 24 g 100 h 40 7 a 28 b 48 c 21 d 12 e 7 f 28 g 120 h 48 Part e is smaller. 8 Any three pairs of values of and so that + = 5. Original Material Cambridge University Press
2 Cambridge Essentials Mathematics Core 7 A1.1 Answers 9 10 a a = 6 b b = 14 c c = 7 d d = 9 e e = 5 f f = 8 g g = 33 h h = 80 i i = 20 j j = 20 k k = 27 l l = a 15 b 2 c 48 d 14 e 18 f 3 g 4 h Original Material Cambridge University Press
3 Cambridge Essentials Mathematics Core 7 A1.1 Answers 13 a 20, 22; increasing b 15, 12; decreasing c 412, 312; decreasing d 88, 104; increasing e 398, 418; increasing f 35, 24; decreasing g 20,24; increasing h 64, 128; increasing i , ; increasing j 16, 8; decreasing k 3, 1; decreasing l 8, 13; increasing 14 The missing numbers are: a 16, 26 b 10, 19 c 5, 11, 20 d 13, 23, 33 e 22, 4 f 48, 40, 32 g 3.5, 4.5, 5 h 9.5, 8.5, 7 i 17.5, 20.5, cm, 58 cm, 60.5 cm, 63 cm, 65.5 cm, 68 cm 17 No. This is a rapid rate of growth that after 100 months would make him over 3 m tall. 18 a b 2061 Original Material Cambridge University Press
4 Cambridge Essentials Mathematics Core 7 A1.1 Answers 19 The 4th term is 3 4 = 12. The 10th term is 3 10 = 30. The 50th term is 3 50 = 150. The nth term is 3 n = 3n. 20 a 6, 7, 8, 9 b 11, 12, 13, 14 c 101, 102, 103, 104 d 0, 1, 2, 3 e 2, 4, 6, 8 f 5, 10, 15, 20 g 10, 20, 30, 40 h 11, 22, 33, 44 i 1.5, 2.5, 3.5, 4.5 j 3.5, 4.5, 5.5, 6.5 k 0.5, 1.5, 2.5, 3.5 l 10.5, 11.5, 12.5, Original Material Cambridge University Press
5 Cambridge Essentials Mathematics Core 7 A1.2 Answers A1.2 Answers 1 a i 15 ii 55 iii 35 iv b i 4 ii 6 iii 7 iv a x 4 x b x x 6 c x 11x d x x Original Material Cambridge University Press
6 Cambridge Essentials Mathematics Core 7 A1.2 Answers a 18 6 b c d 0 7 x x 3 2 x 9 4x x 3x a 2 b Any one of + 0, 0, 1, 1 c 5 14 a 6 b 3 c 2 d a 5 b 9 c d 99 e 14 f 4 g 15 h 4 i 20 j 3 k 5 then + 7 x 8x a b 17 a b 18 a b 19 a 2, 5, 8, 11, 14, 17, 20, 23, 26, 29 b Yes. 239 is the 80th term because = 239. c 16 Original Material Cambridge University Press
7 Cambridge Essentials Mathematics Core 7 A2.1 Answers A2.1 Answers 1 a 14 b 3 c 40 d 16 e 36 f 96 g 39 h 10 i 6.5 j 9 k 12 l 4 2 a 60 b 60 c 3 d 27 e 14 f 2 g 16 h 16 i 13 j 13 k 17 l 1 3 a 8 b 10 c 15 d 13 e 5 f 17 g 13 h 3 i 1 j 10 k 0 l 0 4 a 27 b 34 c 16 d 27 e 4 f 6 g 2 h 4 i 20 5 a 4 b 5 c 3 d 9 e 8 f 10 g 5 h 9 i 2 6 a 11 b 3 c 18 d 18 e 19 f 9 g 33 h 34 i 2 j 2 k 5 l 2 7 a x + 5 b x + 7 c x + 12 d 2x e 3x f x g 3x h x i 5x j x k 9x l 3x 8 a = = 67 b = = 139 c = = 257 d = = 586 e = = 265 Original Material Cambridge University Press
8 Cambridge Essentials Mathematics Core 7 A2.1 Answers 9 The additions and subtractions can be done in any order. This allows us to choose the order that makes the calculation as simple as possible. 10 a 288 b 275 c 167 d 39 e 30 f a 2x + 3 b 2x + 10 c 3x + 2 d x e 4x + 4 f 5x 12 a x + 2y b 2x c 2x + 2y d x + y + 8 e 2x + 3 f x + 5y 3 g 2xy h 5xy i 2xyz j xy 4y k 1 xy +3xyz l p 2pqr 13 a x + 7 b x + y c n 5 d p q 14 a k 2 b t a 4x b 2x + 10 c 4y a 3y + 6 b 6g + 3 c 3n + 7 d 4b + 4 e 6t + 10 f 7m 1 g 6u + 8v h 12h + 1 i 6a 17 a B and F b E and F c A and E d B and D e D, E and F f B, D and F g 4x a B =12 b B = 7 c B = a s = 4 b s = 7 c s = 6 d d = a F = 30 b F = 48 c F = 6.5 d m = 6 21 The formula may be written as N + R A = 2 or any equivalent form. Extra entries in the table should also satisfy this formula. Original Material Cambridge University Press
9 Cambridge Essentials Mathematics Core 7 A2.1 Answers a d = 1.22 b d = 1.25 c d = 0.83 Original Material Cambridge University Press
10 Cambridge Essentials Mathematics Core 7 A2.2 Answers A2.2 Answers 1 a 2 b 4 c 10 d 1 e 0 f 0 2 a x + 2 b x/3 c 5x d x 9 e y 4 f 8g g t + 11 h n/5 3 a 6x + 5 b 4x 3 c x/3 7 d x/8 + 1 e 5(x + 4) f (x 5)/10 4 a b c d e f g h 5 a x = 10 b x = 4 c x = 39 d x = 2 6 a Subtract 6 b Divide by 5 c Add 1 d Multiply by 2 e 5 f 6 g 3 h + 9 i 10 j 3.4 k l a b x = 44 Original Material Cambridge University Press
11 Cambridge Essentials Mathematics Core 7 A2.2 Answers 8 a i ii iii x = 26 b i ii iii k = 23 c i ii iii t = 12 d i ii iii x = 2 e i ii iii n = 12 f i ii iii y = 16 g i ii iii q = 3 h i ii iii t = 5.5 i i ii iii m = 9.5 j i ii iii a = 42 k i ii iii r = 150 l i ii iii k = 82 Original Material Cambridge University Press
12 Cambridge Essentials Mathematics Core 7 A3.1 Answers A3.1 Answers 1 a 4x b x 6 c x + 14 d x 3 e 5 x 2 a x 11 5 b 2(x + 7) c 3 x x d 3(10 x) e a y 10 b 3y 5 y c 3 4 d 17 5y e 20 y 9 f 3y a Subtract 5 and multiply by 2. b Multiply by 5 and subtract from 21. c Divide by 3 and add 4. d Add 8 and divide the answer by 4. e Add 1 then multiply by 4 and subtract the answer from 35. f Subtract 3 and divide the answer into a 21 b 21 c 6 d 7 e 28 f 15 g 30 h 3 i 7 6 a 1 2 b c d 1 1 e f a 2x b x 4 c x d x e x f x + 3 Original Material Cambridge University Press
13 Cambridge Essentials Mathematics Core 7 A3.1 Answers 8 a x b x c x d 4x 4 4x e f x a b c d e f a 170 b 16 c 48 d 404 e 220 f 5 12 a 133 b 14 c 15 d 248 e 36 f 100 Original Material Cambridge University Press
14 Cambridge Essentials Mathematics Core 7 A3.1 Answers 13 a 3x 17 = 16 3x = 33 x = 11 b 2( x + 12) = 68 x + 12 = 34 x = 22 c x = x = 14 5 d 5x = x = 45 x = 9 e x = 6 66 = x 19 x = 85 f x = = 5( x 32) 15 = x 32 x = 47 g 2 = x h 9 = 3(4 x) i x = x = x + 10 x = 2 3 = 4 x x = 1 2x = x + 4 x = 4 14 a 80 b 4 c 6 d 280 e 19 f a 1 2 b 2 3 c 3 5 d 3 4 e 4 5 f a 6x + 34 = 76 b x = 7 c 21 cm 17 a 3x + 60 o = 180 o b x = 40 o c 40 o, 60 o and 80 o 18 a 4x + 40 o = 180 o b x = 35 o c 35 o, 75 o and 70 o 19 Always true. Two examples of x + y = y + x 20 a Sometimes true. Pupil s example where x = y for the true case and another showing x y y x. b Always true. Two pupil examples showing x y = (y x). c Always true. Two pupil examples showing xy = yx. d Sometimes true. Pupil example where x = y for the true case and another showing x/y y/x. Original Material Cambridge University Press
15 Cambridge Essentials Mathematics Core 7 A3.1 Answers 20 e Never true. Two pupil examples showing x + y + 1 (1 y x). f Always true. Two pupil examples showing (1 x) (1 + y) = (x + y). 21 a 3xy b 6xz c 4xy d 2pq e 7pr 2 f 2pq + pr g 5ab h 6w i 8pq 22 a i 60 ii 60 iii 48 iv 48 v 105 vi 105 vii 88 viii 88 b (xy)z = x(yz) 23 All of them! 24 a 300 b 900 c 35 d 38 e 66 f a 5pqr b 7pqr c 9pqr 26 b 15q c 28t d 24r e 36n f 30k 27 b 26w c 11v d x e 19y f 8a 28 a 12x b 38m c 39h d 29p e 30n f 3pq 29 5(x + 3) = 5x (y 4) = 3y = 24(20 1) = = = 456 Original Material Cambridge University Press
16 Cambridge Essentials Mathematics Core 7 A3.1 Answers 32 a = 32 (30 1) = = 928 b = 14 (50 1) = = 686 c = 18 (10 0.1) = = a 4x + 20 b 6n 24 c t d h e p f 99 9b g 3x + xy h 5r rt i kn + 2k 34 a 6x + 3 b 12n 8 c 20 10k d j e 18 10e f 21d + 28 g 2ab + 5a h 9g 3gt i 10z + 6yz 35 4(x + y + 5) = 4x + 4y a 5p + 5q + 15 b 6a + 3b 18 c 48 4m + 8n d 20h k e 30 18c 24d + 12e f 6np + 9nq 9n 37 a 5x + 15 b 7n 12 c 14 10t d 8a + 10 e 9w + 7q f 7h a 19t + 8 b 5m + 12n + 18 c k + 6h d x + 3y 39 a 4 b 8 c 4 d 3 e 9 f 7 40 a p 6 b 2(p + p 6) c 6p 18 d 6p 18 = 48 e p = 11. Rachel has 32 marbles. Original Material Cambridge University Press
17 Cambridge Essentials Mathematics Core 7 A3.2 Answers A3.2 Answers 1 a b y = 3x 2 a b i y = x + 5 ii x x a b i 13 ii 18 iii 35 c i y = x 3 ii x x 3 Original Material Cambridge University Press
18 Cambridge Essentials Mathematics Core 7 A3.2 Answers 4 a i y = 2x + 3 ii x 2x + 3 b i 3 ii 5 iii 11 iv 23 c i 7 ii 30 iii 8.5 iv 23.5 d 5 a In the rule y = x + 3 2, only the 3 is doubled but it should be the value of x + 3 that is doubled. b y = 2(x + 3) c 26 d 22 6 a i y = x + 3 ii y = 10 x iii y = 2x 5 iv y = 20 2x v y = x 1 vi y = 2(x 5) b i ii Original Material Cambridge University Press
19 Cambridge Essentials Mathematics Core 7 A3.2 Answers 6 b iii iv v vi 7 a x x, x 2(x 2) or x 8 x b i x 8 x ii x x iii x 2(x 2) Original Material Cambridge University Press
20 Cambridge Essentials Mathematics Core 7 A4.1 Answers A4.1 Answers 1 a A(2, 4), B(2, 3), C(2, 1.5), D(2, 1.5) b The x-coordinate is always 2. c x = 2 2 a x = 2 b x = 1 c x = a A( 2, 4), B(1, 4), C(2.5, 4), D(4.5, 4) b y = 4 c i y = 2 ii y = a i y = 5 ii x = 3 b (3, 5) c i (7, 3) ii ( 2, 4) iii (3.5, 1) iv ( 1.5, 9) 5 a, c, d b e The line y = x + 1 is parallel to the line y = x but it crosses the y-axis at 1 not 0. Original Material Cambridge University Press
21 Cambridge Essentials Mathematics Core 7 A4.1 Answers 6 a i ( 4, 2) ii (0, 2) iii (2, 4) b, c, d, e 7 a A(4, 4) b B( 7, 5), C(5, 5), D( 7, 7) 8 a (3, 1) b (3, 5) c (7, 5) 9 a (3, 3) b ( 2, 2) c (5, 5) d ( 4, 4) e (3, 4) f ( 2, 6) 10 a x y Original Material Cambridge University Press
22 Cambridge Essentials Mathematics Core 7 A4.1 Answers 10 b y = 2x y = x d (0, 0) 11 a When x = 0, y = mx = m 0 = 0 for any value of m. So, (0, 0) must lie on y = mx i.e., y = mx must pass through the origin. b i 6 ii 8 iii 10 iv 1 c d As m is increased, the graph becomes steeper. Original Material Cambridge University Press
23 Cambridge Essentials Mathematics Core 7 A4.1 Answers 12 a x y b c i ( 1.5, 0) ii (0, 3) 13 a x y b, c d The line crosses the x-axis at (5, 0) and the y-axis at (0, 5). Original Material Cambridge University Press
24 Cambridge Essentials Mathematics Core 7 A4.1 Answers 14 a i ( 2, 6) ii (1, 3) iii (5, 1) b, c d The line crosses the x-axis at (4, 0) and the y-axis at (0, 4). 15 a A y = 10 x B y = 8 x C y = 2 x b c (1) (5, 5) (2) (4, 4) (3) (1, 1) 16 a y = 4 b x = 2 c y = x d y = x + 1 e y = 2 x Original Material Cambridge University Press
25 Cambridge Essentials Mathematics Core 7 A4.2 Answers A4.2 Answers 1 a 60% b i 80% ii 40% iii 50% c 35 d 7 e 31.5 (accept 31 or 32 from the graph). 2 a i 8 in ii 36 in iii 22 in iv 13 in b i 30 cm ii 50 cm iii 75 cm iv 40 cm c 46 cm, 19 in, 49.2 cm, 21 in 3 a 68 o F b 30 o C c 32 o F d 23 o F e 37 o C 4 a b i 15 kg = 33 lb ii 45 kg = 99 lb iii 37 lb = 17 kg iv 101 lb = 46 kg 5 a x = 4 b i x = 2 ii x = 1.5 iii x = 0 iv x = 1 6 a i x = 2 ii x = 8 iii x = 10 iv x = 5 v x = 0 vi x = 12 b x + 5 = 4 2 c x + 21 = x + 5 = x = 3 (from the graph) Original Material Cambridge University Press
26 Cambridge Essentials Mathematics Core 7 A4.2 Answers 7 a b c i x = 2 ii x = 2.5 iii x = 1.5 iv x = 3 8 a b c i x = 3 ii x = 0 iii x = 3 iv x = 1 v x = 4 vi x = 4 d x = 6 2 x + 3 = 1 2 x = 5 9 a b c t = 6 d t = 2 Original Material Cambridge University Press
27 Cambridge Essentials Mathematics Core 7 A5.1 Answers A5.1 Answers 1 a m + 3 b m 5 c 2m d 1 m 2 or 2 m e 7 + m or m + 7 f 11 m 2 a mn b m + n c n + m or m + n d m n e n m f mn 3 a n + 2 b n + 1 c n 1 d n 2 4 a p + 2 b p 1 5 a n 1, n + 2, n + 5 b n + 2 c 3n + 6 d 3(n + 2) e n + 2 f 6 6 a n + 1 b n c 3n a i 2t + 3 ii 2t + 2 iii 8t + 12 iv 2t + 3 b The median and the mean have the same value. 8 a 2 units b (a + 2, 3) c 4 units 9 a a 1 units b (2a 1, 5) c 2(a 1) units 10 a (4, 2) b (x + 3, 2) c (x + 3, y 5) 11 right 4, down b 12 a y = 6x 5 b y = x c y = 7(x 4) d y = x T = n 3 14 y = x Original Material Cambridge University Press
28 Cambridge Essentials Mathematics Core 7 A5.1 Answers 15 a b c 16 a 30 b m = 3n 17 a Pattern number (n) Number of matchsticks (m) b 21 c m = 2n + 1 d Pattern a 16, 19 b 31 c nth term = 3n a 75, 70 b 50 c nth term = 100 5n Original Material Cambridge University Press
29 Cambridge Essentials Mathematics Core 7 A5.2 Answers A5.2 Answers 1 a n = 5 b t = 55 c p = 6 d z = 5 e h = 8 f m = 6 2 a x = 11 b x = 7 c x = 6 d x = 7 e x = 9 f x = 25 3 a p = 1/2 b t = 2/3 c n = 3/5 d a = 2/3 e y = 12/17 f g = 5/8 4 a r = d x = 1 3 b h = e s = c d = f k = a g = 3.5 b w = 2.25 c x = 0.6 d b = 8.9 e m = 1.5 f y = a 5x + 3 = 33; x = 6 b 4x + 6 = 34; x = 7 7 a x + 3 b 3x c 8x + 3 d 8x + 3 = 99 e x = 12 f 15 years 8 a, b c 4x + 4 = 100 d x = 24 e Original Material Cambridge University Press
30 Cambridge Essentials Mathematics Core 7 A5.2 Answers 9 400, 401, 402, 403, a i ACB = 2x ii BAC = 3x iii 6x = 180 b x = 30 c ABC = 30 o, ACB = 60 o, BAC = 90 o 11 a x + (x + 80) + (x 23) = 360 b x = 101 3x + 57 = 360 c POQ = 101 o, POR = 181 o, QOR = 78 o 12 a x = 5 b x = 15 c x = 9 d x = 6 e x = 3 f x = 4 13 a h = 5 b r = 7 c d = 10 d g = 6 e p = 11 f v = 5 14 a x = 17 b u = 55 c t = 19 d y = 40 e a = 37 f e = a 5 ( x + 3) 2 5( x + 3) b = 30 2 c x = 9 d 13 cm 16 a x = 5.4 b x 20 6 = 11.5 x = 10 x = a The 15th term b 3 c 4(n + 9) = 45 gives n = 2.25 and the solution needs to be a whole number to 18 (5, 4) represent a term. Original Material Cambridge University Press
31 Cambridge Essentials Mathematics Core 7 A5.3 Answers A5.3 Answers 1 a i 275 F ii 356 F iii 410 F b i 160 C ii 200 C iii 225 C 2 a, b, d c i 160 km ii 1280 km e i 880 km ii 1152 km iii 1536 km f i 600 miles ii 650 miles iii 850 miles Original Material Cambridge University Press
32 Cambridge Essentials Mathematics Core 7 A5.3 Answers 3 a, b c i 800 mm ii 1025 mm iii 1325 mm d i 37 inches ii 44 inches iii 49 inches 4 a 2:30 p.m. b 30 minutes c 9 km d 1 hour and 30 minutes e 5:30 p.m. f 4 hours 5 a Callum b 25 m c 5 m d Alex e Alex swam faster over the last 10 m. 6 a 5 m b 1 second c 2.5 m d The third bounce 7 a The speed is constant at 5 m/s. b The highest point at the top of the first slope. c BC shows a rapid increase in speed from 5 m/s to 20 m/s over 5 seconds. d CD shows a reduction in speed from 20 m/s to 12.5 m/s over 5 seconds. e 27.5 m/s 8 a The rocket is launched from rest and reaches a speed of 70 m/s in 3 seconds. b 70 m/s c The rocket slows down to a stop in 6 seconds. d B. The greatest height is reached when the rocket has temporarily stopped at the top of its flight path. Original Material Cambridge University Press
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