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
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