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Solutions — Full Answer Key

Ecolint Campus des NationsMathematics
Year 8 · Algebra

Solutions · Full Answer Key

Pack A answers · Pack B answers · Problem-solving worked solutions

Pack A — Answers

Bronze
1.19
2.28x28x
3.4x+124x + 12
4.3x2−2x3x^2 - 2x
5.5(2x+3)5(2x + 3)
6.3x(2x+3)3x(2x + 3)
7.3x3x
8.3x+123x + 12
9.−3-3
10.7p+7q7p + 7q
11.x=12x = 12
12.x=13x = 13
13.x=13x = 13
14.x=28x = 28
15.x=5x = 5
16.x=3x = 3
17.x=6x = 6
18.Yes
19.x=8x = 8
20.x=−5x = -5
Silver
21.41
22.9x+29x + 2
23.4x(5x−y)4x(5x - y)
24.3x3x
25.2x+32x + 3
26.−5x+2-5x + 2
27.2b3\dfrac{2b}{3}
28.c=8,d=−2c = 8, d = -2
29.x(x+6)x(x + 6)
30.(a) x−120x - 120 (b) 3(x−120)3(x - 120)
31.x=5x = 5
32.x=4x = 4
33.x=5x = 5
34.x=18x = 18
35.x=9x = 9
36.x=16x = 16
37.n=4n = 4
38.x=−5x = -5
39.x=6x = 6
40.x=6x = 6
Gold
41.−8x+6-8x + 6
42.8x(3x+2)8x(3x + 2)
43.2x+32x + 3
44.Both equal x2+6xx^2 + 6x ✓.
45.ab(a+b)ab(a + b)
46.40
47.(x+3)(4+y)(x + 3)(4 + y)
48.8x\dfrac{8}{x}
49.7x12\dfrac{7x}{12}
50.P=6x+4P = 6x + 4; P(4)=28P(4) = 28
51.x=11x = 11
52.x=22x = 22
53.x=30x = 30
54.220
55.x=20x = 20
56.31, 32, 33
57.x=20x = 20
58.x=5x = 5
59.x=−10x = -10
60.x=0x = 0
Platinum
61.6x2−2x+106x^2 - 2x + 10
62.3(x2+4x+3)=3(x+1)(x+3)3(x^2 + 4x + 3) = 3(x + 1)(x + 3)
63.2x−32(x+?)\dfrac{2x - 3}{2(x + ?)} — see working
64.(a) y=32y = \tfrac{3}{2} (b) z=2z = 2
65.x=8x = 8
66.x=20x = 20
67.(a−b)(a+b)(a - b)(a + b)
68.A=x2+7x+10A = x^2 + 7x + 10; A(8)=130A(8) = 130 m²
69.x=4x = 4
70.x+116\dfrac{x + 11}{6}
71.x=−23x = -23
72.x=−8x = -8
73.x=1000x = 1000 chf
74.Pen 1.20 chf; pencil 1.00 chf
75.Son 8, man 32
76.x=2x = 2
77.11 and 12
78.x=12x = 12
79.x=76x = \tfrac{7}{6}
80.x=3x = 3

Pack B — Answers

Bronze
1.−2-2
2.30x30x
3.6x+306x + 30
4.5x2−4x5x^2 - 4x
5.6(2x+3)6(2x + 3)
6.5x(2x+5)5x(2x + 5)
7.3x3x
8.5x+355x + 35
9.−10-10
10.13p+2q13p + 2q
11.x=8x = 8
12.x=15x = 15
13.x=9x = 9
14.x=30x = 30
15.x=5x = 5
16.x=4x = 4
17.x=17x = 17
18.Yes
19.x=−6x = -6
20.x=−4x = -4
Silver
21.−1-1
22.5x+15x + 1
23.3x(5x−3y)3x(5x - 3y)
24.3x3x
25.2x+32x + 3
26.−8x+7-8x + 7
27.3x2\dfrac{3x}{2}
28.c=6,d=2c = 6, d = 2
29.x(x+9)x(x + 9)
30.(a) x−75x - 75 (b) 3(x−75)3(x - 75)
31.x=5x = 5
32.x=12x = 12
33.x=5x = 5
34.x=25x = 25
35.x=5x = 5
36.x=7x = 7
37.n=3n = 3
38.x=−3x = -3
39.x=5x = 5
40.x=7x = 7
Gold
41.−11x+7-11x + 7
42.9x(3x+2)9x(3x + 2)
43.3x+23x + 2
44.Both equal x2+10xx^2 + 10x ✓.
45.pq(p−q)pq(p - q)
46.37
47.(x−2)(5−y)(x - 2)(5 - y)
48.11x\dfrac{11}{x}
49.7x10\dfrac{7x}{10}
50.P=8x+4P = 8x + 4; P(4)=36P(4) = 36
51.x=7x = 7
52.x=14x = 14
53.x=24x = 24
54.240
55.x=500x = 500
56.39, 40, 41
57.x=10x = 10
58.x=7x = 7
59.x=−5x = -5
60.Same — 0
Platinum
61.4x2−7x+214x^2 - 7x + 21
62.2(x2+5x+4)=2(x+1)(x+4)2(x^2 + 5x + 4) = 2(x + 1)(x + 4)
63.See working
64.(a) y=32y = \tfrac{3}{2} (b) z=2z = 2
65.x=4x = 4
66.x=500x = 500
67.(5−x)(5+x)(5 - x)(5 + x)
68.A=x2+10x+21A = x^2 + 10x + 21; A(8)=165A(8) = 165 m²
69.Same.
70.x+2312\dfrac{x + 23}{12}
71.x=−11x = -11
72.x=11x = 11
73.x=1000x = 1000 chf
74.Pen 0.90 chf; pencil 0.90 chf
75.Son 6, man 30
76.x=8x = 8
77.12 and 13
78.x=18x = 18
79.x=2x = 2
80.x=4x = 4

Problem-solving — Worked Solutions

1Problem 1
Answer
(a) Equivalent (b) NOT equivalent (c) Equivalent (d) Equivalent
Full working
(a) 3(x+4)=3x+123(x + 4) = 3x + 12 ✓.

(b) (x+2)2=x2+4x+4(x + 2)^2 = x^2 + 4x + 4, not x2+4x^2 + 4.

(c) 2x−6−x+5=x−12x - 6 - x + 5 = x - 1 ✓.

(d) 6x2=3x\tfrac{6x}{2} = 3x ✓.
2Problem 2
Answer
(a) 0 (b) 5 (c) −1−5=15\tfrac{-1}{-5} = \tfrac{1}{5} (d) 64
Full working
(a) −6+6=0-6 + 6 = 0.

(b) 9−4=59 - 4 = 5.

(c) Numerator −1-1, denominator −5-5. Result 15\tfrac{1}{5}.

(d) (2×−3−2)2=(−8)2=64(2 \times -3 - 2)^2 = (-8)^2 = 64.
3Problem 3
Answer
(a) 6(x+2)6(x + 2) (b) 4x(x−2)4x(x - 2) (c) 6ab(2a−3b)6ab(2a - 3b) (d) (x−1)(5+y)(x - 1)(5 + y)
Full working
(a) HCF 6.

(b) HCF 4x4x.

(c) HCF 6ab6ab.

(d) Common factor (x−1)(x - 1).
4Problem 4
Answer
(a) 2ab\tfrac{2a}{b} (b) 2x−32x - 3 (c) 9x10\tfrac{9x}{10} (d) x−1112\tfrac{x - 11}{12}
Full working
(a) Cancel 4ab4ab from numerator and denominator.

(b) Factorise numerator: 3x(2x−3)3x(2x - 3). Divide by 3x3x: 2x−32x - 3.

(c) Common denominator 10: 5x+4x10\tfrac{5x + 4x}{10}.

(d) Common denominator 12: 3(x−3)−2(x+1)12=x−1112\tfrac{3(x - 3) - 2(x + 1)}{12} = \tfrac{x - 11}{12}.
5Problem 5
Answer
(a)(i) x−120x - 120 (ii) 3(x−120)3(x - 120) (b) Elena = 220 songs
Full working
(a) Sabrina = x−120x - 120. Pete = 3×3 \times Sabrina = 3(x−120)3(x - 120).

(b) 3(x−120)=300⇒x−120=100⇒x=2203(x - 120) = 300 \Rightarrow x - 120 = 100 \Rightarrow x = 220.
6Problem 6
Answer
(a) P=6x+8P = 6x + 8; A=(2x+1)(x+3)A = (2x + 1)(x + 3) (b) A=2x2+7x+3A = 2x^2 + 7x + 3 (c) P=32P = 32; A=63A = 63
Full working
(a) P=2(2x+1)+2(x+3)=6x+8P = 2(2x + 1) + 2(x + 3) = 6x + 8. A=(2x+1)(x+3)A = (2x + 1)(x + 3).

(b) Expand: A=2x2+6x+x+3=2x2+7x+3A = 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3.

(c) At x=4x = 4: P=32P = 32; A=32+28+3=63A = 32 + 28 + 3 = 63.
7Problem 7
Answer
(a) x2+8x+15x^2 + 8x + 15 (b) x2+3x−28x^2 + 3x - 28 (c) 2x2+5x−32x^2 + 5x - 3 (d) x2+4x+4x^2 + 4x + 4
Full working
Use the area model / FOIL.

(a) x2+3x+5x+15=x2+8x+15x^2 + 3x + 5x + 15 = x^2 + 8x + 15.

(b) x2+7x−4x−28=x2+3x−28x^2 + 7x - 4x - 28 = x^2 + 3x - 28.

(c) 2x2+6x−x−3=2x2+5x−32x^2 + 6x - x - 3 = 2x^2 + 5x - 3.

(d) (x+2)2=x2+4x+4(x + 2)^2 = x^2 + 4x + 4.
8Problem 8
Answer
E=P3+dE = \tfrac{P}{3} + d
Full working
Sabrina = P/3P/3. Sabrina = Elena −d- d, so Elena =P/3+d= P/3 + d.
9Problem 9
Answer
(a) 6x=1806x = 180, x=30x = 30 (b) 60°,100°,20°60°, 100°, 20°
Full working
(a) Sum 180°: 2x+3x+10+x−10=6x=1802x + 3x + 10 + x - 10 = 6x = 180, x=30x = 30.

(b) Angles: 60°,100°,20°60°, 100°, 20°. Check 60+100+20=180°60 + 100 + 20 = 180° ✓.
10Problem 10
Answer
(d) 522=502+2⋅50⋅2+22=2500+200+4=270452^2 = 50^2 + 2 \cdot 50 \cdot 2 + 2^2 = 2500 + 200 + 4 = 2704
Full working
(a) (a+b)(a+b)=a2+ab+ab+b2=a2+2ab+b2(a + b)(a + b) = a^2 + ab + ab + b^2 = a^2 + 2ab + b^2 ✓.

(b) (a−b)(a−b)=a2−ab−ab+b2=a2−2ab+b2(a - b)(a - b) = a^2 - ab - ab + b^2 = a^2 - 2ab + b^2.

(c) (a+b)(a−b)=a2−ab+ab−b2=a2−b2(a + b)(a - b) = a^2 - ab + ab - b^2 = a^2 - b^2.

(d) Use 52=50+252 = 50 + 2: 522=2500+200+4=270452^2 = 2500 + 200 + 4 = 2704. ✓
11Problem 11
Answer
(a) Test x=6x = 6: (6+6)/2=6(6+6)/2 = 6 vs (12+6)/3=6(12+6)/3 = 6. They are equal at x=6x = 6 (b) x=6x = 6
Full working
Set them equal: x+62=2x+63\tfrac{x + 6}{2} = \tfrac{2x + 6}{3}. Cross: 3(x+6)=2(2x+6)3(x + 6) = 2(2x + 6). 3x+18=4x+123x + 18 = 4x + 12. x=6x = 6.

For x>6x > 6, the second is larger; for x<6x < 6, the first is larger.
12Problem 12
Answer
(a) See working (b) 411\tfrac{4}{11} (c) 17\tfrac{1}{7}
Full working
(a) x=0.ab‾x = 0.\overline{ab}. Then 100x=ab.ab‾100x = ab.\overline{ab}. Subtract xx: 99x=ab99x = ab, so x=ab99x = \tfrac{ab}{99}.

(b) 0.36‾=3699=4110.\overline{36} = \tfrac{36}{99} = \tfrac{4}{11} (divide by 9).

(c) 0.142857‾=142857999999=170.\overline{142857} = \tfrac{142857}{999999} = \tfrac{1}{7} (since 7×142857=9999997 \times 142857 = 999999).
13Problem 13
Answer
(a) x=6x = 6 (b) x=10x = 10 (c) x=7x = 7 (d) x=25x = 25
Full working
(a) 4x=24⇒x=64x = 24 \Rightarrow x = 6.

(b) x−4=6⇒x=10x - 4 = 6 \Rightarrow x = 10.

(c) 3x=21⇒x=73x = 21 \Rightarrow x = 7.

(d) x+3=28⇒x=25x + 3 = 28 \Rightarrow x = 25.
14Problem 14
Answer
(a) 8n+6=38,n=48n + 6 = 38, n = 4 (b) Long 11 cm, short 8 cm (c) 2(11)+2(8)=382(11) + 2(8) = 38 ✓
Full working
(a) P=2(3n−1)+2(n+4)=8n+6=38⇒n=4P = 2(3n - 1) + 2(n + 4) = 8n + 6 = 38 \Rightarrow n = 4.

(b) Long: 3(4)−1=113(4) - 1 = 11 cm. Short: 4+4=84 + 4 = 8 cm.

(c) 22+16=3822 + 16 = 38 ✓.
15Problem 15
Answer
x=220x = 220
Full working
3(x−120)=300⇒x−120=100⇒x=2203(x - 120) = 300 \Rightarrow x - 120 = 100 \Rightarrow x = 220.
16Problem 16
Answer
(a) y=3/2y = 3/2 (b) z=2z = 2 (c) 13 cm
Full working
(a) 3y=y+3⇒y=323y = y + 3 \Rightarrow y = \tfrac{3}{2}, so long = 4.54.5 cm.

(b) z=3z−4⇒z=2z = 3z - 4 \Rightarrow z = 2, so short = 2 cm.

(c) P=2(4.5)+2(2)=9+4=13P = 2(4.5) + 2(2) = 9 + 4 = 13 cm.
17Problem 17
Answer
(a) 797.30 chf (b) New price 743.40; difference 53.90 chf
Full working
(a) 0.74x=590⇒x=590/0.74=797.300.74 x = 590 \Rightarrow x = 590/0.74 = 797.30 chf.

(b) 590×1.26=743.40590 \times 1.26 = 743.40. Difference =797.30−743.40=53.90= 797.30 - 743.40 = 53.90 chf.
18Problem 18
Answer
Number = 5
Full working
3x−72=4⇒3x−7=8⇒3x=15⇒x=5\tfrac{3x - 7}{2} = 4 \Rightarrow 3x - 7 = 8 \Rightarrow 3x = 15 \Rightarrow x = 5.
19Problem 19
Answer
(a) x=15x = 15 chf (b) Effective rises from 15.20 to 15.40 chf/ticket
Full working
(a) 25x+5=380⇒25x=375⇒x=1525x + 5 = 380 \Rightarrow 25x = 375 \Rightarrow x = 15 chf.

(b) Doubling fee to 10: total 375+10=385375 + 10 = 385 chf, per ticket =385/25=15.40= 385/25 = 15.40 chf. Before, per-ticket effective was 380/25=15.20380/25 = 15.20.
20Problem 20
Answer
(a) 4s+4=3(s+4)4s + 4 = 3(s + 4) (b) Son 8, man 32
Full working
(a) Currently: son ss, man 4s4s. In 4 years: son s+4s + 4, man 4s+44s + 4. Equation: 4s+4=3(s+4)4s + 4 = 3(s + 4).

(b) 4s+4=3s+12⇒s=84s + 4 = 3s + 12 \Rightarrow s = 8. Man = 32.
21Problem 21
Answer
5 L
Full working
Acid mass =1= 1 L (20% of 5). After adding ww L water, total volume =5+w= 5 + w. Acid is still 1 L → fraction =1/(5+w)=0.10= 1/(5 + w) = 0.10. So 5+w=10⇒w=55 + w = 10 \Rightarrow w = 5 L.
22Problem 22
Answer
(a) 4x+8=36,x=74x + 8 = 36, x = 7 (b) 10, 13, 13 cm (c) 10+13>1310 + 13 > 13 ✓ — valid
Full working
(a) (x+3)+(2x−1)+(x+6)=4x+8=36⇒x=7(x + 3) + (2x - 1) + (x + 6) = 4x + 8 = 36 \Rightarrow x = 7.

(b) 10, 13, 13.

(c) Pairs: 10+13=23>1310 + 13 = 23 > 13 ✓; 13+13=26>1013 + 13 = 26 > 10 ✓. Valid (isosceles).
23Problem 23
Answer
(a) Small: 3/n3/n; Large: 11/(5n−4)11/(5n - 4) (b) n=3n = 3 (c) Large
Full working
(a) Small: 3/n3/n chf/bag. Large: 11/(5n−4)11/(5n - 4) chf/bag.

(b) Equal value: 3n=115n−4⇒3(5n−4)=11n⇒15n−12=11n⇒4n=12⇒n=3\tfrac{3}{n} = \tfrac{11}{5n - 4} \Rightarrow 3(5n - 4) = 11n \Rightarrow 15n - 12 = 11n \Rightarrow 4n = 12 \Rightarrow n = 3.

(c) Try n=4n = 4: small 0.750.75 chf/bag; large 11/16≈0.6911/16 \approx 0.69 chf/bag. Large is better.
24Problem 24
Answer
x≈5.0x \approx 5.0 (exact: 5)
Full working
x=5x = 5: 25+5=3025 + 5 = 30 ✓. So x=5x = 5 exactly. (If a non-integer solution is wanted, e.g. x2+x=31x^2 + x = 31 gives x≈5.1x \approx 5.1.)

For the actual equation: f(5)=30f(5) = 30 exactly, so x=5x = 5 is the positive solution.