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Solutions — Full Answer Key

Ecolint Campus des NationsMathematics
Year 8 · 8.8 Equations

Solutions · Full Answer Key

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

Pack A — Answers

Bronze
1.x=12x = 12
2.x=13x = 13
3.x=13x = 13
4.x=28x = 28
5.x=5x = 5
6.x=3x = 3
7.x=6x = 6
8.Yes
9.x=8x = 8
10.x=−5x = -5
Silver
11.x=5x = 5
12.x=4x = 4
13.x=5x = 5
14.x=18x = 18
15.x=9x = 9
16.x=16x = 16
17.n=4n = 4
18.x=−5x = -5
19.x=6x = 6
20.x=6x = 6
Gold
21.x=11x = 11
22.x=22x = 22
23.x=30x = 30
24.220
25.x=20x = 20
26.31, 32, 33
27.x=20x = 20
28.x=5x = 5
29.x=−10x = -10
30.x=0x = 0
Platinum
31.x=−23x = -23
32.x=−8x = -8
33.x=1000x = 1000 chf
34.Pen 1.20 chf; pencil 1.00 chf
35.Son 8, man 32
36.x=2x = 2
37.11 and 12
38.x=12x = 12
39.x=76x = \tfrac{7}{6}
40.x=3x = 3

Pack B — Answers

Bronze
1.x=8x = 8
2.x=15x = 15
3.x=9x = 9
4.x=30x = 30
5.x=5x = 5
6.x=4x = 4
7.x=17x = 17
8.Yes
9.x=−6x = -6
10.x=−4x = -4
Silver
11.x=5x = 5
12.x=12x = 12
13.x=5x = 5
14.x=25x = 25
15.x=5x = 5
16.x=7x = 7
17.n=3n = 3
18.x=−3x = -3
19.x=5x = 5
20.x=7x = 7
Gold
21.x=7x = 7
22.x=14x = 14
23.x=24x = 24
24.240
25.x=500x = 500
26.39, 40, 41
27.x=10x = 10
28.x=7x = 7
29.x=−5x = -5
30.Same — 0
Platinum
31.x=−11x = -11
32.x=11x = 11
33.x=1000x = 1000 chf
34.Pen 0.90 chf; pencil 0.90 chf
35.Son 6, man 30
36.x=8x = 8
37.12 and 13
38.x=18x = 18
39.x=2x = 2
40.x=4x = 4

Problem-solving — Worked Solutions

1Problem 1
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.
2Problem 2
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 ✓.
3Problem 3
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.
4Problem 4
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.
5Problem 5
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.
6Problem 6
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.
7Problem 7
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.
8Problem 8
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.
9Problem 9
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.
10Problem 10
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).
11Problem 11
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.
12Problem 12
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.