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Solutions — Full Answer Key

Ecolint Campus des NationsMathematics
Year 8 · 8.3 Proportions

Solutions · Full Answer Key

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

Pack A — Answers

Bronze
1.1.50 chf
2.3.00 chf
3.Yes — y/x is constant (= 4).
4.k=5k = 5
5.54
6.x=8x = 8
7.y=−6y = -6
8.y=5xy = 5x
9.The origin (0,0)(0, 0).
10.y=4xy = 4x (proportional)
Silver
11.54 m in 3 min; 90 m in 5 min
12.5 minutes
13.144 km
14.7 eggs
15.Yes — k=3k = 3.
16.Gradient = 7. The gradient of y=kxy = kx is exactly kk, the constant of proportionality.
17.(a) 4 L/min (b) V=4tV = 4t (c) 28 L
18.12.50 chf
19.b=55b = 55
20.y=2.5xy = 2.5x; y=15y = 15
Gold
21.(a) y=4.5xy = 4.5 x (b) y=45y = 45 (c) x=20x = 20
22.A: yes (k=3k = 3). B: no.
23.(a) 12.50 chf (b) 8 tickets (c) Yes — passes through origin, C/nC/n constant.
24.(a) 45 chf (b) 500 km (c) Straight line through origin, gradient 0.18
25.Alex (8 chf/h vs Brigit 7.50 chf/h).
26.(a) 1 cm : 33.3 cm (b) 72 cm
27.(a) m=2.4ℓm = 2.4 \ell (b) 180 g (c) 100 cm
28.(a) 320 cal (b) 5000 steps (c) Approximately, but actual calories depend on weight, gradient, speed, etc.
29.(a) k=1.5k = 1.5 (b) y=30y = 30 (c) x=20x = 20
30.(a) 14 km/L (b) 392 km (c) 4200 km
Platinum
31.(a) V=6tV = 6t (b) ≈33.3≈ 33.3 min (c) V=(6−2)t=4tV = (6-2)t = 4t → 5050 min
32.(a) w=10w = 10, ℓ=20\ell = 20 (b) 200 m² (c) Area becomes 800 m² — quadruples (linear scale 2 → area scale 4).
33.A is proportional (y=2xy = 2x); B is not (passes through (0, -4)). Same shade when 2x=3x−42x = 3x - 4, i.e. x=4x = 4, y=8y = 8.
34.Yes — y=k1xy = k_1 x and x=k2zx = k_2 z ⇒ y=k1k2zy = k_1 k_2 z, so y∝zy \propto z with constant k1k2k_1 k_2.
35.k=4,p=12k = 4, p = 12
36.(a) 24 km/h (b) d=24td = 24t (t in h) (c) 42 km (d) Line through origin slope 24
37.(a) 40, 60, 100 cm (b) Yes — scaling all by the same factor preserves ratios.
38.(a) k=4k = 4 (b) y=28y = 28 (c) See working
39.E.g. data (1, 3) and (2, 6) gives ratio 3, but adding (3, 8) breaks proportionality.
40.Perimeter: yes (P=4sP = 4s). Area: no (A=s2A = s^2, a power relationship).

Pack B — Answers

Bronze
1.1.50 chf
2.4.50 chf
3.No — y/x is not constant (3, 4.5, 9).
4.k=7k = 7
5.48
6.x=8x = 8
7.y=−6y = -6
8.y=4xy = 4x
9.The origin (0,0)(0, 0).
10.y=4xy = 4x
Silver
11.75 m in 3 min; 175 m in 7 min
12.7.5 minutes
13.300 km
14.5 eggs
15.Yes — k=4k = 4.
16.Gradient = 3. Same general rule.
17.(a) 2.5 L/min (b) V=2.5tV = 2.5t (c) 22.5 L
18.17.50 chf
19.b=37.5b = 37.5
20.y=0.4xy = 0.4x; y=6y = 6
Gold
21.(a) y=7xy = 7 x (b) y=84y = 84 (c) x=13x = 13
22.A: yes. B: no.
23.(a) 8.50 chf (b) 11 (with 6 chf left) (c) Yes.
24.(a) 55 chf (b) ≈ 409 km (c) Gradient 0.22
25.Alex (9 chf/h vs Brigit ≈ 8.33 chf/h).
26.(a) 1 cm : 40 cm (b) 50 cm
27.(a) m=2.5ℓm = 2.5 \ell (b) 187.5 g (c) 96 cm
28.See Pack A.
29.(a) k=4/3k = 4/3 (b) y=80/3≈26.67y = 80/3 ≈ 26.67 (c) x=22.5x = 22.5
30.(a) 12 km/L (b) 336 km (c) 2880 km
Platinum
31.See Pack A.
32.(a) w=15w = 15, ℓ=30\ell = 30 (b) 450 m² (c) Quadruples to 1800 m².
33.A proportional. Same at 3x=4x−53x = 4x - 5, x=5x = 5, y=15y = 15.
34.Same.
35.k=5,p=10k = 5, p = 10
36.(a) 27 km/h (b) d=27td = 27t (c) 60.75 km (d) slope 27
37.See Pack A.
38.(a) k=3k = 3 (b) y=21y = 21 (c) See working
39.Similar — use a parabola or piecewise example.
40.Same.

Problem-solving — Worked Solutions

1Problem 1
Answer
(a) 18 m/min (b) d=18td = 18t (c) 216 m (d) 30 minutes
Full working
(a) Rate =36÷2=18= 36 \div 2 = 18 m/min.

(b) d=ktd = kt with k=18k = 18, so d=18td = 18t.

(c) d=18×12=216d = 18 \times 12 = 216 m.

(d) t=540÷18=30t = 540 \div 18 = 30 minutes.
2Problem 2
Answer
(a) 138.24 km (b) 429 km (c) 172.8% (d) No — geometric (multiplicative), not proportional.
Full working
Weekly distances: 80,96,115.2,138.2480, 96, 115.2, 138.24.

(a) Week 4 =80×1.23=138.24= 80 \times 1.2^3 = 138.24 km.

(b) Total ≈429\approx 429 km.

(c) 138.24/80=1.728=172.8%138.24 / 80 = 1.728 = 172.8\%.

(d) Not proportional — week-vs-distance is a geometric (exponential) sequence, not a constant-ratio (linear) one. Plotting gives a curve, not a straight line through the origin.
3Problem 3
Answer
(a) Ratios all equal 12 km/L (b) d=12fd = 12f (c) 336 km (d) 25 L
Full working
(a) 60/5=120/10=180/15=240/20=1260/5 = 120/10 = 180/15 = 240/20 = 12. Constant ratio → proportional.

(b) d=12fd = 12f (km/L).

(c) d(28)=336d(28) = 336 km.

(d) f=300/12=25f = 300/12 = 25 L.
4Problem 4
Answer
(a) 60 g flour, 40 g sugar, 0.75 eggs per person (b) 720 g flour, 480 g sugar, 9 eggs (c) Eggs
Full working
(a) Per person: 480/8=60480/8 = 60 g flour; 320/8=40320/8 = 40 g sugar; 6/8=0.756/8 = 0.75 eggs.

(b) For 12 people: flour =720= 720 g; sugar =480= 480 g; eggs =9= 9.

(c) Max people from each: flour 1000/60≈16.71000/60 \approx 16.7; sugar 800/40=20800/40 = 20; eggs 10/0.75≈13.310/0.75 \approx 13.3. **Eggs are the limiting ingredient** — max 13 people.
5Problem 5
Answer
(a) Yes (b) No (has yy-intercept 4) (c) No (curve, not linear) (d) Yes
Full working
Direct proportion has the form y=kxy = kx (line through origin, no offset, power 1).

(a) E=12hE = 12h ✓ — proportional.

(b) F=4+1.5dF = 4 + 1.5d — line with yy-intercept 4, not through origin → **not** proportional.

(c) A=s2A = s^2 — curve (parabola), not linear → not proportional.

(d) C=0.30nC = 0.30n ✓ — proportional.
6Problem 6
Answer
(a) No (b) No (c) V/S=s/6V/S = s/6 — grows linearly with ss.
Full working
(a) V=s3V = s^3 is a cube relationship; doubling ss gives V×8V \times 8, not V×2V \times 2. Not proportional.

(b) S=6s2S = 6s^2 — quadratic. Doubling ss gives S×4S \times 4. Not proportional.

(c) V/S=s3/(6s2)=s/6V/S = s^3 / (6s^2) = s/6. As s→∞s \to \infty, this grows without bound. **Bigger cubes have proportionally more volume per surface area** — this is the "square-cube law" of biology and engineering.
7Problem 7
Answer
(a) C=0.92UC = 0.92U (b) 230 CHF (c) 500 USD (d) Yes
Full working
(a) C=0.92UC = 0.92U where CC is CHF and UU is USD.

(b) C=0.92×250=230C = 0.92 \times 250 = 230 CHF.

(c) U=460/0.92=500U = 460 / 0.92 = 500 USD.

(d) Yes — passes through origin (0 USD = 0 CHF) and is linear with constant rate k=0.92k = 0.92.
8Problem 8
Answer
(a) 1, 2, 3, 4 (b) No (c) t=60/vt = 60/v
Full working
(a) t=d/v=60/vt = d/v = 60/v: t=1,2,3,4t = 1, 2, 3, 4 h for v=60,30,20,15v = 60, 30, 20, 15.

(b) Not directly proportional — as vv doubles, tt halves. Direct proportion would mean both double together.

(c) **Inverse proportion**: t=k/vt = k/v with k=60k = 60. The product vt=k=60vt = k = 60 is constant.
9Problem 9
Answer
(a) 600 people/year (b) P=12000+600tP = 12000 + 600t (c) 18 000 (d) No — exponential, not proportional.
Full working
(a) Growth =15000−12000=3000= 15000 - 12000 = 3000 over 5 years → 600 per year.

(b) P=12000+600tP = 12000 + 600t. P0=12000P_0 = 12000, k=600k = 600.

(c) P(10)=12000+6000=18000P(10) = 12000 + 6000 = 18000.

(d) Percentage growth (e.g. 5% per year) makes the population follow P=P0(1.05)tP = P_0(1.05)^t — exponential, not proportional to tt.
10Problem 10
Answer
(a) 2.20, 2.10, 1.90 chf/kg (b) Not a straight line through the origin — bulk discount (c) 5 kg sack (d) Bulk discounts / lower per-unit packaging cost.
Full working
(a) 1 kg: 2.20 chf/kg. 2.5 kg: 5.25/2.5=2.105.25/2.5 = 2.10. 5 kg: 9.50/5=1.909.50/5 = 1.90 chf/kg.

(b) Plotting (1, 2.20), (2.5, 5.25), (5, 9.50) gives points that are roughly on a straight line, but not perfectly through origin — the per-kg rate decreases as size grows, so the points actually bend slightly.

(c) 5 kg sack at 1.90 chf/kg is the best value.

(d) Larger sacks have lower per-unit packaging cost and incentivise bulk buying — common retail strategy. Strict direct proportion would imply equal per-kg cost.
11Problem 11
Answer
(a) C=0.126dC = 0.126d (b) ≈ 40.32 chf (c) ≈ 44.80 chf (d) ≈ 8.40 chf
Full working
(a) Petrol per km: 7/100=0.077/100 = 0.07 L/km. Cost per km: 0.07×1.80=0.1260.07 \times 1.80 = 0.126 chf/km. So C=0.126dC = 0.126d.

(b) C(320)=0.126×320=40.32C(320) = 0.126 \times 320 = 40.32 chf.

(c) New per-km cost: 0.07×2.00=0.140.07 \times 2.00 = 0.14 chf/km. For 320 km: 44.8044.80 chf.

(d) Old (1.80): 0.126×600=75.600.126 \times 600 = 75.60 chf. New (2.00): 0.14×600=84.000.14 \times 600 = 84.00 chf. Difference: 8.408.40 chf extra.
12Problem 12
Answer
(a) No — piecewise linear with a kink at 20 h (b) Two segments (c) 26 hours
Full working
(a) Not proportional — for h≤20h \leq 20: pay =18h= 18h (proportional in this range). For h>20h > 20: pay =360+25(h−20)= 360 + 25(h - 20), which has a different gradient and a non-zero constant.

(b) Two straight-line segments: from (0, 0) to (20, 360) with slope 18, then from (20, 360) to (30, 610) with slope 25. There is a "kink" at h=20h = 20.

(c) 510 chf is above the 20-hour threshold (360 chf). Extra: 510−360=150510 - 360 = 150 at 25 chf/h → 150/25=6150/25 = 6 extra hours. Total: 20+6=2620 + 6 = 26 hours.