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Solutions — Full Answer Key

Ecolint Campus des NationsMathematics
Year 8 · 8.2 Ratio and Rates

Solutions · Full Answer Key

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

Pack A — Answers

Bronze
1.4:94:9; HCF = 4
2.3:2:53:2:5
3.25%
4.1:31:3
5.360 g
6.75%
7.0.25 chf
8.12
9.125 chf
10.93.75 chf
Silver
11.3:4:53:4:5
12.24 chf and 36 chf
13.(a) 13:713:7 (b) 35%
14.7.50 chf
15.18 bars
16.80 km/h ≈ 1.33 km/min
17.(a) 72 L (b) 720 L
18.2.50 chf/kg
19.25%
20.4320 chf
Gold
21.25%
22.797.30 chf
23.15:2815:28
24.Male = 650; Female = 1000
25.93.75 chf
26.Shop B (≈ 0.23 chf/bar vs 0.25 chf/bar)
27.−60-60 m
28.A and C
29.(a) 90 km/h (b) 25 m/s
30.100 chf
Platinum
31.1:21:2
32.10% decrease
33.100 chf
34.16
35.(a) 60 (b) 33.3%
36.£13 770
37.254.80 EUR
38.5 litres
39.£800, £1200, £2000
40.£480

Pack B — Answers

Bronze
1.3:43:4; HCF = 6
2.2:3:52:3:5
3.30%
4.2:32:3
5.200 g
6.35%
7.0.30 chf
8.8
9.92 chf
10.72 chf
Silver
11.3:2:53:2:5
12.30 chf and 50 chf
13.(a) 3:23:2 (b) 40%
14.6.00 chf
15.20 bars
16.90 km/h = 1.5 km/min
17.(a) 96 L (b) 960 L
18.3 chf/kg
19.25%
20.5830 chf
Gold
21.32%
22.600.00 chf
23.6:56:5 (or 1.2:11.2:1)
24.Male = 400; Female = 500
25.216 chf
26.Shop B (0.16 chf/bar vs 0.25 chf/bar)
27.−96-96 m
28.A and C
29.(a) 72 km/h (b) 20 m/s
30.96 chf
Platinum
31.2:12:1
32.20% increase
33.100 chf
34.32
35.(a) 100 (b) 33.3%
36.£20 655
37.288.12 EUR
38.4 litres
39.£1200, £1800, £3000
40.£490

Problem-solving — Worked Solutions

1Problem 1
Answer
(a) 650 (b) 1000 (c) 50%
Full working
Total parts =20+13=33= 20 + 13 = 33. One part =1650÷33=50= 1650 \div 33 = 50 competitors. (a) Male finishers =13×50=650= 13 \times 50 = 650. (b) Female finishers =20×50=1000= 20 \times 50 = 1000. (c) Female finishers as % of all 2000: 10002000×100=50%\frac{1000}{2000} \times 100 = 50\%.
2Problem 2
Answer
(a) 797.30 chf (b) New price = 743.40 chf; difference = 53.90 chf
Full working
(a) Sale price is 100%−26%=74%100\% - 26\% = 74\% of original. Original =590÷0.74≈797.297→= 590 \div 0.74 \approx 797.297 \rightarrow **797.30 chf**.

(b) Raising 590 by 26%: 590×1.26=743.40590 \times 1.26 = 743.40 chf. This is **less** than the original 797.30 chf. Difference =797.30−743.40=53.90= 797.30 - 743.40 = 53.90 chf. A decrease of 26% followed by an increase of 26% does not return to the start because the increase is taken from a smaller base — combined multiplier 0.74×1.26=0.93240.74 \times 1.26 = 0.9324, about 6.76% below original.
3Problem 3
Answer
Plane: 80 kg (≈ 48%). Train: 15.316 kg (≈ 9%). Car (solo): 165.356 kg (100%). Train ≈ 9.3%, plane ≈ 48.4%, car solo 100%.
Full working
Plane: 80 kg per passenger (round-trip). Train: round-trip =2×547=1094= 2 \times 547 = 1094 km. CO₂ =1094×0.014=15.316= 1094 \times 0.014 = 15.316 kg. Car: round-trip =2×617=1234= 2 \times 617 = 1234 km. CO₂ =1234×0.134=165.356= 1234 \times 0.134 = 165.356 kg. Largest = car (165.36 kg) → 100%. Plane =48.4%= 48.4\%. Train =9.3%= 9.3\%. Train is by far the lowest-carbon option.
4Problem 4
Answer
(a) 1000 mL (b) 1500 mL of blue and 2500 mL of yellow (c) 2.88 L, limited by yellow
Full working
(a) Blue : Yellow =3:5= 3:5. Each part: blue 600 mL means 1 part =200= 200 mL, so yellow =5×200=1000= 5 \times 200 = 1000 mL.

(b) Total parts =8= 8. For 4 L =4000= 4000 mL: 1 part =500= 500 mL. Blue =1500= 1500 mL; yellow =2500= 2500 mL.

(c) With 1.2 L blue (3 parts) max paint = 1.23×8=3.2\frac{1.2}{3} \times 8 = 3.2 L. With 1.8 L yellow (5 parts) max = 1.85×8=2.88\frac{1.8}{5} \times 8 = 2.88 L. Limiting colour: **yellow** (smaller max) → green =2.88= 2.88 L. Uses 1.08 L blue (under 1.2 ✓) and 1.80 L yellow (all of it).
5Problem 5
Answer
(a) 260 EUR (b) 300 chf (c) 254.80 EUR
Full working
(a) 250×1.04=260250 \times 1.04 = 260 EUR.

(b) 312÷1.04=300312 \div 1.04 = 300 chf.

(c) Without commission you would receive 260 EUR. Commission removes 2%: 260×0.98=254.80260 \times 0.98 = 254.80 EUR.
6Problem 6
Answer
(a) 4:3:24:3:2 (b) 300 g flour, 225 g sugar, 150 g butter, 5 eggs (c) 25 brownies
Full working
(a) 120:90:60120:90:60. HCF = 30: 4:3:24:3:2.

(b) Scale factor =15÷6=2.5= 15 \div 6 = 2.5. Flour: 300300 g. Sugar: 225225 g. Butter: 150150 g. Eggs: 55.

(c) Flour per brownie =120÷6=20= 120 \div 6 = 20 g. 500÷20=25500 \div 20 = 25 brownies.
7Problem 7
Answer
(a) 110\tfrac{1}{10} (b) 14135\tfrac{14}{135}
Full working
(a) Boys split 1:91:9, so left-handed fraction = 11+9=110\tfrac{1}{1 + 9} = \tfrac{1}{10}.

(b) Whole school = 9 parts (4 boys + 5 girls). Left-handed boys: 49×110=490\tfrac{4}{9} \times \tfrac{1}{10} = \tfrac{4}{90}. Left-handed girls: 59×112=5108\tfrac{5}{9} \times \tfrac{1}{12} = \tfrac{5}{108}. Common denominator 540: 490=24540\tfrac{4}{90} = \tfrac{24}{540} and 5108=25540\tfrac{5}{108} = \tfrac{25}{540}. Sum: 49540\tfrac{49}{540}. Hmm — let's recompute. Boys 49\frac{4}{9}, of whom 110\frac{1}{10} left-handed → 490\frac{4}{90} of school. Girls 59\frac{5}{9}, of whom 112\frac{1}{12} → 5108\frac{5}{108}. To combine: 490+5108\frac{4}{90} + \frac{5}{108}. LCM(90, 108) = 540: 24540+25540=49540\frac{24}{540} + \frac{25}{540} = \frac{49}{540}. (Approx 9.1%9.1\%.)
8Problem 8
Answer
(a) 1077 chf (b) 500 chf (c) 1163.16 chf
Full working
(a) 1000×1.077=10771000 \times 1.077 = 1077 chf.

(b) Pre-tax price = 538.90÷1.077=500538.90 \div 1.077 = 500 chf.

(c) Discounted pre-tax: 1200×0.90=10801200 \times 0.90 = 1080. With VAT: 1080×1.077=1163.161080 \times 1.077 = 1163.16 chf.
9Problem 9
Answer
(a) 1.28, 1.18, 1.12 chf per 100 g (b) 1 kg pack (c) Two paid + one free → 1500 g for 11.80 → 0.787 chf / 100 g (best value)
Full working
(a) 250 g: 3.20÷2.5=1.283.20 \div 2.5 = 1.28 chf/100 g. 500 g: 5.90÷5=1.185.90 \div 5 = 1.18 chf/100 g. 1 kg: 11.20÷10=1.1211.20 \div 10 = 1.12 chf/100 g.

(b) 1 kg pack at 1.12 chf/100 g is cheapest per gram.

(c) Buy two 500 g packs (2 × 5.90 = 11.80) get one free → 1500 g for 11.80 chf → 11.80÷15≈0.78711.80 \div 15 \approx 0.787 chf/100 g. Easily the best deal.
10Problem 10
Answer
(a) 864 (b) ≈ 1007 (c) 2035
Full working
(a) 800×1.08=864800 \times 1.08 = 864.

(b) After 4 years: 800×1.084≈800×1.3605≈1088800 \times 1.08^4 \approx 800 \times 1.3605 \approx 1088. Hmm — let me recompute: 1.082=1.16641.08^2 = 1.1664; 1.084=1.16642≈1.36051.08^4 = 1.1664^2 \approx 1.3605; so 2030 has 800×1.3605≈1088800 \times 1.3605 \approx 1088 trees.

(c) Solve 800×1.08n>1500⇒1.08n>1.875800 \times 1.08^n > 1500 \Rightarrow 1.08^n > 1.875. Take logs (or iterate): 1.088≈1.8511.08^8 \approx 1.851; 1.089≈2.0001.08^9 \approx 2.000. So after 9 years = **2035**.
11Problem 11
Answer
(a) Alex 24 km/h; Brigit ≈ 24.5 km/h (b) ≈ 48 : 49 (c) ≈ 145.5 km apart
Full working
(a) Alex: 60÷2.5=2460 \div 2.5 = 24 km/h. Brigit: 1 h 50 min=1161\,h\,50\,min = \tfrac{11}{6} h. Speed =45÷116=45×611=27011≈24.5= 45 \div \tfrac{11}{6} = 45 \times \tfrac{6}{11} = \tfrac{270}{11} \approx 24.5 km/h.

(b) Ratio =24:27011=264:270=44:45= 24 : \tfrac{270}{11} = 264 : 270 = 44 : 45.

(c) Combined separation rate ≈48.5\approx 48.5 km/h. In 3 h: ≈145.5\approx 145.5 km. (Computed: 24×3+24.5×3=72+73.5=145.524 \times 3 + 24.5 \times 3 = 72 + 73.5 = 145.5 km.)
12Problem 12
Answer
(a) Plan X ≈ 1157.63 chf; Plan Y ≈ 1156.66 chf (b) Plan X by ≈ 0.97 chf — almost equal!
Full working
(a) Plan X multiplier: 1.053=1.1576251.05^3 = 1.157625. Value: 1000×1.157625=1157.631000 \times 1.157625 = 1157.63 chf. Plan Y multiplier: 1.08×1.04×1.03=1.1566561.08 \times 1.04 \times 1.03 = 1.156656. Value: 1156.661156.66 chf.

(b) Difference: 1157.63−1156.66≈0.971157.63 - 1156.66 \approx 0.97 chf in favour of Plan X. They are surprisingly close because the **product** of rates matters more than their order or precise distribution; both plans give an average annual rate near 5%.