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A Level H1 Mathematics Numbers Ratio Proportion Quiz
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Questions
A-Level Maths H1 Quiz - Numbers Ratio Proportion
Name: ________________________
Class: ________________________
Date: ________________________
Score: ______ / 40
Duration: 45 minutes
Total Marks: 40
Instructions:
- Answer ALL questions.
- Show all working clearly. Marks are awarded for method.
- Unless otherwise stated, give numerical answers to 3 significant figures.
- You may use an approved graphing calculator.
- The number of marks is given in brackets [ ] at the end of each question or part question.
Section A: Short Answer (10 marks)
Answer all questions in this section.
1. Express the ratio 0.75 : 1.25 in its simplest integer form. [1]
2. A sum of money is divided between Ali, Bala, and Chen in the ratio 3 : 5 : 7. If Chen receives $84 more than Ali, find the total sum of money. [2]
3. The population of a town increased from 24,000 to 27,600 over a year. Find the percentage increase. [1]
4. A car travels 240 km in 3 hours 20 minutes. Find its average speed in km/h. [2]
5. The cost of 8 identical pens is $14.40. Find the cost of 15 such pens. [2]
Section B: Structured Questions (10 marks)
Answer all questions in this section.
6. A map has a scale of 1 : 25,000. Two towns are 8.4 cm apart on the map. Find the actual distance between the towns in kilometres. [2]
7. A company produces two types of gift hampers, Standard and Deluxe. The ratio of the number of Standard hampers to Deluxe hampers produced in a day is 7 : 3.
(a) On a day when 450 hampers are produced in total, find the number of Deluxe hampers produced. [2]
(b) The profit on each Standard hamper is 20. On the day described in part (a), find the total profit. [2]
(c) On another day, the total profit from the sale of all hampers is $5,760. Given that the ratio of Standard to Deluxe hampers produced remains 7 : 3, find the total number of hampers produced on that day. [2]
8. An investment of $5,000 earns compound interest at a rate of 3.5% per annum, compounded annually. Find the value of the investment after 4 years. [2]
9. A rectangular field has a perimeter of 400 m. The length and width are in the ratio 3 : 2.
(a) Find the length and width of the field. [2]
(b) Calculate the area of the field in hectares. (1 hectare = 10,000 m²) [2]
Section C: Structured Questions (10 marks)
Answer all questions in this section.
10. An investment of 6,500. [2]
11. A solution is made by mixing liquid A and liquid B in the ratio 2 : 5 by volume. Liquid A costs 2.00 per litre.
(a) Find the cost of 1 litre of the mixture. [2]
(b) How many litres of the mixture can be made for $50? [1]
12. The scale of a model car is 1 : 18. The model has a surface area of 250 cm². Find the surface area of the actual car in m². [2]
13. A manufacturer produces cylindrical containers with no lid. Each container has a fixed volume of 500π cm³. The material for the base costs 0.05 per cm².
Let the radius of the base be r cm and the height be h cm.
(a) Express h in terms of r. [2]
(b) Show that the total cost, $C, of material for one container is given by: [2]
Section D: Application and Problem Solving (10 marks)
Answer all questions in this section.
14. A manufacturer produces cylindrical containers with no lid. Each container has a fixed volume of 500π cm³. The material for the base costs 0.05 per cm².
Let the radius of the base be r cm and the height be h cm.
(a) Use differentiation to find the value of r that minimises the cost. Verify that this value gives a minimum cost. [3]
(b) Hence find the minimum cost of material for one container, giving your answer to the nearest cent. [2]
15. A sum of money is divided between three siblings in the ratio 2 : 3 : 4. The largest share is $1,200. Find the total sum of money. [2]
16. A car uses 12 litres of petrol to travel 150 km. Find the petrol consumption in litres per 100 km. [2]
17. A shop offers a 15% discount on a television set. The discounted price is $680. Find the original price. [2]
18. A piece of rope is cut into three pieces in the ratio 2 : 5 : 8. The longest piece is 24 m longer than the shortest piece. Find the original length of the rope. [2]
19. A map has a scale of 1 : 50,000. A forest is represented by an area of 12 cm² on the map. Find the actual area of the forest in square kilometres. [2]
20. The value of a machine depreciates by 20% each year. Its value after 2 years is $6,400. Find its original value. [2]
END OF QUIZ
Answers
A-Level Maths H1 Quiz - Numbers Ratio Proportion
ANSWER KEY AND MARKING SCHEME
Total Marks: 40
Section A: Short Answer (10 marks)
1. Express the ratio 0.75 : 1.25 in its simplest integer form. [1]
Answer: 3 : 5
Working: 0.75 : 1.25 = 75 : 125 = 3 : 5 (divide both by 25)
Marking: B1 for 3 : 5 (accept equivalent integer ratio)
2. A sum of money is divided between Ali, Bala, and Chen in the ratio 3 : 5 : 7. If Chen receives $84 more than Ali, find the total sum of money. [2]
Answer: $315
Working: Let the shares be 3x, 5x, and 7x. Chen − Ali = 7x − 3x = 4x = 84 x = 21 Total = 3x + 5x + 7x = 15x = 15 × 21 = 315
Marking: M1 for setting up 4x = 84 or equivalent; A1 for $315
3. The population of a town increased from 24,000 to 27,600 over a year. Find the percentage increase. [1]
Answer: 15%
Working: Increase = 27,600 − 24,000 = 3,600 Percentage increase = (3,600 ÷ 24,000) × 100% = 15%
Marking: B1 for 15%
4. A car travels 240 km in 3 hours 20 minutes. Find its average speed in km/h. [2]
Answer: 72 km/h
Working: 3 hours 20 minutes = 3⅓ hours = 10/3 hours Speed = Distance ÷ Time = 240 ÷ (10/3) = 240 × 3/10 = 72 km/h
Marking: M1 for converting time to hours correctly; A1 for 72 km/h
5. The cost of 8 identical pens is $14.40. Find the cost of 15 such pens. [2]
Answer: $27.00
Working: Cost of 1 pen = 1.80 Cost of 15 pens = 15 × 27.00
Marking: M1 for finding unit cost; A1 for $27.00
Section B: Structured Questions (10 marks)
6. A map has a scale of 1 : 25,000. Two towns are 8.4 cm apart on the map. Find the actual distance between the towns in kilometres. [2]
Answer: 2.1 km
Working: Actual distance = 8.4 × 25,000 = 210,000 cm = 2,100 m = 2.1 km
Marking: M1 for multiplying by scale factor; A1 for 2.1 km
7. A company produces two types of gift hampers, Standard and Deluxe. The ratio of the number of Standard hampers to Deluxe hampers produced in a day is 7 : 3.
(a) On a day when 450 hampers are produced in total, find the number of Deluxe hampers produced. [2]
Answer: 135
Working: Total parts = 7 + 3 = 10 Deluxe = (3/10) × 450 = 135
Marking: M1 for using ratio correctly; A1 for 135
(b) The profit on each Standard hamper is 20. On the day described in part (a), find the total profit. [2]
Answer: $6,480
Working: Standard hampers = 450 − 135 = 315 Profit = 315 × 20 = 2,700 = $6,480
Marking: M1 for finding both quantities and multiplying by respective profits; A1 for $6,480
(c) On another day, the total profit from the sale of all hampers is $5,760. Given that the ratio of Standard to Deluxe hampers produced remains 7 : 3, find the total number of hampers produced on that day. [2]
Answer: 400
Working: Let number of Standard = 7k, Deluxe = 3k Total profit = 7k × 20 = 84k + 60k = 144k 144k = 5,760 k = 40 Total hampers = 10k = 400
Marking: M1 for expressing profit in terms of k; A1 for 400
8. An investment of $5,000 earns compound interest at a rate of 3.5% per annum, compounded annually. Find the value of the investment after 4 years. [2]
Answer: 5,740 to 3 s.f.)
Working: A = 5,000 × (1 + 0.035)⁴ = 5,000 × 1.035⁴ = 5,000 × 1.147523... = 5,738
Marking: M1 for using compound interest formula; A1 for 5,740)
9. A rectangular field has a perimeter of 400 m. The length and width are in the ratio 3 : 2.
(a) Find the length and width of the field. [2]
Answer: Length = 120 m, Width = 80 m
Working: Let length = 3x, width = 2x Perimeter = 2(3x + 2x) = 10x = 400 x = 40 Length = 3 × 40 = 120 m, Width = 2 × 40 = 80 m
Marking: M1 for expressing perimeter in terms of x; A1 for both correct values
(b) Calculate the area of the field in hectares. (1 hectare = 10,000 m²) [2]
Answer: 0.96 hectares
Working: Area = 120 × 80 = 9,600 m² In hectares = 9,600 ÷ 10,000 = 0.96 hectares
Marking: M1 for calculating area in m²; A1 for 0.96
Section C: Structured Questions (10 marks)
10. An investment of 6,500. [2]
Answer: 8 years
Working: 5,000 × 1.035ⁿ > 6,500 1.035ⁿ > 1.3 n × ln(1.035) > ln(1.3) n > ln(1.3) ÷ ln(1.035) = 0.262364... ÷ 0.034401... = 7.626... Complete years = 8
Marking: M1 for setting up inequality and using logarithms; A1 for 8
11. A solution is made by mixing liquid A and liquid B in the ratio 2 : 5 by volume. Liquid A costs 2.00 per litre.
(a) Find the cost of 1 litre of the mixture. [2]
Answer: 17/7 ≈ $2.428...
Working: In 7 litres of mixture, 2 litres of A and 5 litres of B. Cost of 7 litres = 2 × 2.00 = 10.00 = 17.00 ÷ 7 = 2.43
Marking: M1 for finding cost of combined volume; A1 for $2.43
(b) How many litres of the mixture can be made for $50? [1]
Answer: 20.6 litres (to 3 s.f.) or 350/17 ≈ 20.588...
Working: Litres = 17/7) = 50 × 7/17 = 350/17 ≈ 20.6 litres
Marking: A1 for 20.6 litres
12. The scale of a model car is 1 : 18. The model has a surface area of 250 cm². Find the surface area of the actual car in m². [2]
Answer: 8.1 m²
Working: Scale factor for area = 18² = 324 Actual area = 250 cm² × 324 = 81,000 cm² = 81,000 ÷ 10,000 m² = 8.1 m²
Marking: M1 for using square of scale factor; A1 for 8.1 m²
13. A manufacturer produces cylindrical containers with no lid. Each container has a fixed volume of 500π cm³. The material for the base costs 0.05 per cm².
Let the radius of the base be r cm and the height be h cm.
(a) Express h in terms of r. [2]
Answer: h = 500/r²
Working: Volume = πr²h = 500π r²h = 500 h = 500/r²
Marking: M1 for setting up volume equation; A1 for h = 500/r²
(b) Show that the total cost, $C, of material for one container is given by: [2]
Answer: Shown
Working: Area of base = πr² Area of curved surface = 2πrh = 2πr(500/r²) = 1,000π/r Cost = 0.08 × πr² + 0.05 × (1,000π/r) = 0.08πr² + 50π/r
Marking: M1 for area of base and curved surface; A1 for correct expression
Section D: Application and Problem Solving (10 marks)
14. A manufacturer produces cylindrical containers with no lid. Each container has a fixed volume of 500π cm³. The material for the base costs 0.05 per cm².
Let the radius of the base be r cm and the height be h cm.
(a) Use differentiation to find the value of r that minimises the cost. Verify that this value gives a minimum cost. [3]
Answer: r = 6.79 cm (to 3 s.f.)
Working: C = 0.08πr² + 50πr⁻¹ dC/dr = 0.16πr − 50πr⁻² = 0.16πr − 50π/r² Set dC/dr = 0: 0.16πr = 50π/r² 0.16r³ = 50 r³ = 50/0.16 = 312.5 r = ∛312.5 = 6.786... ≈ 6.79 cm
d²C/dr² = 0.16π + 100π/r³ At r = 6.79, d²C/dr² > 0, therefore minimum.
Marking: M1 for differentiating correctly; M1 for setting dC/dr = 0 and solving; A1 for r = 6.79 with verification
(b) Hence find the minimum cost of material for one container, giving your answer to the nearest cent. [2]
Answer: $34.73
Working: C = 0.08π(6.786...)² + 50π/6.786... = 0.08π(46.053...) + 50π/6.786... = 11.576... + 23.152... = 34.728... ≈ $34.73
Marking: M1 for substituting r into cost formula; A1 for 34.70–$34.73)
15. A sum of money is divided between three siblings in the ratio 2 : 3 : 4. The largest share is $1,200. Find the total sum of money. [2]
Answer: $2,700
Working: Let shares be 2x, 3x, 4x. Largest share = 4x = 1,200 x = 300 Total = 2x + 3x + 4x = 9x = 9 × 300 = 2,700
Marking: M1 for setting up 4x = 1,200; A1 for $2,700
16. A car uses 12 litres of petrol to travel 150 km. Find the petrol consumption in litres per 100 km. [2]
Answer: 8 litres per 100 km
Working: Consumption = (12 litres / 150 km) × 100 km = 8 litres per 100 km
Marking: M1 for setting up proportion; A1 for 8
17. A shop offers a 15% discount on a television set. The discounted price is $680. Find the original price. [2]
Answer: $800
Working: Let original price be $x. x × (1 − 0.15) = 680 0.85x = 680 x = 680 ÷ 0.85 = 800
Marking: M1 for 0.85x = 680; A1 for $800
18. A piece of rope is cut into three pieces in the ratio 2 : 5 : 8. The longest piece is 24 m longer than the shortest piece. Find the original length of the rope. [2]
Answer: 60 m
Working: Let pieces be 2x, 5x, 8x. Longest − Shortest = 8x − 2x = 6x = 24 x = 4 Total length = 2x + 5x + 8x = 15x = 15 × 4 = 60 m
Marking: M1 for 6x = 24; A1 for 60 m
19. A map has a scale of 1 : 50,000. A forest is represented by an area of 12 cm² on the map. Find the actual area of the forest in square kilometres. [2]
Answer: 3 km²
Working: Scale factor for area = 50,000² = 2.5 × 10⁹ Actual area in cm² = 12 × 2.5 × 10⁹ = 3 × 10¹⁰ cm² 1 km² = 10¹⁰ cm² Actual area = 3 × 10¹⁰ ÷ 10¹⁰ = 3 km²
Marking: M1 for using square of scale factor; A1 for 3 km²
20. The value of a machine depreciates by 20% each year. Its value after 2 years is $6,400. Find its original value. [2]
Answer: $10,000
Working: Let original value be $V. V × (1 − 0.2)² = 6,400 V × 0.8² = 6,400 V × 0.64 = 6,400 V = 6,400 ÷ 0.64 = 10,000
Marking: M1 for V × 0.64 = 6,400; A1 for $10,000
END OF ANSWER KEY