OxfordAQA International GCSE Mathematics: Ratio, rates, best buys and direct and inverse proportion (9260) – Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – examination boards hold copyright in their own papers. Use these alongside the official past papers from your board or school.

These questions cover section 3.1.3 (Ratio and proportion), statements N16 to N20, in the OxfordAQA International GCSE Mathematics specification (9260), Version 5.1. All of it is Core content, which can also be assessed at Extension. Learn the methods first in the study guide or the revision notes. The course hub, the checklist and the free diagnostics are linked for the next step. Show every step of your working.

Questions

1. (a) Write the ratio 45 : 75 in simplest form. [1] (b) Write 3.2 m : 80 cm in simplest form. [2] (c) In a class the ratio of boys to girls is 4 : 5. What fraction of the class are girls? [1]

2. Divide Rs 1260 in the ratio 3 : 4 : 5. [3]

3. Tom and Uma share some money in the ratio 7 : 4. Tom receives 63 dollars more than Uma. Find the total amount shared. [4]

4. The ratio x : y is 5 : 2 and the ratio y : z is 3 : 4. (a) Find the ratio x : y : z. [2] (b) The sum x + y + z is 232. Find z. [2]

5. A bag holds red and green beads in the ratio 4 : 7. After 20 green beads are taken out, the ratio of red to green is 2 : 3. How many beads were in the bag at first? [4]

6. Aisha earns 11.60 dollars an hour for her first 36 hours each week. Any extra hours are paid at 1.25 times this rate. One week she works 44 hours. Find her total pay for the week. [4]

7. Orange juice is sold in three sizes: 750 ml for 1.35 dollars, 1.5 litres for 2.46 dollars and 2.25 litres for 3.60 dollars. Which size is the best buy? Show your working. [4]

8. A cyclist rides 42 km in 2 hours 20 minutes at a steady speed. (a) Find the speed in km/h. [2] (b) How long would she take to ride 63 km at the same speed? Give your answer in hours and minutes. [2]

9. (a) Here are some values of x and y.

x	4	10	14
y	6	15	21

Show that y is directly proportional to x. [2] (b) A printer prints 180 pages in 6 minutes. Assuming a constant rate, find the time to print 750 pages. [2]

10. A team of 9 painters can paint a building in 14 days. After 4 days, 3 painters leave and are not replaced. The others work at the same rate. How many days in total does the job take? [5]

11. A laptop costs 640 dollars. The price falls by 15% and later rises by 20%. (a) Find the final price. [2] (b) Find the overall percentage change in the price. [2]

12. A café makes a drink by mixing concentrate and water in the ratio 2 : 7. (a) What fraction of the drink is concentrate? [1] (b) How much concentrate is needed to make 90 litres of the drink? [2] (c) Concentrate is sold in 1.5 litre bottles at 6.30 dollars and in 4 litre bottles at 15.20 dollars. Which is the better buy? [3] (d) Using only the better-value bottles, find the cost of the concentrate for 90 litres of the drink. [2]

Answers

1. (a) 45 : 75, divide by 15, gives 3 : 5 [1] (b) 3.2 m = 320 cm [1]. 320 : 80 = 4 : 1 [1] (c) The girls are 5 parts out of 4 + 5 = 9 parts, so 5/9 [1] [4] Examiner insight: Convert to one unit before simplifying and leave units out of the final ratio. Writing 3.2 : 80 and simplifying gives a wrong ratio.

2. Total parts 3 + 4 + 5 = 12 [1]. One part is 1260 ÷ 12 = 105 [1]. Shares: 3 × 105 = 315, 4 × 105 = 420, 5 × 105 = 525, so Rs 315, Rs 420, Rs 525 [1] [3] Examiner insight: Adding your three shares to get 1260 is a quick check, and it shows the examiner that the answer is complete.

3. The difference is 7 - 4 = 3 parts [1]. 3 parts = 63, so 1 part = 21 [1]. Total parts 7 + 4 = 11 [1]. Total = 11 × 21 = 231 dollars [1] [4] Examiner insight: When no total is given, find the value of one part from the difference. Answers that divide 63 by 11 or by 7 are a common slip.

4. (a) Make y the same in both ratios: the common multiple of 2 and 3 is 6. x : y = 15 : 6 [1]. y : z = 6 : 8, so x : y : z = 15 : 6 : 8 [1] (b) Total parts 15 + 6 + 8 = 29 [1]. One part is 232 ÷ 29 = 8, so z = 8 × 8 = 64 [1] [4] Examiner insight: Part (b) is answerable from the ratio found in part (a). State the three-part ratio clearly so that your later working is easy to follow.

5. Let the beads be 4k red and 7k green [1]. After the change: 4k : (7k - 20) = 2 : 3, so 3 × 4k = 2 × (7k - 20) [1]. 12k = 14k - 40, so k = 20 [1]. Red 80, green 140, so 220 beads [1] [4] Examiner insight: The question asks for the original total, not k. Check by testing 80 : (140 - 20) = 80 : 120 = 2 : 3.

6. Basic pay: 36 × 11.60 = 417.60 [1]. Overtime rate: 1.25 × 11.60 = 14.50 [1]. Overtime hours 44 - 36 = 8, and 8 × 14.50 = 116 [1]. Total 417.60 + 116 = 533.60 dollars [1] [4] Examiner insight: Work out the number of overtime hours explicitly. Applying 1.25 to all 44 hours is a typical wrong method.

7. Convert to litres: 0.75 L, 1.5 L, 2.25 L [1]. Cost per litre: 1.35 ÷ 0.75 = 1.80 [1]; 2.46 ÷ 1.5 = 1.64 [1] and 3.60 ÷ 2.25 = 1.60. The cheapest per litre is the 2.25 litre size [1] [4] Examiner insight: A best-buy answer needs a figure for every option on a common basis and a clear conclusion. Naming the winner without figures leaves the comparison unsupported.

8. (a) 2 h 20 min = 7/3 h [1]. Speed = 42 ÷ 7/3 = 42 × 3/7 = 18 km/h [1] (b) Time = 63 ÷ 18 = 3.5 h [1] = 3 hours 30 minutes [1] [4] Examiner insight: Convert minutes to a fraction of an hour by dividing by 60. Entering 2.20 for 2 h 20 min gives a wrong speed, and the units in the final answer should be hours and minutes as asked.

9. (a) y ÷ x: 6 ÷ 4 = 1.5, 15 ÷ 10 = 1.5, 21 ÷ 14 = 1.5 [1]. The ratio is constant, so y is directly proportional to x [1] (b) The printer prints 180 ÷ 6 = 30 pages a minute [1]. Time = 750 ÷ 30 = 25 minutes [1] [4] Examiner insight: On a “show that” question, give the calculated values and a conclusion. One check is not enough to support a general claim.

10. Total work: 9 × 14 = 126 painter-days [1]. Done in 4 days: 9 × 4 = 36 [1]. Remaining: 126 - 36 = 90 [1]. Painters now: 9 - 3 = 6, so 90 ÷ 6 = 15 more days [1]. Total 4 + 15 = 19 days [1] [5] Examiner insight: A common slip is to give 15 as the final answer. Reread what the question asks for, which is the total time.

11. (a) Multipliers 0.85 and 1.20 [1]. 640 × 0.85 × 1.20 = 652.80 dollars [1] (b) 0.85 × 1.20 = 1.02 [1], an overall increase of 2% [1] [4] Examiner insight: Subtracting 15% from 20% to get 5% is not valid, because the two percentages apply to different amounts. Multiply the multipliers.

12. (a) Concentrate is 2 parts out of 2 + 7 = 9, so 2/9 [1] (b) 90 × 2/9 [1] = 20 litres [1] (c) 6.30 ÷ 1.5 = 4.20 dollars per litre [1]. 15.20 ÷ 4 = 3.80 dollars per litre [1]. The 4 litre bottle is the better buy [1] (d) 20 ÷ 4 = 5 bottles [1]. Cost = 5 × 15.20 = 76 dollars [1] [8] Examiner insight: In a multi-part question, later parts rely on earlier answers. Label each part, show the unit cost for both bottles, and keep the figures consistent between parts.

Where marks are usually lost

• Leaving the units unconverted before simplifying a ratio such as metres and centimetres.
• Dividing by the number of shares rather than by the total number of parts.
• Stopping at the value of one part, or at k, instead of answering the question asked.
• Giving 5/4 or 4/5 when a question asks for the fraction of the whole, which needs the total of the parts as the denominator.
• Comparing prices without reducing both to a cost per unit, or giving only one unit price.
• Forgetting to state which option is the best buy.
• Writing 2 h 20 min as 2.20 hours in a speed calculation.
• Treating an inverse proportion as direct, so that fewer workers finish sooner.
• Overtime applied to all hours instead of only the extra hours.
• Adding percentage changes instead of multiplying multipliers.

Next steps

• Review the revision notes for the methods
• Reread the study guide for any weak part
• Open the OxfordAQA IGCSE Mathematics hub
• Tick off the topic on the printable checklist
• Try the free diagnostics to find what to revise next
• Book a free trial class

Official syllabus

OxfordAQA International GCSE Mathematics (9260), Version 5.1, for exams in May/June 2018 onwards, published by OxfordAQA. Section 3.1.3, Ratio and proportion, statements N16 to N20.