Ratio, Proportion and Rates Of Change

50 maths questions with worked answers with answers for OCR 5 Ratio, Proportion and Rates Of Change in OCR GCSE Maths (spec J560). Practise with flashcards, a topic quiz, a random quiz across the course, 5 crosswords, or read every question and answer below. Not official OCR material.

1. Question 1 · Simplifying ratios · OCR 5.01a

   Simplify 24 : 36.

   Answer:

   HCF is 12.
   2 : 3.

2. Question 2 · Simplifying ratios · OCR 5.01a

   Simplify 1.5 : 4.5.

   Answer:

   Multiply by 2: 3 : 9.
   1 : 3.

3. Question 3 · Ratio form · OCR 5.01a

   Write 5 : 8 in the form 1 : n.

   Answer:

   Divide both parts by 5.
   1 : 1.6.

4. Question 4 · Simplifying ratios · OCR 5.01a

   Simplify 12 : 18 : 30.

   Answer:

   HCF is 6.
   2 : 3 : 5.

5. Question 5 · Ratios · OCR 5.01a

   Simplify the ratio 18:30.

   Answer:

   Divide both parts by their highest common factor, 6.
   18:30 = 3:5.

6. Question 6 · Ratios · OCR 5.01a

   Simplify the ratio 2.4:3.6.

   Answer:

   Multiply both parts by 10: 24:36.
   Divide both by 12: 2:3.

7. Question 7 · Ratios · OCR 5.01a

   Write 3 m:75 cm in simplest form.

   Answer:

   Convert 3 m to 300 cm: 300:75.
   Divide both parts by 75: 4:1.

8. Question 8 · Sharing · OCR 5.01a

   Red:blue is 3:4 and there are 28 blue counters. How many are red?

   Answer:

   Four blue parts are 28, so one part is 28 ÷ 4 = 7.
   Three red parts = 3 × 7 = 21.

9. Question 9 · Sharing · OCR 5.01a

   A:B = 5:2 and A exceeds B by 21. Find A and B.

   Answer:

   The difference is 5 − 2 = 3 parts; one part = 21 ÷ 3 = 7.
   A = 5 × 7 = 35 and B = 2 × 7 = 14.

10. Question 10 · Sharing · OCR 5.01b

    Share £72 in the ratio 5 : 3.

    Answer:

    8 parts, so 1 part = £9.
    £45 and £27.

11. Question 11 · Sharing · OCR 5.01b

    Share 150 g in the ratio 1 : 2 : 3.

    Answer:

    6 parts, so 1 part = 25 g.
    25 g, 50 g and 75 g.

12. Question 12 · Sharing · OCR 5.01b

    Divide 96 in the ratio 1:3:4.

    Answer:

    Total parts = 1 + 3 + 4 = 8; one part is 96 ÷ 8 = 12.
    Shares: 12, 36 and 48.

13. Question 13 · Sharing · OCR 5.01b

    Paint uses red:white in the ratio 2:3. How much white paint is needed for 8 litres of red?

    Answer:

    Two red parts are 8 litres, so one part is 4 litres.
    Three white parts = 3 × 4 = 12 litres.

14. Question 14 · Ratio links · OCR 5.01b

    If a:b = 2:3 and b:c = 6:5, find a:b:c.

    Answer:

    Double a:b = 2:3 to make b = 6: a:b = 4:6.
    Combine with b:c = 6:5 to get 4:6:5.

15. Question 15 · Ratio links · OCR 5.01b

    If x:y = 4:7, express xy as a fraction.

    Answer:

    x:y = 4:7 means x corresponds to 4 parts and y to 7.
    Therefore xy = 47.

16. Question 16 · Fractions in ratio · OCR 5.01c

    Red and blue counters are in the ratio 3 : 5. What fraction of the counters are red?

    Answer:

    There are 3 + 5 = 8 parts.
    Red = 38.

17. Question 17 · Fractions in ratio · OCR 5.01c

    25 of a class are boys. What is the ratio of boys to girls?

    Answer:

    Girls are 35.
    Boys : girls = 2 : 3.

18. Question 18 · Fractions in ratio · OCR 5.01c

    In a bag, 14 of the sweets are red and the rest are green. There are 36 green sweets. How many sweets are there?

    Answer:

    Green is 34 of the bag, so 14 is 36 ÷ 3 = 12.
    Total = 12 × 4 = 48 sweets.

19. Question 19 · Fractions in ratio · OCR 5.01c

    Paint is mixed red : white in the ratio 2 : 7. What fraction is white?

    Answer:

    Total parts = 9.
    79.

20. Question 20 · Ratio and fractions · OCR 5.01c

    There are 4 red and 6 green counters. What fraction are green?

    Answer:

    Total counters = 4 + 6 = 10.
    Green fraction = 610 = 35.

21. Question 21 · Ratio and fractions · OCR 5.01c

    In a group the ratio of adults to children is 2:3. What fraction are children?

    Answer:

    Total parts = 2 + 3 = 5.
    Children make up 3 parts, so the fraction is 35.

22. Question 22 · Ratio and fractions · OCR 5.01c

    A class has 12 girls and 18 boys. Express girls as a fraction of boys.

    Answer:

    Girls as a fraction of boys = 1218.
    Divide by 6 to get 23.

23. Question 23 · Ratio fractions · OCR 5.01c

    Flour:sugar is 3:2. What fraction of their combined mass is sugar?

    Answer:

    Total parts = 3 + 2 = 5.
    Sugar accounts for 2 parts: 25.

24. Question 24 · Ratio fractions · OCR 5.01c

    A rope is cut in ratio 7:5. What fraction is the longer piece?

    Answer:

    Total parts = 7 + 5 = 12.
    The longer piece is 7 parts, so its fraction is 712.

25. Question 25 · Ratio problems · OCR 5.01d

    Boys and girls are in the ratio 4 : 5. There are 35 girls. How many boys are there?

    Answer:

    5 parts = 35, so 1 part = 7.
    4 × 7 = 28 boys.

26. Question 26 · Ratio problems · OCR 5.01d

    Red and blue beads are in the ratio 3 : 7. There are 28 more blue than red. How many beads are there?

    Answer:

    The difference is 4 parts = 28, so 1 part = 7.
    10 parts = 70 beads.

27. Question 27 · Ratio and multiplication · OCR 5.01d

    Write the ratio 3 : 8 as a multiplicative relationship.

    Answer:

    The second is 83 times the first.
    b = 83a.

28. Question 28 · Ratio problems · OCR 5.01d

    Ali and Ben share money in the ratio 2 : 3. Ben gets £18. How much does Ali get?

    Answer:

    3 parts = £18, so 1 part = £6.
    Ali gets £12.

29. Question 29 · Ratio and equations · OCR 5.02a

    The ratio x : y is 2 : 5 and x + y = 63. Find x.

    Answer:

    7 parts = 63, so 1 part = 9.
    x = 18.

30. Question 30 · Ratio and equations · OCR 5.02a

    a : b = 3 : 4 and b = 20. Find a.

    Answer:

    4 parts = 20, so 1 part = 5.
    a = 15.

31. Question 31 · Direct proportion · OCR 5.02a

    5 pens cost £3.40. Find the cost of 8 pens.

    Answer:

    1 pen costs 3.40 ÷ 5 = 68p.
    8 × 0.68 = £5.44.

32. Question 32 · Recipes · OCR 5.02a

    A recipe for 4 people uses 300 g of rice. How much rice is needed for 10 people?

    Answer:

    For 1 person: 75 g.
    10 × 75 = 750 g.

33. Question 33 · Best buy · OCR 5.02a

    Which is better value: 500 g for £1.20 or 750 g for £1.70?

    Answer:

    Per 100 g: 24p and 22.7p.
    750 g for £1.70 is better value.

34. Question 34 · Currency · OCR 5.02a

    £1 = €1.15. Change £240 to euros.

    Answer:

    Multiply by 1.15.
    240 × 1.15 = €276.

35. Question 35 · Currency · OCR 5.02a

    £1 = $1.25. Change $85 to pounds.

    Answer:

    Divide by 1.25.
    85 ÷ 1.25 = £68.

36. Question 36 · Proportion graphs · OCR 5.02a

    What does the graph of y = kx look like if y is directly proportional to x?

    Answer:

    It is a straight line.
    It passes through the origin.

37. Question 37 · Direct proportion · OCR 5.02a

    y is directly proportional to x. When x = 4, y = 14. Find y when x = 10.

    Answer:

    y = kx, so k = 14 ÷ 4 = 3.5.
    y = 3.5 × 10 = 35.

38. Question 38 · Inverse proportion · OCR 5.02b

    4 workers take 9 days to build a wall. How long would 6 workers take?

    Answer:

    Total work = 4 × 9 = 36 worker-days.
    36 ÷ 6 = 6 days.

39. Question 39 · Inverse proportion · OCR 5.02b

    y is inversely proportional to x. When x = 3, y = 8. Find y when x = 6.

    Answer:

    y = kx, so k = 24.
    y = 246 = 4.

40. Question 40 · Inverse proportion · OCR 5.02b

    A journey takes 3 hours at 40 mph. How long would it take at 60 mph?

    Answer:

    Distance = 120 miles.
    120 ÷ 60 = 2 hours.

41. Question 41 · Inverse proportion · OCR 5.02b · Higher only

    y is inversely proportional to x². If y = 12 when x = 2, find y when x = 4.

    Answer:

    Write y = kx². Then k = 12 × 2² = 48.
    When x = 4, y = 48 ÷ 4² = 3.

42. Question 42 · Inverse proportion · OCR 5.02b

    Six workers take 4 days at the same rate. How long would 3 workers take?

    Answer:

    The total work is 6 × 4 = 24 worker-days.
    For 3 workers, time = 24 ÷ 3 = 8 days.

43. Question 43 · Compound interest · OCR 5.03a

    £5000 is invested at 2% compound interest per year. Find its value after 3 years.

    Answer:

    5000 × 1.02³ = 5000 × 1.061208.
    £5306.04.

44. Question 44 · Growth and decay · OCR 5.03a

    A culture of 200 bacteria doubles every hour. How many are there after 5 hours?

    Answer:

    200 × 2⁵.
    200 × 32 = 6400.

45. Question 45 · Growth and decay · OCR 5.03a · Higher only

    A car loses 20% of its value each year. After 2 years it is worth £9600. What did it cost new?

    Answer:

    9600 ÷ 0.8².
    9600 ÷ 0.64 = £15 000.

46. Question 46 · Iterative processes · OCR 5.03a · Higher only

    A population is modelled by Pₙ₊₁ = 1.1Pₙ with P₀ = 1000. Find P₂.

    Answer:

    P₁ = 1100.
    P₂ = 1.1 × 1100 = 1210.

47. Question 47 · Percentages · OCR 5.03a

    A salary of £400 is increased by 5%, then by a further 10%. Find the new salary.

    Answer:

    First increase: £400 × 1.05 = £420.
    Second increase: £420 × 1.10 = £462.

48. Question 48 · Percentages · OCR 5.03a

    A value is reduced by 10% twice. What single percentage decrease is this?

    Answer:

    Two decreases give a multiplier of 0.9 × 0.9 = 0.81.
    81% remains, so the decrease is 19%.

49. Question 49 · Growth · OCR 5.03a

    £200 gains 10% interest each year, compounded. Find its value after two years.

    Answer:

    Each year's multiplier is 1.10.
    After two years: £200 × 1.10 × 1.10 = £242.

50. Question 50 · Growth · OCR 5.03a

    A quantity of 1,000 falls by 10% each year. Find it after two years.

    Answer:

    Each year's multiplier is 0.90.
    After two years: 1,000 × 0.90 × 0.90 = 810.

More OCR GCSE Maths topics

• OCR 1 Number Operations and Integers
• OCR 2 Fractions, Decimals and Percentages
• OCR 3 Indices and Surds
• OCR 4 Approximation and Estimation
• OCR 5 Ratio, Proportion and Rates Of Change
• OCR 6 Algebra
• OCR 7 Graphs of Equations and Functions
• OCR 8 Basic Geometry
• OCR 9 Congruence and Similarity
• OCR 10 Mensuration
• OCR 11 Probability
• OCR 12 Statistics

Other Maths courses: AQA GCSE Maths, Edexcel GCSE Maths.