Real Numbers Mixed Review

🟢 Section A: Foundation Review

These questions check your understanding of the basic skills from the chapter.

1. State whether each number is natural, whole, integer, rational or irrational. Give every category that applies.
   
   a) 7
   b) 0
   c) −4
   d) 3/5
   e) √3
2. Write all the factors of 24.
3. State whether 37 is prime or composite.
4. Write 84 as a product of prime factors.
5. Find the HCF of 18 and 30.
6. Find the LCM of 8 and 12.
7. Calculate:
   
   −7 + 12
8. Calculate:
   
   −6 × −4
9. Calculate:
   
   18 − 3 × 4
10. Calculate:
    
    (18 − 3) × 4
11. Simplify:
    
    12/18
12. Calculate:
    
    3/4 + 1/8
13. Calculate:
    
    5/6 − 1/3
14. Calculate:
    
    2/5 × 15/4
15. Calculate:
    
    3/7 ÷ 6/7
16. State the value of the digit 6 in 24.361.
17. Arrange in ascending order:
    
    0.7, 0.67, 0.706, 0.76
18. Round 8.473 to two decimal places.
19. Convert 3/5 to a decimal.
20. Convert 0.72 to a percentage.

🔵 Section B: Developing Review

These questions combine skills and require more careful selection of methods.

21. Find 35% of 480.
22. Express 42 as a percentage of 60.
23. Increase US$250 by 18%.
24. A bicycle costing US$320 is discounted by 15%. Find the sale price.
25. A population increases from 800 to 920. Find the percentage increase.
26. A shirt costs US$72 after a 20% discount. Find its original price.
27. Calculate:
    
    5³
28. Calculate:
    
    √144
29. Calculate:
    
    ∛216
30. Simplify:
    
    √48
31. Simplify:
    
    3√12
32. Simplify:
    
    4√3 + 7√3
33. Simplify:
    
    √12 + √27
34. Simplify:
    
    √6 × √8
35. Rationalise:
    
    5/√2
36. Write the next three terms:
    
    7, 12, 17, 22, ...
37. Find the next two terms:
    
    3, 9, 27, 81, ...
38. Find the missing term:
    
    4, 9, 14, __, 24
39. State whether 342₅ is a valid Base 5 number.
40. Convert 10110₂ to Base 10.

🟠 Section C: Intermediate Review

41. Convert 132₄ to Base 10.
42. Convert 45₁₀ to Base 2.
43. Convert 83₁₀ to Base 5.
44. Calculate in Base 2:
    
    1011₂ + 110₂
45. Calculate in Base 5:
    
    243₅ + 132₅
46. Calculate in Base 2:
    
    11010₂ − 1011₂
47. Simplify the ratio:
    
    18 : 30
48. Divide US$420 in the ratio 3 : 4.
49. A fruit-drink mixture uses concentrate and water in the ratio 2 : 7.
    
    How much water is required for 6 litres of concentrate?
50. Five exercise books cost US$12.50. Find the cost of eight exercise books at the same rate.
51. A car travels 210 km in 3 hours. Find its average speed.
52. Write 5 600 000 in standard form.
53. Write 0.000072 in standard form.
54. Write 3.45 × 10⁵ as an ordinary number.
55. Calculate and write the answer in standard form:
    
    (3 × 10⁴)(2 × 10³)
56. Estimate:
    
    19.7 × 4.12
57. Estimate:
    
    398 ÷ 8.1
58. Round 0.006784 to two significant figures.
59. A mass is recorded as 18 kg to the nearest kilogram. Write its error interval.
60. A length is recorded as 7.4 m to one decimal place. Write its error interval.

🔴 Section D: Exam-Style Challenge

These questions require several stages or combine more than one Real Numbers skill.

61. Three traffic lights change every 30 seconds, 45 seconds and 75 seconds.
    
    They change together at 08:00:00.
    
    At what time will they next change together?
62. A school has 168 Mathematics books and 252 Science books. The books are placed into the greatest possible number of identical sets, with no books left over.
    
    a) How many sets can be formed?
    b) How many Mathematics books are in each set?
    c) How many Science books are in each set?
63. A shop increases the price of an item by 20% and later reduces the new price by 20%.
    
    The original price is US$150.
    
    Find the final price and explain why it is not US$150.
64. A sum of US$800 is invested at 6% compound interest per year. Find the value after three years.
65. A square garden has an area of 288 m². Find the exact side length in simplest surd form.
66. Simplify completely:
    
    2√75 − √27 + 3√12
67. Rationalise and simplify:
    
    12/√3
68. The terms of a pattern are:
    
    2, 5, 11, 23, 47, ...
    
    a) Describe the term-to-term rule.
    b) Find the next two terms.
69. In Base 5:
    
    243₅ + 134₅
    
    Give your answer in Base 5.
70. A number is written as 24_b and has a Base 10 value of 18. Find the base b.
71. Flour and sugar are mixed in the ratio 5 : 2. A baker uses 1.4 kg of sugar.
    
    a) Find the mass of flour required.
    b) Find the total mass of the mixture.
72. A car uses 18 litres of fuel to travel 225 km.
    
    a) Find the distance travelled per litre.
    b) Estimate how far the car can travel using 30 litres at the same rate.
73. The distance from Earth to a satellite is 3.84 × 10⁵ km.
    
    Another satellite is 7.2 × 10⁴ km away.
    
    Find the difference between the distances in standard form.
74. A rectangular field has a recorded length of 24 m and a recorded width of 15 m, each correct to the nearest metre.
    
    Find:
    a) the lower bound for the area;
    b) the upper bound for the area.
75. A value x is recorded as 6.8 correct to one decimal place.
    
    Another value y is recorded as 3.2 correct to one decimal place.
    
    Find:
    a) the lower bound of x + y;
    b) the upper bound of x + y.