Mathematics Practice Problems - Fractions, Percentages, Scientific Notation, and Expanding Brackets

Practice various mathematics topics such as fractions, percentages, scientific notation, and expanding brackets with sample problems and solutions.

• Mathematics
• Fractions
• Percentages
• Scientific Notation
• Algebra

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1. Scientific Notation (You can use a calculator for questions with this symbol) A grain of sand weighs 6.45 x 10-5 kilograms. A jar holds 4 x 106 grains of sand. Calculate the weight of the sand in the jar. Answer: 258kg

2. Scientific Notation There are approximately 7.9 x 109 people on Earth. Roughly 8% of people have blue eyes. Estimate the number of people on Earth who have blue eyes. Write your answer in scientific notation. Answer: 6.32 x 108

3. Fractions Calculate 240 360 3.14 30. Answer: 62.8

4. Fractions Calculate 31 2+ 22 9 . 103 18 or 5 13 Answer: 18

5. Fractions Calculate 51 4 24 5 . 49 20 or 2 9 Answer: 20

6. Fractions Calculate 21 3 14 5 . 21 5 Answer:

7. Fractions Calculate 91 2 33 8 . 76 27 Answer:

8. Fractions Calculate2 5(13 5+ 21 2). 41 25 Answer:

9. Percentages Calculate 12.5% of 56. Answer: 7

10. Percentages A group of friends go out for lunch. The total of their bill with a 12% tip comes to 76.16. Find the cost of their bill before the tip was added. Answer: 68

11. Percentages Lisa buys a cottage in 2018 valued at 82,000. The value of the cottage appreciated by 3.8% per annum. What was the value of the cottage in 2021? Round your answer to 2 significant figures. Answer: 92,000

12. Expanding Brackets Expand and simplify (3x + 1)(2x 7). Answer: 6x2 19x - 7

13. Expanding Brackets Expand and simplify (2x + 3)(x2 5x + 6). Answer: 2x3 7x2 3x + 18

14. Expanding Brackets Expand and simplify 4(2x + 5) 3(2x + 7). Answer: 2x - 1

15. Factorising Factorise 12p2 + 66pq. Answer: 6p(2p + 11q)

16. Factorising Factorise 9n2 - 100. Answer: (3n + 10)(3n 10)

17. Factorising Factorise x2 + 5x - 24. Answer: (x + 8)(x 3)

18. Factorising Factorise 3x2 + 14x + 8. Answer: (3x + 2)(x + 4)

19. Straight Lines Find the equation of the line joining the points (-5,2) and (1,4). Answer: y = 1 3x + 11 3

20. Straight Lines Find the equation of the line from the diagram below: Answer: y = -4x - 3

21. Surds Simplify 75 + 2 3 27. Answer: 4 3

22. Indices ?7 ?2 ?3 . Simplify Answer: c6

23. Indices Expand and simplify (5y4)2. Answer: 25y8

24. Indices Simplify, writing your answer with a positive index: ? 5 1 2 2 ? 1 ?2 Answer:

25. Arcs & Sectors Calculate the length of the major arc AB: A 1150 B O 6cm Answer: 25.7cm

26. Arcs & Sectors The length of major arc AB is 108cm. The radius of the sector is 20cm. Find the size of the shaded angle. 20cm A B Answer: 3090

27. Volume An object is made of a cone sitting on top of a cylinder, as shown: 20cm 8cm Answer: 1850cm3 14cm Find the total volume of this object to 3 significant figures.

28. Volume An object is made of a cylinder with a hemispherical indent as shown: 8cm 9cm Answer: 573cm3 10cm Find the total volume of this object to 3 significant figures.

29. Completing the Square Write x2 + 14x + 50 in the form (x + p)2 + q. Answer: (x + 7)2 + 1

30. Pythagoras Theorem In the diagram below, radius OB = 9cm and length AB = 14cm. h O B A Calculate the length of h to 1 decimal place. Answer: 14.7cm

31. Pythagoras Theorem 8cm 15cm 17cm Is the triangle above right-angled? Justify your answer. Answer: Yes since 172 = 152 + 82

32. Pythagoras Theorem A 12cm 7cm B 20cm Calculate the length of the space diagonal AB. Answer: 24.4cm

33. Angle Properties A B 88 O D 52o C Calculate the size of the shaded angle ABC. Answer: 840

34. Angle Properties C A 41 B 46 O D Calculate the size of the shaded angle COD. Answer: 1420

35. Angle Properties D E C F B A ABCDE is a regular pentagon. Find the size of the shaded angle EFA. Answer: 360

36. Algebraic Fractions 14 9?2. 8 Simplify 3? 12? 7 Answer:

37. Algebraic Fractions 3? 2?+7 2?+7 3? 1. Simplify 3? 3? 1 Answer:

38. Algebraic Fractions ?+5as a single fraction in its simplest form. 5 3 Write ?+2 2?+19 ?+2 ?+5 Answer:

39. Equations and Inequalities Solve the inequality 15 - 3(2x - 4) < 25. Answer: ? >1 3

40. Equations and Inequalities 4. Solve the equation 2? 3 2 =? 4 Answer: x = 12 5

41. Changing the Subject of a Formula Change the subject of this formula to n: ? =3?2 ? ?? 3 Answer: n =

42. Simultaneous Equations Solve the system of simultaneous equations: 2x + 4y = -2 5x 3y = 21 Answer: x = 3, y = -2

43. Simultaneous Equations The cost of 10 textbooks and 8 boxes of pens is 105. The cost of 4 textbooks and 10 boxes of pens is 59. Create an equation each for the information above and then find the cost of one textbook and of one box of pens. Answer: one textbook costs 8.50, one box of pens costs 2.50

44. Further Trigonometry Find the size of length PQ. P 24cm 750 480 Q R Answer: 18.5cm

45. Further Trigonometry Find the size of angle PQR. P 44cm 930 R 53cm Q Answer: 560

46. Further Trigonometry Find the size of length PQ. Q P 7cm 8.5cm 870 R Answer: 10.7cm

47. Further Trigonometry Find the size of angle RPQ. P 22cm 12cm Q 16cm R Answer: 450

48. Further Trigonometry Find the area of triangle PQR. P 14cm 99o 18cm R Q Answer: 124.4cm2

49. Further Trigonometry Find the size of length SR. P 6cm S 170 210 Q R Answer: 5.8cm

50. Statistics Find the mean and standard deviation of this data set: 27 25 32 28 Answer: mean = 28, s.d. = 2.9

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