Average: Detailed Notes, Examples, and Practice Questions

 

Average: Detailed Notes, Examples, and Practice Questions

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1. Simple Average (Arithmetic Mean)

Definition

The simple average (arithmetic mean) is the sum of all values divided by the number of values.

Formula

$$\text{Average}=\frac{\text{Sum of all values}}{\text{Number of values}}$$

Example

If the ages of 5 children are 8, 10, 12, 14, 16,

$$\text{Average age}=\frac{8+10+12+14+16}{5}=\frac{60}{5}=12$$

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2. Weighted Average

Definition

A weighted average is an average where each value is multiplied by a weight reflecting its importance or frequency. It is especially useful when combining groups with different sizes or importance23467.

Formula

$$\text{Weighted Average}=\frac{\sum _{i=1}^n{w}_i{x}_i}{\sum _{i=1}^n{w}_i}$$

Where:

• $x_i$ = value of each observation

• $w_i$ = weight of each observation

Steps to Calculate

1. Multiply each value by its weight.

2. Add all the products.

3. Divide by the sum of the weights6.

Example 1 (Byjus, Vedantu, Cuemath)

A class has 10 students with an average score of 80, and 15 students with an average of 60.
Weighted average = $\frac{10\times 80+15\times 60}{10+15}=\frac{800+900}{25}=\frac{1700}{25}=68$25.

Example 2

Data: 4, 7, 5, 9 with weights 1, 2, 3, 2
Weighted average = $\frac{4\times 1+7\times 2+5\times 3+9\times 2}{1+2+3+2}=\frac{4+14+15+18}{8}=\frac{51}{8}=6.375$2.

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3. Applications in Real-Life Problems

A. Grouped Data

• Used when different groups have different sizes (e.g., average marks of two classes).

B. Speed

• Average speed = Total distance / Total time (not the mean of speeds if time/distance varies).

C. Data Interpretation

• Used in analyzing survey results, grades, financial data, etc.

D. Mixtures

• Used to find the average value when mixing items of different prices or concentrations.

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4. Common Mistakes to Avoid

• Do not use the simple average when the groups have different sizes.

• For rates (like speed), always use total distance/total time, not the mean of the speeds.

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5. Examples

Simple Average Example:
Find the average of 12, 15, 18, 20, 25.

$$\frac{12+15+18+20+25}{5}=\frac{90}{5}=18$$

Weighted Average Example:
Section A: 20 students, average marks = 60
Section B: 30 students, average marks = 80
Combined average = $\frac{20\times 60+30\times 80}{20+30}=\frac{1200+2400}{50}=\frac{3600}{50}=72$

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6. Practice Questions (with answers for the first 20)

Questions 1–20 (with answers)

1. Find the average of 8, 10, 12, 14, 16.
   Ans: 12

2. Find the average of 25, 30, 35, 40.
   Ans: 32.5

3. Find the average of 5 numbers: 20, 25, 30, 35, 40.
   Ans: 30

4. The average of 6, 8, 10, 12 is ___.
   Ans: 9

5. The average of 10, 20, 30 is ___.
   Ans: 20

6. Section A: 15 students, average marks = 70. Section B: 25 students, average marks = 80. Find the overall average.
   Ans: $\frac{15\times 70+25\times 80}{40}=\frac{1050+2000}{40}=\frac{3050}{40}=76.25$

7. A batsman scored 20, 35, 40, 25, 30 runs in 5 innings. Find his average score.
   Ans: 30

8. Find the average of first 5 even numbers.
   Ans: (2 + 4 + 6 + 8 + 10)/5 = 6

9. The average weight of 4 children is 30 kg. The total weight is ___.
   Ans: 120 kg

10. If the average of 5 numbers is 18, what is their total sum?
    Ans: 90

11. Section A: 10 students, average = 60. Section B: 20 students, average = 75. Find combined average.
    Ans: $\frac{10\times 60+20\times 75}{30}=\frac{600+1500}{30}=\frac{2100}{30}=70$

12. Find the average of 12, 15, 18, 21, 24.
    Ans: 18

13. The average of 7, 14, 21, 28, 35 is ___.
    Ans: 21

14. Find the average of 9, 11, 13, 15, 17.
    Ans: 13

15. Section X: 12 students, average = 50. Section Y: 18 students, average = 80. Find overall average.
    Ans: $\frac{12\times 50+18\times 80}{30}=\frac{600+1440}{30}=\frac{2040}{30}=68$

16. The average of 3 numbers is 24. Two numbers are 18 and 27. Find the third number.
    Ans: Total = 24 × 3 = 72; Third = 72 – (18 + 27) = 27

17. The average of 4 numbers is 15. If one number is 12, what is the sum of the other three?
    Ans: Total = 60; Sum of other three = 60 – 12 = 48

18. The average age of 5 students is 16 years. If one student leaves, the average becomes 15. Find the age of the student who left.
    Ans: Total age = 80; Remaining = 60; Age left = 20

19. The average of 5 consecutive numbers is 25. Find the numbers.
    Ans: 23, 24, 25, 26, 27

20. The average of 6 numbers is 12. If one number is 18, what is the average of the remaining 5?
    Ans: Total = 72; Remaining = 54; Average = 10.8

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Questions 21–100 (Practice Only)

21. Find the average of 15, 20, 25, 30, 35.

22. Find the average of 7, 14, 21, 28.

23. The average of 3, 6, 9, 12, 15 is ___.

24. Find the average of 11, 13, 15, 17, 19.

25. Section A: 8 students, average = 50. Section B: 12 students, average = 70. Find combined average.

26. Find the average of first 6 odd numbers.

27. The average of 5 numbers is 40. What is their total sum?

28. If the average of 4 numbers is 18, what is their sum?

29. Section A: 25 students, average = 60. Section B: 35 students, average = 80. Find overall average.

30. Find the average of 2, 4, 6, 8, 10, 12.

31. The average of 10, 20, 30, 40, 50 is ___.

32. Find the average of 9, 12, 15, 18, 21.

33. Section X: 30 students, average = 65. Section Y: 20 students, average = 75. Find combined average.

34. The average of 8, 16, 24, 32 is ___.

35. The average of 5 numbers is 22. If one number is 32, what is the average of the other four?

36. The average of 6 numbers is 18. If one number is 24, what is the average of the other five?

37. The average age of 7 students is 14. If one leaves, the average becomes 13. Find the age of the student who left.

38. The average of 5 consecutive numbers is 31. Find the numbers.

39. The average of 4 numbers is 20. If one number is 28, what is the sum of the other three?

40. Find the average of 13, 17, 21, 25, 29.

41. Find the average of 5, 10, 15, 20, 25.

42. Section A: 16 students, average = 60. Section B: 24 students, average = 80. Find overall average.

43. The average of 7, 14, 21, 28, 35, 42 is ___.

44. The average of 10, 12, 14, 16, 18 is ___.

45. Find the average of 3, 6, 9, 12, 15, 18.

46. Section X: 40 students, average = 75. Section Y: 60 students, average = 85. Find combined average.

47. The average of 8, 16, 24, 32, 40 is ___.

48. The average of 5 numbers is 30. If one number is 40, what is the average of the other four?

49. The average of 6 numbers is 24. If one number is 36, what is the average of the other five?

50. The average age of 8 students is 15. If one leaves, the average becomes 14. Find the age of the student who left.

51. The average of 5 consecutive numbers is 41. Find the numbers.

52. The average of 4 numbers is 30. If one number is 38, what is the sum of the other three?

53. Find the average of 17, 19, 21, 23, 25.

54. Find the average of 20, 30, 40, 50, 60.

55. Section A: 18 students, average = 65. Section B: 22 students, average = 85. Find combined average.

56. Find the average of first 7 even numbers.

57. The average of 6 numbers is 36. What is their total sum?

58. If the average of 5 numbers is 22, what is their sum?

59. Section A: 12 students, average = 70. Section B: 18 students, average = 90. Find overall average.

60. Find the average of 3, 6, 9, 12, 15, 18.

61. The average of 15, 30, 45, 60, 75 is ___.

62. Find the average of 8, 16, 24, 32, 40, 48.

63. Section X: 15 students, average = 55. Section Y: 25 students, average = 75. Find combined average.

64. The average of 12, 24, 36, 48 is ___.

65. The average of 5 numbers is 28. If one number is 38, what is the average of the other four?

66. The average of 6 numbers is 30. If one number is 36, what is the average of the other five?

67. The average age of 9 students is 18. If one leaves, the average becomes 17. Find the age of the student who left.

68. The average of 5 consecutive numbers is 51. Find the numbers.

69. The average of 4 numbers is 40. If one number is 48, what is the sum of the other three?

70. Find the average of 21, 23, 25, 27, 29.

71. Find the average of 10, 20, 30, 40, 50, 60.

72. Section A: 20 students, average = 80. Section B: 30 students, average = 90. Find overall average.

73. The average of 9, 18, 27, 36, 45, 54 is ___.

74. The average of 14, 16, 18, 20, 22 is ___.

75. Find the average of 5, 10, 15, 20, 25, 30.

76. Section X: 50 students, average = 85. Section Y: 50 students, average = 95. Find combined average.

77. The average of 8, 16, 24, 32, 40, 48, 56 is ___.

78. The average of 5 numbers is 36. If one number is 46, what is the average of the other four?

79. The average of 6 numbers is 42. If one number is 54, what is the average of the other five?

80. The average age of 10 students is 16. If one leaves, the average becomes 15. Find the age of the student who left.

81. The average of 5 consecutive numbers is 61. Find the numbers.

82. The average of 4 numbers is 50. If one number is 58, what is the sum of the other three?

83. Find the average of 23, 25, 27, 29, 31.

84. Find the average of 30, 40, 50, 60, 70.

85. Section A: 28 students, average = 75. Section B: 32 students, average = 85. Find combined average.

86. Find the average of first 8 odd numbers.

87. The average of 7 numbers is 56. What is their total sum?

88. If the average of 5 numbers is 32, what is their sum?

89. Section A: 16 students, average = 80. Section B: 24 students, average = 100. Find overall average.

90. Find the average of 4, 8, 12, 16, 20, 24.

91. The average of 20, 40, 60, 80, 100 is ___.

92. Find the average of 12, 24, 36, 48, 60, 72.

93. Section X: 18 students, average = 65. Section Y: 22 students, average = 85. Find combined average.

94. The average of 14, 28, 42, 56 is ___.

95. The average of 5 numbers is 38. If one number is 48, what is the average of the other four?

96. The average of 6 numbers is 48. If one number is 54, what is the average of the other five?

97. The average age of 12 students is 19. If one leaves, the average becomes 18. Find the age of the student who left.

98. The average of 5 consecutive numbers is 71. Find the numbers.

99. The average of 4 numbers is 60. If one number is 68, what is the sum of the other three?

100. Find the average of 31, 33, 35, 37, 39.

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Practice these questions to master simple and weighted averages, and their applications in real-life problems and data interpretation.