AVERAGES topic Concept and Formulae Questions Average's by Wisdom Helps Averages Topic Wisdom Helps

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Problems on — Averages

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Arithmetic Mean (A.M.) — Examples

Formula: [Arithmetic Mean = \( \bar{x} = \dfrac{\sum x_i}{n} \)]

Q1. The average of five numbers is 18. Four of them are 12, 20, 22 and 15. What is the fifth number?

  • (A) 20
  • (B) 21
  • (C) 22
  • (D) 18

Correct answer: (B) 21

Explanation (stepwise):

  1. Total sum = average × number = \(18 \times 5 = 90\). [use \( \sum x = n\bar{x} \)]
  2. Sum of four given = \(12+20+22+15 = 69\).
  3. Fifth = total sum − given sum = \(90 - 69 = 21\).
  4. Shortcut: Multiply the average by n, subtract known values.

Q2. The average score of 12 students is 75. Two students scored 90 and 80. What is the average of the remaining 10 students?

  • (A) 72
  • (B) 73
  • (C) 74
  • (D) 75

Correct answer: (B) 73

  1. Total sum = \(12 \times 75 = 900\). [\( \sum x = n\bar{x} \)]
  2. Sum of two = \(90 + 80 = 170\).
  3. Remaining sum = \(900 - 170 = 730\).
  4. Average of remaining = \(730 \div 10 = 73\).
  5. Shortcut: Remove known values from total then divide by remaining count.

Q3. What is the arithmetic mean of the first 10 natural numbers (1 to 10)?

  • (A) 5
  • (B) 5.5
  • (C) 6
  • (D) 6.5

Correct answer: (B) 5.5

  1. Sum of 1 to 10 = \( \dfrac{10\cdot(1+10)}{2} = 55 \). [Use sum of arithmetic sequence \( \dfrac{n(a_1+a_n)}{2}\) ]
  2. Average = \(55 \div 10 = 5.5\).
  3. Shortcut: For consecutive numbers average = middle value = \( (1+10)/2\).

Q4. Find the average of the numbers 3, 7, 7, 12, and 21.

  • (A) 9
  • (B) 10
  • (C) 11
  • (D) 12

Correct answer: (B) 10

  1. Sum = \(3+7+7+12+21 = 50\).
  2. Average = \(50 \div 5 = 10\). [\( \bar{x} = \dfrac{\sum x_i}{n}\)]
  3. Shortcut: Group easy pairs to add quickly (3+7=10,7+12=19,10+19+21=50).

Weighted Mean — Examples

Formula: [Weighted Mean = \( \dfrac{\sum w_i x_i}{\sum w_i} \)]

Q1. A student gets 80 in test A (weight 30%) and 70 in test B (weight 70%). What is the weighted score?

  • (A) 72
  • (B) 73
  • (C) 74
  • (D) 75

Correct answer: (B) 73

  1. Weighted = \(0.30\times80 + 0.70\times70 = 24 + 49 = 73\). [\( \dfrac{\sum w_ix_i}{\sum w_i}\) with \(\sum w_i=1\) here]
  2. Shortcut: Convert weights to decimals and multiply, or use percentages directly.

Q2. Group A: 5 students average 78. Group B: 7 students average 84. Combined average of all students?

  • (A) 80
  • (B) 81.5
  • (C) 82
  • (D) 82.5

Correct answer: (B) 81.5

  1. Total sum = \(5\times78 + 7\times84 = 390 + 588 = 978\).
  2. Combined average = \(978 \div (5+7) = 978 \div 12 = 81.5\).
  3. [Weighted mean with weights = group sizes: \( \dfrac{5\cdot78 + 7\cdot84}{5+7}\)].

Q3. A mixture has 60% of item A priced ₹50 and 40% of item B priced ₹80. What is the weighted price?

  • (A) ₹60
  • (B) ₹62
  • (C) ₹64
  • (D) ₹65

Correct answer: (B) ₹62

  1. Weighted price = \(0.6\times50 + 0.4\times80 = 30 + 32 = 62\). [\( \dfrac{\sum w_ix_i}{\sum w_i}\)]
  2. Note: Weights are by fraction (60% = 0.6).

Q4. A car travels 40% of a journey at 30 km/h and 60% at 60 km/h (percentages are by distance). The average speed for whole journey is:

  • (A) 45 km/h
  • (B) 48 km/h
  • (C) 50 km/h
  • (D) 42 km/h

Correct answer: (B) 48 km/h

  1. Since percentages are by distance, use weighted mean by distance: \(0.4\times30 + 0.6\times60 = 12 + 36 = 48\).
  2. [Weighted Mean = \( \dfrac{\sum w_iv_i}{\sum w_i}\) — here weights are distance fractions summing to 1]
  3. Important note: If distances were equal, arithmetic weight works; if times were equal, you must use harmonic mean. Always check whether weights are by distance or time.

Geometric Mean (G.M.) — Examples

Formula: [Geometric Mean of n positive numbers = \( (x_1 x_2 \dots x_n)^{1/n} \)]

Q1. The geometric mean of 8 and 18 is:

  • (A) 10
  • (B) 12
  • (C) 13
  • (D) 14

Correct answer: (B) 12

  1. G.M. = \( \sqrt{8\times18} = \sqrt{144} = 12\). [\( (x_1x_2)^{1/2}\)]
  2. Shortcut: Multiply the two numbers and take square root.

Q2. G.M. of 2, 8 and 32 is:

  • (A) 6
  • (B) 7
  • (C) 8
  • (D) 9

Correct answer: (C) 8

  1. Product = \(2\times8\times32 = 512\).
  2. G.M. = \(512^{1/3} = 8\). [\( (x_1x_2x_3)^{1/3}\)]

Q3. The geometric mean of two positive numbers is 15. If one number is 9, the other is:

  • (A) 20
  • (B) 24
  • (C) 25
  • (D) 27

Correct answer: (C) 25

  1. G.M. = \( \sqrt{9\times x} = 15\) → \(9x = 225\) → \(x = 225/9 = 25\).
  2. [Use \(x = \dfrac{(\text{G.M.})^2}{\text{given}}\) when solving for one value with two numbers.]

Q4. The geometric mean of 4 and 64 is:

  • (A) 14
  • (B) 16
  • (C) 18
  • (D) 20

Correct answer: (B) 16

  1. G.M. = \( \sqrt{4\times64} = \sqrt{256} = 16\).
  2. Shortcut: Recognize powers of 2: \(4=2^2,\,64=2^6\) → G.M. \(=2^{(2+6)/2}=2^4=16\).

Harmonic Mean (H.M.) — Examples

Formula: [Harmonic Mean = \( H = \dfrac{n}{\sum_{i=1}^n \dfrac{1}{x_i}} \)]

Q1. The harmonic mean of 3 and 6 is:

  • (A) 3
  • (B) 4
  • (C) 4.5
  • (D) 5

Correct answer: (B) 4

  1. H = \( \dfrac{2}{\frac{1}{3}+\frac{1}{6}} = \dfrac{2}{\frac{1}{2}} = 4\).
  2. [Use \(H = \dfrac{n}{\sum 1/x_i}\).]

Q2. H.M. of 4, 6 and 12 equals:

  • (A) 5
  • (B) 6
  • (C) 7
  • (D) 8

Correct answer: (B) 6

  1. Compute reciprocals: \(1/4 + 1/6 + 1/12 = 0.25 + 0.16666... + 0.08333... = 0.5\).
  2. H = \(3 \div 0.5 = 6\). [\(H = \dfrac{n}{\sum 1/x_i}\)]

Q3. A car travels equal distances at 40 km/h and 60 km/h. The average speed for the whole trip is:

  • (A) 45 km/h
  • (B) 48 km/h
  • (C) 50 km/h
  • (D) 52 km/h

Correct answer: (B) 48 km/h

  1. For equal distances, use harmonic mean of speeds: \(H = \dfrac{2}{\frac{1}{40}+\frac{1}{60}} = \dfrac{2}{0.025+0.0166667} \approx 48\).
  2. [Harmonic mean is used when averaging rates over equal quantities (e.g., equal distances).]
  3. Shortcut: For two speeds \(v_1, v_2\) over equal distance, \(H = \dfrac{2v_1v_2}{v_1+v_2}\).

Q4. H.M. of 2 and 3 is:

  • (A) 2
  • (B) 2.4
  • (C) 2.5
  • (D) 3

Correct answer: (B) 2.4

  1. H = \( \dfrac{2}{\frac{1}{2}+\frac{1}{3}} = \dfrac{2}{\frac{5}{6}} = \dfrac{12}{5} = 2.4\).
  2. [Use \(H = \dfrac{n}{\sum 1/x_i}\).]

Median — Examples

Definition: The median is the middle value after sorting. [If n odd → middle; if n even → average of two middle values.]

Q1. Median of the numbers 7, 3, 9, 5 and 1 is:

  • (A) 3
  • (B) 5
  • (C) 7
  • (D) 9

Correct answer: (B) 5

  1. Sort: 1, 3, 5, 7, 9 → middle (3rd) = 5.
  2. [Median rule: odd n → element at position \( (n+1)/2\).]

Q2. Median of 4, 8, 6, 10, 2, 14, 9, 12 is:

  • (A) 8
  • (B) 8.5
  • (C) 9
  • (D) 9.5

Correct answer: (B) 8.5

  1. Sort: 2,4,6,8,9,10,12,14. For n=8 (even), median = average of 4th and 5th = \( (8+9)/2 = 8.5\).
  2. [Even n → median = average of positions \(n/2\) and \(n/2+1\).]

Q3. Median of data 5, 5, 7, 7, 7, 8, 9 is:

  • (A) 6
  • (B) 7
  • (C) 7.5
  • (D) 8

Correct answer: (B) 7

  1. n = 7 (odd). Sorted list already shown; middle is 4th element = 7.
  2. [Median for odd n is element at position \((n+1)/2\).]

Q4. Median of 2, 3, 3, 5, 6, 8, 11, 13, 13, 14 is:

  • (A) 6.5
  • (B) 7
  • (C) 7.5
  • (D) 8

Correct answer: (B) 7

  1. n = 10 (even). Middle two positions = 5th and 6th values: 6 and 8.
  2. Median = \( (6+8)/2 = 7\).

Mode — Examples

Definition: Mode is the most frequent value(s) in the dataset. (There can be one mode, more than one, or none.)

Q1. Mode of 2, 3, 3, 4, 5 is:

  • (A) 2
  • (B) 3
  • (C) 4
  • (D) 5

Correct answer: (B) 3

  1. 3 appears twice, others once → mode = 3.
  2. Shortcut: Tally frequencies; highest frequency value → mode.

Q2. Mode of 5, 7, 7, 8, 8, 8, 9 is:

  • (A) 5
  • (B) 7
  • (C) 8
  • (D) 9

Correct answer: (C) 8

  1. 8 appears three times (most frequent) → mode = 8.

Q3. Mode of 1, 2, 2, 3, 4, 2, 5 is:

  • (A) 1
  • (B) 2
  • (C) 3
  • (D) 4

Correct answer: (B) 2

  1. 2 appears three times → mode = 2.

Q4. Mode of 10, 12, 11, 12, 13, 12, 14 is:

  • (A) 11
  • (B) 12
  • (C) 13
  • (D) 14

Correct answer: (B) 12

  1. 12 appears three times; others appear once → mode = 12.

Notes & Tips for teaching/quick revision:

  • Always identify what is being averaged (values, rates, distances, or weights) — that decides whether to use A.M., W.M., or H.M.
  • Keep formulas visible on the board:
    [A.M. = \( \dfrac{\sum x}{n}\)], [W.M. = \( \dfrac{\sum w_ix_i}{\sum w_i}\)], [H.M. = \( \dfrac{n}{\sum 1/x_i}\)], [G.M. = \( (\prod x_i)^{1/n}\)].
  • Shortcuts: use \(n\bar{x} = \sum x\) to move between sum and average quickly.
  • For competitive exams, practise recognising whether data are grouped, paired, or weighted — that saves time in selecting the right formula.
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