Lessons / Quantitative aptitude
Quantitative aptitude · Day 6

Averages

An average is the level everything would sit at if you evened it out. Work with deviations from that level instead of adding raw numbers.

Practise (11 questions) Present as slidesCheat sheet: Summations and series

Why this topic matters

Five friends have Rs 10, 20, 30, 40 and 50. If they pool it and share equally, each gets Rs 30. The average is the 'equal share'.

Averages appear on their own and inside DI, mixtures and ages. Most questions are one identity: total = average × count.

Average = total / count, so total = average × count

The average is what everyone would have if the total were shared equally.

The key move in almost every question is to go to totals: total = average × count. Totals can be added and subtracted; averages cannot.

10A20B30C40D50E
Average 30: the bars above 30 give exactly what the bars below need
Example 1The average of 6 numbers is 24. If one number is removed, the average of the rest is 22. What number was removed?
  1. Total of 6 = 6 × 24 = 144.Go to totals.
  2. Total of 5 = 5 × 22 = 110.
  3. Removed = 144 - 110 = 34.

Someone joins or leaves

When a person joins and the average changes, compare totals. Or think of the newcomer as 'giving' or 'taking' from everyone.

A new person raises the average of 10 people by 2: the newcomer brings the old average plus 2 for each of the 11 people now in the group.

Example 1The average age of 30 students is 15. When the teacher's age is included, the average becomes 16. Teacher's age?
  1. Old total = 30 × 15 = 450.
  2. New total = 31 × 16 = 496.31 people now.
  3. Teacher = 496 - 450 = 46.

Combined averages and consecutive numbers

Two groups together: combined average = (n1 a1 + n2 a2) / (n1 + n2). It lies between the two averages, closer to the bigger group.

Consecutive numbers (or any evenly spaced list): the average is the middle term, which is also (first + last)/2.

Example 1Class A has 30 students averaging 60 marks; class B has 20 averaging 70. Combined average?
  1. Totals: 1800 and 1400.Average × count.
  2. 3200 / 50 = 64.
Example 2The average of 7 consecutive odd numbers is 27. Find the largest.
  1. The middle (4th) number is 27.Evenly spaced: the average is the middle.
  2. Largest = 27 + 3 × 2 = 33.

Formula summary

Average = sum / count
Assumed mean (deviation) method
Pick a round number, add the + and - gaps, divide by count, add back.
Someone joins, average rises by d
Newcomer = old average + (new count) × d.
Someone leaves, average rises by d
Leaver = old average - (new count) × d.
Consecutive numbers
Average = middle term = (first + last)/2.
Overlapping groups (first 6, last 6 of 11)
Middle term = sum of both groups - total.

Shortcuts and tricks

Deviation method

Use it when: Five 2- or 3-digit numbers clustered together.

  1. Assume 100: 92, 97, 105, 108, 88 → -8, -3, +5, +8, -12 = -10.
  2. -10/5 = -2 → 98.
ExampleAverage of 92, 97, 105, 108, 88
  1. 98.

New member shifts the average

Use it when: Teacher joins a class, batsman plays one more innings.

  1. Everyone, old and new, gets +d. The newcomer paid for all of it.
Example30 students average 45. With the teacher the average is 46. Teacher's age?
  1. 45 + 31 × 1.

Common mistakes

Averaging two averages when the groups have different sizes.
Weight by group size.

Before you move on, you should be able to...

  • always move to totals
  • find the age or mark of someone who joins or leaves
  • combine two groups' averages

Practice questions (11)

Try each one before opening the solution. Or practise them one by one so your score is saved.

easy Average of 5 consecutive odd numbers is 61. Largest?
  1. Middle 61, step 2.

Answer: 65

easy 40 students average 50, another 60 average 60. Combined average?
  1. (2000 + 3600)/100.

Answer: 56

easy Average weight of 8 people rises 2.5 kg when one weighing 65 kg is replaced. New person's weight?
  1. 65 + 8 × 2.5.

Answer: 85 kg

easy Average of first 50 even numbers?
  1. n + 1.

Answer: 51

medium Average of 11 numbers is 60. First six average 58, last six average 63. The 6th number?
  1. 348 + 378 - 660.

Answer: 66

medium A batsman scores 85 in his 17th innings and raises his average by 3. New average?
  1. 85 - 16 × 3.

Answer: 37

medium A batsman's average after 16 innings is 36. After the 17th his average is 38. What did he score in the 17th?
  1. Total after 16 = 576. After 17 = 646.Average × innings.
  2. 646 - 576 = 70.

Answer: 70

hard The average of 4 numbers is 42. The first is twice the second, the second is twice the third, and the fourth is 24 more than the third. Find the fourth.
  1. Third = x: 4x + 2x + x + (x + 24) = 168.
  2. 8x = 144 → x = 18.
  3. Fourth = 42.

Answer: 42

hard The average of 30 marks was 60, but an 80 was read as 50. Correct average?
  1. Total short by 30 → +1 on average.

Answer: 61

hard Replacing two men aged 20 and 24 by two women raises the average of 10 people by 2. Average age of the women?
  1. Total rises 20.
  2. Women = 44 + 20 = 64, average 32.

Answer: 32

hard A batsman averages 40 in 20 innings. His highest score beats his lowest by 100. Leaving both out, the average is 38. Highest score?
  1. 800 - 18 × 38 = 116 = H + L.
  2. H - L = 100 → H = 108.

Answer: 108

Class plan (2 hours, for trainers)

A tested order for teaching this topic in one 2-hour session. Present mode follows the same order.

10 minWarm-up.
20 minAverage as equal share; totals.
25 minRemoving, replacing, joining.
20 minCombined groups; consecutive numbers.
10 minReplacement: a wrong mark corrected.
30 minPractice set.
5 minRecap.
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