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Average — The One Number That Tells the Whole Story

Chapter Outline & Learning Goals

By the end of this chapter, you will be able to:

1. Define average and compute it for any set of numbers.
2. Use the assumed average shortcut for fast calculation.
3. Solve problems on change in average when members join or leave a group.
4. Handle weighted averages (mixtures of two groups).
5. Crack average speed problems (the biggest trap in exams).
6. Apply averages to ages, marks, runs, and consecutive numbers.

The Hook: The Cricketer Who "Improved" by Scoring Zero

A batsman plays 10 innings and averages 40 runs. In his 11th innings, he gets out for a duck — zero runs. His average *falls*. But here's the strange part: if he had scored exactly 40, his average would have stayed the same. Score 41, and it rises — but by less than 1 run.

One number, zero, dragged down the work of ten innings. That is the power — and the danger — of the average. Competitive exams (SSC CGL, IBPS PO, RRB NTPC) exploit this exact behaviour again and again. Master it, and average questions become some of the fastest marks in the paper.

1. What Is an Average?

The average (arithmetic mean) of a set of numbers is the equal share each member would get if the total were distributed equally.

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

If 4 friends have ₹10, ₹20, ₹30, ₹40, the total is ₹100. Shared equally, each gets ₹25. That ₹25 is the average — notice that nobody actually has ₹25. The average is a representative, not a real member.

Key properties to remember:

• The average always lies between the smallest and largest value.
• If every value increases by *k*, the average increases by *k*.
• If every value is multiplied by *k*, the average is multiplied by *k*.
• Sum = Average × Number of terms (this reverse formula solves half of all exam questions).

🧠 Active Recall: Without looking back — if the average of 6 numbers is 15, what is their sum? If each number is doubled, what happens to the average?

2. The Assumed Average Trick (Your Speed Weapon)

Adding 10 numbers like 47, 52, 49, 51, 48… is slow. Instead, assume an average — say 50 — and work only with deviations:

• Deviations: −3, +2, −1, +1, −2 → net deviation = −3
• Correction = −3 ÷ 5 = −0.6
• True average = 50 − 0.6 = 49.4

You never added the big numbers at all. In exams, pick a round number near the values, and the arithmetic becomes mental.

🧠 Active Recall: Find the average of 61, 64, 59, 62, 60 using 60 as the assumed average. (Answer: net deviation +6, so 60 + 6/5 = 61.2)

3. When Members Join or Leave

This is the most-tested pattern. The golden rule:

> New member's value = Old average ± (change in average × new number of members)

Example: The average weight of 8 people is 50 kg. A new person joins and the average becomes 51 kg. New person's weight?

• Total increase = 51 × 9 − 50 × 8 = 459 − 400 = 59 kg
• Shortcut: 50 + (1 × 9) = 59 kg. The newcomer must carry his own 50 *plus* 1 kg extra for each of the 9 people.

Replacement pattern: If a person of weight *x* is replaced and the average rises by *k* in a group of *n*, the new person weighs x + nk.

🧠 Active Recall: Average age of 11 players is 22. A player aged 28 is replaced and the average drops by 1. How old is the replacement? (Answer: 28 − 11 = 17)

4. Weighted Average — When Groups Merge

Two classes: Class A has 20 students averaging 60 marks; Class B has 30 students averaging 70 marks. Combined average is not 65 — the bigger group pulls harder.

$$\bar{x} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2} = \frac{20(60) + 30(70)}{50} = \frac{3300}{50} = 66$$

Alligation shortcut: The combined average divides the gap (60 to 70) in the inverse ratio of group sizes. The distances from the two averages are in the ratio 30 : 20 = 3 : 2, so the combined average sits 3/5 of the way from 60 (equivalently, 2/5 of the way from 70): 60 + (3/5 × 10) = 60 + 6 = 66. Same answer, five seconds.

🧠 Active Recall: 15 workers earn an average of ₹400, 5 earn an average of ₹600. Combined average? (Answer: ₹450)

5. Average Speed — The Classic Trap

A car goes from A to B at 40 km/h and returns at 60 km/h. Average speed is not 50 km/h! The car spends *more time* travelling slowly, so the true average is lower.

For equal distances:

$$\text{Average speed} = \frac{2xy}{x+y} = \frac{2 \times 40 \times 60}{100} = 48 \text{ km/h}$$

This is the harmonic mean of the two speeds. Remember: average speed = total distance ÷ total time — never the average of speeds (unless the *times* are equal).

🧠 Active Recall: A train covers half a journey at 30 km/h and half at 70 km/h. Average speed? (Answer: 2×30×70/100 = 42 km/h)

6. Averages of Consecutive Numbers

• Average of first *n* natural numbers = (n + 1)/2
• Average of first *n* even numbers = n + 1
• Average of first *n* odd numbers = n
• Average of any set of consecutive numbers = (first + last)/2 — it's simply the middle term.

Example: Average of 1 to 99 = (1 + 99)/2 = 50. Done in one line.

🃏 Memory Trick: "The Generous Newcomer"

Imagine the average as a party where everyone must hold the same number of balloons. When a newcomer arrives and the average rises by 1, picture him walking in and handing 1 balloon to every single person — including himself. So he must have brought: (old average) + (1 × new count) balloons. If the average *falls*, he's a balloon thief — he takes from everyone. This one image solves every joining/leaving/replacement question without memorising formulas.

Summary Table

🔁 Review Reminder

Review this chapter after 2 days. Redo the Active Recall questions from memory, and re-derive the "Generous Newcomer" logic once. Then attempt Practice Set 1 — if you score above 80%, move to the harder sets; if not, re-read Sections 3 and 5 first.