Averages

Average = sum of values ÷ number of values. The average is the "level" you'd get if all values were equal.

Core identities:

• Sum = average × count.
• Average of first n natural numbers = (n+1)/2.
• Average of first n even numbers = n+1; first n odd numbers = n.
• Average of squares of first n naturals = (n+1)(2n+1)/6.

The "shift" rule (most useful):
If average of n numbers is A, and a new number x is added, the new average is (nA + x)/(n+1). The change in average per existing member is (x − A)/(n+1).

Replacement: if one number a is replaced by b in a set of n numbers with old average A, new average = A + (b − a)/n.

Example: average weight of 10 students is 45 kg. One student of weight 50 kg leaves and a new student joins, making the average 44 kg. Weight of new student = 50 + (44 − 45) × 10 = 50 − 10 = 40 kg.

Weighted average: when sub-groups have different sizes, weight each average by its group size.
Average mark of 30 boys = 60, of 20 girls = 70. Class average = (30×60 + 20×70)/50 = (1800 + 1400)/50 = 64.

Average speed for two-way trip (same distance both ways): use harmonic mean, not arithmetic mean!
Avg speed = 2xy/(x+y) where x and y are the two speeds. E.g. 40 km/h going and 60 km/h returning ⟹ avg = 2·40·60/100 = 48 km/h (not 50).

Age problems:

• Average age of family stays the same when a member of average age leaves or joins.
• If average increases by Δ when a new member of age x joins, x = (old average) + (new count) × Δ.

Example 1 — Innings cricket pattern:
A batsman's average after 16 innings is 27. After the 17th innings his average becomes 29. What did he score in the 17th innings?
Method: New total − Old total = 17 × 29 − 16 × 27 = 493 − 432 = 61 runs.

Shortcut: Score = new avg + (increase × old count) = 29 + 2×16 = 61.

Example 2 — Average of replacement:
Average age of 20 boys is 16 yrs. The teacher (age 36) joins. New average = (20×16 + 36)/21 = 356/21 ≈ 16.95 yrs.

Example 3 — Two-stream average:
A factory has 200 male workers with avg salary ₹15,000 and 100 female workers avg ₹18,000. Overall avg = (200×15000 + 100×18000)/300 = (30,00,000 + 18,00,000)/300 = ₹16,000.

Example 4 — Speed (two-way trip):
Cycle to college at 12 km/h, return at 18 km/h. Average speed = 2·12·18/30 = 14.4 km/h.

Example 5 — Family ages:
Average age of a family of 5 is 25 years. If the youngest is 7 years old, what was the family average 7 years ago?
Trick: 7 years ago, the youngest didn't exist, so average is of the other 4 members. Each of them was 7 years younger.
Total now = 125. Excluding youngest = 125 − 7 = 118. 7 years ago = 118 − 4×7 = 90, divided by 4 = 22.5 years.

Common trap: students average the two speeds arithmetically (= (12+18)/2 = 15) on a two-way trip. Always use harmonic mean for "same distance" cases. For "same time" cases, arithmetic mean is correct.