Averages

Average = total ÷ how many. Rearranged, total = average × how many — and that second form is what actually solves the paper. Whenever an average changes, go back to the totals.

Four types cover the topic:

1. Consecutive numbers. The average is the middle term, i.e. (first + last)/2. No addition needed.
2. A misread value. The total changes by correct − wrong, so the average changes by that difference divided by how many.
3. Two groups combined. (n₁a₁ + n₂a₂) ÷ (n₁ + n₂) — a weighted average.
4. A member joins or leaves. New member's value = new total − old total.

The single most common mistake is type 3: taking the plain mean of the two averages. That is only correct when both groups are the same size. If 40 boys average 60 and 10 girls average 80, the class average is much nearer 60 than 70 — the bigger group pulls harder.

Watch out for average speed too: for equal distances it is the harmonic mean 2xy/(x+y), never (x+y)/2, because more time is spent at the slower speed.

Average = total/n · total = average × n · combined = (n₁a₁ + n₂a₂)/(n₁ + n₂) · consecutive → (first + last)/2 · equal distances → 2xy/(x+y)

The average of 10 numbers is 42. One number was read as 26 instead of 62. Find the correct average. → The total was too small by 62 − 26 = 36, so the average was too small by 36 ÷ 10 = 3.6. Correct average = 42 + 3.6 = 45.6. Sanity check: old total 420, correct total 456, and 456 ÷ 10 = 45.6.