Average, Mixture & Alligation • Topic 1 of 4

Average Basics

Average = sum / count, so sum = average x count. When one value changes, the total changes by the same amount, and the average shifts by (change / count). The 'replacement' pattern — a person of weight A leaves and one of weight B joins, changing the average by d for n people — gives B - A = n x d. Always think in totals, not averages, when something is added or removed.

✅ Solved examples

1. Average of 5 numbers is 18. Their sum?
Sum = 18 x 5 = 90.
2. Average of 10 numbers is 25; one number 30 is removed. New average?
Old sum 250; remove 30 -> 220 over 9 = 24.44.
3. A batsman's average in 16 innings is 36. To raise it to 38 in the 17th, what must he score?
Old total 576; new total 38 x 17 = 646; required = 70.
4. The average weight of 8 people rises by 2.5 kg when a 56 kg person is replaced by a new one. New person weight?
Increase in total = 8 x 2.5 = 20; new = 56 + 20 = 76 kg.

✏️ Practice — try these, take hints as needed

1. Average of 7 numbers is 12. Sum?
Sum = avg x n.
12 x 7.
—
84
2. Average of 6 numbers is 20; add a 27. New average?
Old sum 120.
+27 = 147.
/7.
21
3. Average of first 10 natural numbers?
Sum 55.
/10.
—
5.5
4. Average age of 30 students is 14; teacher added makes 15. Teacher age?
Old total 420.
New total 31 x 15 = 465.
Difference.
45
5. Average of 9 results is 50; first 4 average 45, last 4 average 52. Fifth result?
Total 450.
First4 180, last4 208.
450 - 388.
62

📝 Topic test — 8 questions

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