Averages & Mixtures • Topic 2 of 4

Weighted Average

When the groups being combined are different sizes, you cannot just average the averages — you must weight each average by its group size: combined = (n₁a₁ + n₂a₂)/(n₁ + n₂). A class of 30 averaging 70 combined with a class of 10 averaging 90 does not average 80; it averages (30·70 + 10·90)/40 = (2100 + 900)/40 = 75, pulled toward the bigger (lower-scoring) group. The key intuition — and a frequent Quantitative Comparison hook — is that the weighted mean always lands between the two averages, closer to the group with more weight. This same structure appears everywhere on the GRE: GPA (grades weighted by credit hours), average speed (speeds weighted by time), portfolio returns, and blended prices. Whenever a problem gives you two averages and two different counts, reach for the weighted formula, not the plain midpoint.

✅ Solved examples

1. A class of 30 averages 70; a class of 10 averages 90. Combined average?
(30·70 + 10·90)/(30 + 10) = (2100 + 900)/40 = 3000/40 = 75.
2. A student scores 80 on a test worth 40% and 90 on one worth 60%. Weighted average?
0.40·80 + 0.60·90 = 32 + 54 = 86.
3. Quantitative Comparison. Group X (20 people) averages 50; group Y (5 people) averages 80. Quantity A: the combined average. Quantity B: 65. Which is greater?
Combined = (20·50 + 5·80)/25 = (1000 + 400)/25 = 1400/25 = 56. Quantity B (65) is greater — the average leans toward the larger group X.
4. 4 kg of nuts at $6/kg is mixed with 6 kg at $9/kg. Average price per kg?
(4·6 + 6·9)/(4 + 6) = (24 + 54)/10 = 78/10 = $7.80.

✏️ Practice — try these, take hints as needed

1. A group of 12 averages 15 and a group of 8 averages 25. Combined average?
Weight by group size.
(12·15 + 8·25)/20.
(180 + 200)/20.
19
2. Marks: 60 with weight 2 and 90 with weight 1. Weighted average?
(2·60 + 1·90)/(2+1).
(120 + 90)/3.
210/3.
70
3. A course grade is 30% homework (score 95) and 70% exam (score 75). Final grade?
0.30·95 + 0.70·75.
28.5 + 52.5.
Add.
81
4. 3 litres of 20% juice is blended with 2 litres of 45% juice. Concentration of the blend?
Weighted by volume.
(3·20 + 2·45)/5.
(60 + 90)/5.
30%
5. Two groups of 25 and 75 average 40 and 60 respectively. Combined average?
(25·40 + 75·60)/100.
(1000 + 4500)/100.
5500/100.
55

📝 Topic test — 8 questions

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