Averages • Topic 2 of 4

Weighted Average

A weighted average is used when the groups being combined are of different sizes or carry different importance. Instead of a simple mean of the group averages, each group average is weighted by its count: weighted average = (w₁a₁ + w₂a₂ + …)/(w₁ + w₂ + …). The CAT power-tool here is alligation, which rearranges this into a clean ratio. If two groups with averages A₁ and A₂ combine to give a blended average Avg, then w₁ : w₂ = (A₂ − Avg) : (Avg − A₁) — the cheaper distance to the mean, on the opposite side. This turns "in what ratio must milk at ₹40 and milk at ₹60 be mixed to sell at ₹45?" into 15 : 5 = 3 : 1 in seconds. The blended average always lies between the two group averages and sits closer to the heavier group.

✅ Solved examples

1. A class of 20 boys averages 60 marks and 30 girls average 70 marks. Find the class average.
Weighted mean = (20×60 + 30×70)/50 = (1200 + 2100)/50 = 3300/50 = 66 marks.
2. In what ratio must rice at ₹30/kg and rice at ₹45/kg be mixed to get a blend worth ₹40/kg?
Alligation: (45 − 40) : (40 − 30) = 5 : 10 = 1 : 2.
3. A merchant mixes two tea varieties costing ₹120 and ₹180 per kg in the ratio 2 : 3. Find the cost per kg of the mix.
Weighted mean = (2×120 + 3×180)/5 = (240 + 540)/5 = 780/5 = ₹156/kg.
4. The average score of a section was 75. The 40% who were boys averaged 70. Find the girls’ average.
Take 100 students: 40 boys, 60 girls. Total = 75×100 = 7500. Boys = 40×70 = 2800. Girls = 4700/60 ≈ 78.33.

✏️ Practice — try these, take hints as needed

1. A vessel has 30 L of a 20% acid solution and 20 L of a 45% acid solution. Find the concentration of the mix.
Weighted average of concentrations.
(30×20 + 20×45)/50.
(600 + 900)/50.
30%
2. In what ratio must water (₹0) be mixed with milk at ₹50/L to make it ₹40/L?
Treat water as ₹0.
(50 − 40) : (40 − 0).
10 : 40.
1 : 4
3. Two groups of 15 and 25 people average 48 kg and 56 kg. Find the combined average weight.
Weighted mean.
(15×48 + 25×56)/40.
(720 + 1400)/40.
53 kg
4. A blend of nuts at ₹200 and ₹320 per kg sells at ₹260 per kg. Find the mixing ratio.
Alligation.
(320 − 260) : (260 − 200).
60 : 60.
1 : 1
5. A trader has 25 kg of wheat at ₹28/kg and x kg at ₹40/kg, blended at ₹34/kg. Find x.
Alligation gives the ratio.
(40 − 34) : (34 − 28) = 6 : 6 = 1 : 1.
Equal ratio ⇒ x = 25.
25 kg

📝 Topic test — 8 questions

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