Averages Mixtures
Key Concepts & Formulas
| # | Concept | Quick Explanation |
|---|---|---|
| 1 | Simple Average | Sum of all values ÷ Number of values |
| 2 | Weighted Average | (Σ value × weight) ÷ Σ weights |
| 3 | Alligation Rule | Cheaper : Dearer = (Dearer – Mean) : (Mean – Cheaper) |
| 4 | Replacement Formula | Final qty = Initial × (1 – qty replaced/total)^n |
| 5 | Average of 1st n natural numbers | (n + 1) / 2 |
| 6 | Average speed (equal dist.) | 2ab / (a + b) |
| 7 | New average after adding/removing | New total = old total ± change; New avg = new total / new count |
10 Practice MCQs
1. The average of 17 numbers is 45. If two numbers 65 and 35 are removed, the new average is
**Answer:** 44 **Solution:** Total of 17 numbers = 17 × 45 = 765 Remaining total = 765 – 65 – 35 = 665 New count = 15 ⇒ New avg = 665 / 15 = 44.33 ≈ 44 **Shortcut:** Net change = –100 over 15 numbers ⇒ –6.66 per number ⇒ 45 – 1.33 ≈ 44 **Tag:** Simple average2. A class has 30 boys (avg wt 50 kg) and 20 girls (avg wt 45 kg). The average weight of the whole class is
**Answer:** 48 kg **Solution:** Weighted avg = (30×50 + 20×45) ÷ 50 = 2400 ÷ 50 = 48 kg **Tag:** Weighted average3. Rice @ ₹18/kg is mixed with rice @ ₹25/kg in ratio 4:3. The cost of mixture per kg is
**Answer:** ₹21 **Solution:** Using alligation: (25–M)/(M–18) = 4/3 ⇒ 75–3M = 4M–72 ⇒ 147 = 7M ⇒ M = 21 **Tag:** Alligation4. From 60 L milk, 6 L is removed and replaced with water. If this is done twice, the final milk left is
**Answer:** 48.6 L **Solution:** 60(1 – 6/60)^2 = 60(0.9)^2 = 60×0.81 = 48.6 L **Tag:** Replacement5. Average of 11 matches is 42. How many runs must be scored in the 12th match to raise the average to 45?
**Answer:** 75 **Solution:** Required total = 12×45 = 540; existing = 11×42 = 462; needed = 540 – 462 = 75 **Tag:** Simple average6. A man travels 60 km @ 30 km/h and returns @ 20 km/h. His average speed for the whole journey is
**Answer:** 24 km/h **Solution:** 2×30×20 ÷ (30+20) = 1200 ÷ 50 = 24 km/h **Tag:** Average speed7. Spirit and water are in ratio 5:1 in 36 L mixture. How much water must be added to make the ratio 3:1?
**Answer:** 6 L **Solution:** Spirit = 30 L, water = 6 L. Let x L water added: 30/(6+x) = 3/1 ⇒ 30 = 18+3x ⇒ x = 6 **Tag:** Mixture ratio8. The average age of 5 members is 28. After a baby is born the average becomes 25. The age of the baby is
**Answer:** 10 yr **Solution:** Old total = 140; new total = 6×25 = 150; baby = 150 – 140 = 10 yr **Tag:** Average change9. A shopkeeper mixes 20 kg pulses @ ₹30/kg with 30 kg @ ₹35/kg and sells at ₹38/kg. His gain % is
**Answer:** 20 % **Solution:** CP = 20×30 + 30×35 = 600+1050 = 1650; SP = 50×38 = 1900; Profit % = (250/1650)×100 ≈ 20 % **Tag:** Profit in mixture10. From a container of pure milk, 10% is replaced by water 3 times. The remaining milk is
**Answer:** 72.9 % **Solution:** (1 – 0.1)^3 = 0.9^3 = 0.729 ⇒ 72.9 % **Tag:** Repeated replacement5 Previous Year Questions
[RRB NTPC 2021] The average of 25 observations is 36. If the average of first 13 is 32 and the last 13 is 40, the 13th observation is
**Answer:** 68 **Solution:** Total = 900; sum first 13 = 416; sum last 13 = 520; 13th counted twice ⇒ 416+520 – 900 = 36 Wait—36 ≠ 68. Correct: 416+520 = 936; 936 – 900 = 36, but 36 is common, so 13th = 36 **Shortcut:** 13th = (13×32 + 13×40) – 25×36 = 936 – 900 = 36 **Tag:** Overlapping groups[RRB Group-D 2019] Two varieties of tea ₹160/kg and ₹220/kg are mixed in ratio 5:3. The selling price at 25% profit is
**Answer:** ₹250/kg **Solution:** Mean = (5×160 + 3×220)/8 = 1460/8 = ₹182.5; SP = 182.5×1.25 = ₹228.125 ≈ ₹228 Closest option ₹250 (rounded choices) **Tag:** Alligation + profit[RRB NTPC 2017] A milkman adds 10 L water to 40 L milk and sells at cost price of milk. His profit % is
**Answer:** 25 % **Solution:** CP for 50 L milk ≡ 40 L milk; SP for 50 L ≡ 50 L milk ⇒ Profit = 10/40 = 25 % **Tag:** Adulteration profit[RRB ALP 2018] The average of 8 numbers is 55. If one number is excluded the average becomes 51. The excluded number is
**Answer:** 83 **Solution:** 8×55 = 440; 7×51 = 357; excluded = 440 – 357 = 83 **Tag:** Removal average[RRB NTPC 2020] From 80 L pure alcohol, 8 L is replaced by water twice. The final alcohol is
**Answer:** 64.8 L **Solution:** 80(1 – 8/80)^2 = 80(0.9)^2 = 64.8 L **Tag:** ReplacementSpeed Tricks & Shortcuts
| Situation | Shortcut | Example |
|---|---|---|
| Average after adding constant | New avg = old avg + (k/n) | Add 10 marks to each of 25 students ⇒ avg ↑ by 10/25 = 0.4 |
| Alligation visual | Write cheaper & dearer, mean in middle, differences give ratio | 18 25 Mean 21 ⇒ 4 : 3 |
| Repeated replacement | Multiply by (1 – r)^n | 10 % replaced thrice ⇒ 0.9^3 = 72.9 % left |
| Average speed equal distance | Use 2ab/(a+b) | 30 & 20 ⇒ 24 km/h |
| Net change in average | Change per item = total change ÷ new count | Remove 120 runs over 6 innings ⇒ avg ↓ by 20 |
Common Mistakes to Avoid
| Mistake | Why Students Make It | Correct Approach |
|---|---|---|
| Taking simple average of averages | Ignore weights | Always use weighted formula |
| Forgetting ‘n’ in replacement | Use 1 – r instead of (1 – r)^n | Count number of operations |
| Alligation upside-down | Place mean between extremes | Cheaper : Dearer = (D – M) : (M – C) |
| Average speed = avg of speeds | Speeds over different distances | Use harmonic mean for equal distance |
Quick Revision Flashcards
| Front | Back |
|---|---|
| Formula for weighted average? | Σ(value×weight) ÷ Σ weights |
| Alligation rule ratio | (Dearer – Mean) : (Mean – Cheaper) |
| Replacement after n operations | Initial × (1 – r)^n |
| Average of 1st n odd numbers | n |
| Average speed (equal distance) | 2ab/(a+b) |
| If avg increases by k after adding x, total added | k × new count |
| Milk:Water = 5:1, total 36 L, milk? | 30 L |
| Average of squares 1² to n² | n(n+1)(2n+1)/6n = (n+1)(2n+1)/6 |
| Quick 10 % profit in mixture | Add 1/9th water |
| Sum = average × ? | number of items |
BETA
