Average, Mixture & Alligation

Average, Mixture & Alligation

🟢 Lite

Key Rule / Formula

Alligation (mean price): Mean price = (Total cost of cheaper + Total cost of expensive) / Total quantity. Also: Quantity of cheaper / Quantity of expensive = (Price of expensive − Mean price) / (Mean price − Price of cheaper).

Memory Trick

Alligation = “Add Low to High, find the Ratio” — the difference of each price from the mean gives the quantity ratio.

1-Sentence Summary

SSC tests weighted averages and mixture ratios — if you know how to apply alligation, you can solve any mixture question in under 60 seconds.

Quick Example

Q: In what ratio must two sugars costing ₹30/kg and ₹50/kg be mixed to get a mixture worth ₹40/kg? A: 1 : 1 — Difference from mean: (40−30)=10 and (50−40)=10. Ratio = 10:10 = 1:1.

🟡 Standard

Concept

Average is simply the sum of values divided by the number of values. But SSC CGL doesn’t stop at basic averages — it tests weighted averages where different groups have different sizes, replacement problems (removing and adding items), and the relationship between average and total sum. Mixture and Alligation is essentially a shortcut for solving weighted average problems visually. When you mix two items at different prices, the alligation method instantly gives the ratio without setting up equations.

The alligation method works because it equalises the deviations from the mean price. If one item is ₹10 above the mean and another is ₹10 below, mixing equal quantities brings the mean to exactly midway.

Key Points

• Average = Sum of observations / Number of observations.
• When one item replaces another in a group: New average = Old average ± (Change in total / Number of items).
• Alligation gives the ratio of quantities, not the actual amounts.
• If the mean is closer to one price, that ingredient dominates the mixture in higher proportion.
• For more than two ingredients, apply alligation successively — pair the lowest and highest first, then include the middle.

Worked Example

Q: The average weight of 30 students in a class is 48 kg. When 5 new students join, the average becomes 50 kg. Find the average weight of the new students. Approach: Total weight of original 30 = 30 × 48 = 1440 kg. Total weight of all 35 = 35 × 50 = 1750 kg. Weight of 5 new students = 1750 − 1440 = 310 kg. Average = 310 / 5 = 62 kg. Answer: 62 kg

SSC Pattern / Tips

• Replacement questions are common: when x items are removed and y items added, find the value of the removed/added items using total sum balance.
• For successive filling/emptying (tank problems), track the quantity of pure/liquid remaining each step.
• Alligation only works for two groups at a time — for 3+ groups, do it in steps.
• Average speed: harmonic mean approach for equal distances — not simple arithmetic mean.

🔴 Extended

Full Concept

The average concept in Tier 2 goes beyond simple arithmetic. The most challenging variant is the assumed mean method — when you have a large dataset (like ages of 100 people), working with deviations from an assumed mean dramatically reduces calculation time. If values are x₁, x₂, …, xₙ and assumed mean is A, then actual mean = A + (Σdᵢ)/n where dᵢ = xᵢ − A.

Replacement Problems are a Tier 2 specialty: when some items are removed and identical items (or different items) are added, the average changes. The key formula: if average of n items is A, one item with value x is replaced by y, new average = A + (y−x)/n. This “÷n” is the critical step most students miss.

Alligation is the visual form of weighted average. The alligation cross: write the cheaper price on left, expensive on right, mean price in the middle. Draw an X — the difference from mean on each side gives the ratio. This works because: Cheaper quantity × (Mean − Cheaper price) = Expensive quantity × (Expensive price − Mean).

Advanced Mixture Topics: When mixing multiple ingredients (3 or more), use the “repeated alligation” method — always link the current mixture with the next ingredient. Also, watch for “repeated replacement” (successive dilution): if a container has initial quantity Q and fraction f is replaced with water each time, after n replacements, quantity of original = Q × (1−f)ⁿ.

SSC CGL Deep Analysis

• Frequency: 1–2 questions per paper. Mixture/alligation appears nearly every year in some form.
• Difficulty: Easy to medium. Most students can solve alligation questions quickly; the replacement formula trips up 40% of test-takers.
• Recent trend: Tier 2 increasingly combines average with data interpretation — e.g., average of grouped data, finding missing frequency when mean is given.
• Average of fractions: Sometimes the mean itself turns out to be a fraction (e.g., mean price = 47.33). Alligation still works — convert fractions to a common denominator or use decimal differences.
• Total weight in Tier 2: Roughly 2–3% of the quant paper.

High-Scoring Strategy

1. For any average problem with large numbers, use assumed mean to avoid big additions.
2. For replacement problems, always track: what changed? What stayed the same? The total count n is constant.
3. In dilution problems, use the compound formula: Remaining% = (1 − removed_fraction)^n × initial%.
4. When three ingredients are mixed, first find the mixture of two, then mix the result with the third.
5. Memorise: if average of n items is A and one item x is removed, new average = (nA − x)/(n−1).
6. For equal distance at different speeds, use harmonic mean: Average speed = 2ab/(a+b) for speeds a and b over same distance.

SSC-Level Practice

Q1: A shopkeeper mixes 30 kg of rice at ₹40/kg with 50 kg at ₹60/kg and sells the mixture at ₹58/kg. Find his profit percentage. Answer: ≈ 10.48% — Working: Total cost = 30×40 + 50×60 = 1200 + 3000 = 4200. Total quantity = 80 kg. Cost price per kg = 4200/80 = ₹52.5. Selling price = ₹58/kg. Profit per kg = 58 − 52.5 = ₹5.5. Profit% = (5.5/52.5) × 100 ≈ 10.48%.

Q2: A container has 80 litres of milk. 20 litres are removed and replaced with water. This is done 3 times. How much milk remains? Answer: 33.75 litres — Working: Each step removes 20 of 80 litres, so the fraction replaced is 20/80 = 1/4 and the milk retained each step is (1 − 1/4) = 3/4. After n replacements, milk remaining = 80 × (3/4)ⁿ. For n = 3: 80 × (3/4)³ = 80 × 27/64 = 33.75 litres. Step-by-step check: 80 → 60 → 45 → 33.75 litres.

Common Traps

• Trap 1: Using simple average instead of weighted average when groups have different sizes. Alligation IS weighted average — don’t ignore it.
• Trap 2: In replacement problems, forgetting that the denominator is the original number of items (not n−1) when computing the new average. The formula is A + (y−x)/n, not (nA − x + y)/(n).
• Trap 3: In dilution problems, using arithmetic progression instead of geometric progression. Each replacement is a geometric reduction: multiply by (1 − fraction removed), don’t subtract linearly.

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