Mixture and Allegation — Study Notes

Overview

Mixture and Allegation is a moderate-scoring topic in SSC MTS Paper 1, typically yielding 1–2 questions worth 1 mark each. These problems involve combining two or more items with different prices, strengths, or concentrations to form a mixture, or replacing part of a mixture with something else. The allegation rule provides a shortcut formula to find ratios without lengthy equation-solving, making it a time-saver in the exam.

Students must master two core scenarios: **(1) Finding the ratio in which two components are mixed** when the mean price/concentration is given, and **(2) Successive replacement problems** where a container's contents are repeatedly replaced. Allegation problems often appear in a variety of real-world contexts—mixing rice of different prices, diluting milk with water, blending teas, or mixing liquids of different concentrations. The key is to recognize the problem type and apply the formula correctly.

Because these questions involve straightforward calculation once the method is clear, they are high-value targets for quick marks. Spend focused practice on identifying which formula to use and performing accurate arithmetic under time pressure.

Key Concepts

• **Mixture:** A combination of two or more components (liquids, solids, or abstract quantities like prices) in a certain ratio or quantity.
• **Mean or Average Price/Strength:** The effective price or concentration of the final mixture, lying between the individual prices or concentrations of the components.
• **Allegation Rule:** A visual, ratio-based method to find the proportion in which two ingredients are mixed when the mean value is known. It eliminates the need to set up and solve simultaneous equations.
• **Replacement:** Removing a certain quantity of a mixture and replacing it with a pure component (often water or a cheaper item), which dilutes the original mixture. Repeated replacement follows an exponential decay formula.
• **Two-component vs. multi-component mixtures:** SSC MTS focuses almost entirely on two-component problems. Multi-component problems, if they appear, can be broken into pairs.
• **Ratio and proportion foundation:** Allegation is fundamentally about ratios. If two items are mixed in ratio a:b, then the quantity of the first item is a/(a+b) of the total, and the second is b/(a+b) of the total.
• **Units consistency:** All prices or concentrations must be in the same unit before applying allegation.

Formulas / Key Facts

1. **Allegation Formula (Two components):** If two items with values C₁ and C₂ (C₁ < C₂) are mixed to get a mean value M, the ratio in which they are mixed is: **(C₂ − M) : (M − C₁)** Visually, draw a cross: cheaper item gets the difference (dear − mean), dearer item gets (mean − cheap).

2. **Weighted Mean:** If quantities q₁ and q₂ are mixed with prices/concentrations C₁ and C₂, mean M = (C₁ × q₁ + C₂ × q₂) / (q₁ + q₂).

3. **Replacement Formula (Single Replacement):** If a container holds V litres of mixture and x litres are removed and replaced with pure liquid (e.g., water), the final concentration of the original component is: **Final concentration = Initial concentration × (1 − x/V)**

4. **Replacement Formula (n Successive Replacements):** After n replacements of x litres each from a V-litre container: **Final concentration = Initial concentration × (1 − x/V)ⁿ**

5. **Ratio to Quantity Conversion:** If items are in ratio a:b and total quantity is T, then first item = (a/(a+b)) × T, second item = (b/(a+b)) × T.

6. **Percentage to Ratio:** If A is p% and B is q%, the ratio A:B = p:q.

7. **Milk and Water Problems:** When milk and water are mixed in ratio m:w, fraction of milk = m/(m+w), fraction of water = w/(m+w).

8. **Price and Quantity Inverse:** When mixing by price, quantities are inversely proportional to the price differences from the mean.

Worked Examples

**Example 1: Allegation — Mixing Two Rice Varieties** Two types of rice cost ₹40 per kg and ₹60 per kg. They are mixed to sell at ₹50 per kg. In what ratio must they be mixed?

*Solution:* Cheaper rice (C₁) = ₹40, Dearer rice (C₂) = ₹60, Mean (M) = ₹50. Using allegation: Ratio = (C₂ − M) : (M − C₁) = (60 − 50) : (50 − 40) = 10 : 10 = 1 : 1. **Answer: 1:1 ratio.**

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**Example 2: Replacement Problem (Single Replacement)** A 40-litre mixture of milk and water contains 80% milk. 10 litres of the mixture are removed and replaced with pure water. What is the final percentage of milk?

*Solution:* Initial milk = 80% of 40 = 32 litres. After removing 10 litres, milk removed = 80% of 10 = 8 litres. Remaining milk = 32 − 8 = 24 litres. Total mixture = 40 litres (10 removed, 10 water added). Final milk percentage = (24/40) × 100 = 60%. *Or use formula:* Final concentration = 80% × (1 − 10/40) = 80% × (30/40) = 80% × 0.75 = 60%. **Answer: 60%.**

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**Example 3: Successive Replacement** A 100-litre vessel contains pure milk. 10 litres are removed and replaced with water. This process is repeated once more. What is the final quantity of milk?

*Solution:* Use formula: Final concentration = Initial × (1 − x/V)ⁿ. Here, Initial = 100 litres (100%), x = 10, V = 100, n = 2. Final milk = 100 × (1 − 10/100)² = 100 × (0.9)² = 100 × 0.81 = 81 litres. **Answer: 81 litres.**

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**Example 4: Finding Mean Price** Tea costing ₹200 per kg is mixed with tea costing ₹300 per kg in the ratio 3:2. What is the price per kg of the mixture?

*Solution:* Total quantity = 3 + 2 = 5 parts. Mean price M = (200 × 3 + 300 × 2) / 5 = (600 + 600) / 5 = 1200 / 5 = ₹240 per kg. **Answer: ₹240 per kg.**

Common Mistakes

1. **Inverting the allegation ratio:** *Wrong:* Taking (M − C₁) for the cheaper item. *Correct:* Cheaper item gets (C₂ − M), dearer gets (M − C₁). Use the cross-multiplication visual to avoid confusion.

2. **Forgetting to remove the original component in replacement:** *Wrong:* Thinking that adding water increases milk quantity. *Correct:* When x litres are removed from a mixture, the milk removed = (milk fraction) × x. Only then is water added.

3. **Using percentage instead of absolute values in allegation:** *Wrong:* Directly using percentages like 40% and 60% without converting to actual concentrations if needed. *Correct:* Ensure C₁, C₂, and M are in the same unit (rupees per kg, percentage concentration, grams per litre, etc.).

4. **Misapplying successive replacement formula:** *Wrong:* Multiplying (1 − x/V) by n instead of raising to the power n. *Correct:* For n replacements, the factor is (1 − x/V)ⁿ, not n × (1 − x/V).

5. **Confusing ratio and absolute quantity:** *Wrong:* Reporting the ratio as the answer when the question asks for absolute litres or kg. *Correct:* Convert ratio to quantity using the total: if ratio is a:b and total is T, first item = a/(a+b) × T.

Quick Reference

• **Allegation ratio formula:** (Dearer − Mean) : (Mean − Cheaper).
• **Single replacement:** Final = Initial × (1 − removed/total).
• **n replacements:** Final = Initial × (1 − removed/total)ⁿ.
• **Mean price/concentration:** Sum of (price × quantity) divided by total quantity.
• **Draw the allegation cross:** Visually connect cheaper, mean, dearer to avoid ratio inversion.
• **Check units:** All prices/concentrations must match before calculation.