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SSC MTS Mixture & Alligation

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This page covers SSC MTS Mixture & Alligation with complete concept notes, 11 graded practice MCQs, key points and exam-specific tips. Free to study.

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Concept Notes

Mixture & Alligation— Rules & Concept

Core ConceptRead this first — the foundation of the topic
CORE CONCEPT

Alligation is a quick method to find the ratio of mixing two items when we know their individual values and the mean (average) value of the mixture. It works on the principle that gain of one quantity equals loss of another quantity

KEY RULES

The ratio of quantities is inversely proportional to their differences from the mean 2. Cheaper quantity × Its difference from mean = Dearer quantity × Its difference from mean 3. For replacement problems, use: Final strength = Initial strength × (1 - R/C)^n where R = replaced quantity, C = total capacity, n = number of operations

Formula BlockMemorise — at least one formula appears in every paper
• Basic Alligation Formula: Ratio = (Mean - Cheaper value) : (Dearer value - Mean)
• For percentages: If x% and y% solutions are mixed to get z%, then Ratio = (z-x) : (y-z)
• Weighted Average: Mean = (A×a + B×b) / (A+B) where A,B are quantities and a,b are their values
• Replacement Formula: New concentration = Old × (1 - fraction removed)^number of operations
Exam PatternsWhat examiners ask — read before attempting PYQs

SSC CGL typically asks 1-2 questions on mixtures. Common types include: mixing two different priced items, alcohol-water mixtures, milk-water problems, successive mixing, and replacement of mixtures.

ShortcutsUse these to save 30–60 seconds per question

- CROSS METHOD: Draw a cross diagram. Put mean in center, cheaper value top-left, dearer value bottom-left. Differences go on right side.

This gives the mixing ratio directly.

Worked ExampleSolve this step-by-step before moving on
1
Step 1

Apply alligation formula Cheaper tea = Rs 64, Dearer tea = Rs 86, Mean = Rs 68

2
Step 2

Find differences Difference of cheaper from mean = 68 - 64 = 4 Difference of dearer from mean = 86 - 68 = 18

3
Step 3

Write ratio Ratio = 4:18 = 2:9 So tea worth Rs 64 and Rs 86 should be mixed in ratio 2:9. WORKED EXAMPLE 2: A vessel contains 60 litres of milk-water mixture with 80% milk. How much mixture should be removed and replaced with water to make milk 60%?

1
Step 1

Find initial milk quantity Initial milk = 60 × 80% = 48 litres

2
Step 2

Set up equation for final state Let x litres be removed and replaced with water Milk removed = x × 80% = 0.8x Final milk = 48 - 0.8x

3
Step 3

Apply condition Final milk percentage = 60% (48 - 0.8x)/60 = 60/100 48 - 0.8x = 36 0.8x = 12 x = 15 litres SHORTCUT FOR SUCCESSIVE OPERATIONS: When same fraction is removed repeatedly, use: Final = Initial × (1-f)^n where f = fraction removed, n = number of times.

Exam TrapsCommon mistakes students make — avoid these

#1: Students often confuse the direction of ratio in alligation. Remember: the ratio is OPPOSITE to what you might think. If mean is closer to cheaper value, you need MORE of the cheaper item, not less.

Always write (Mean - Cheaper) : (Dearer - Mean) for the ratio of Cheaper : Dearer.

Key Points to Remember

  • Alligation finds mixing ratios when individual values and mean value are known
  • Formula: Ratio = (Mean - Cheaper) : (Dearer - Mean) for quantities cheaper:dearer
  • Cross method: Put mean in center, write differences on opposite sides to get ratio
  • For replacement: Final concentration = Initial × (1 - removed fraction)^operations
  • In successive operations, multiply the reduction factor repeatedly
  • Weighted average formula: Mean = (Q1×V1 + Q2×V2)/(Q1+Q2)
  • The mixing ratio is inversely proportional to distance from mean value
  • When mean is closer to cheaper value, more cheaper quantity is needed
  • For percentage mixtures, treat percentages as values in alligation formula
  • Always check if final answer makes logical sense with given constraints

Exam-Specific Tips

  • Alligation rule: Cheaper quantity : Dearer quantity = (Mean - Cheaper value) : (Dearer value - Mean)
  • For milk-water problems, concentration typically reduces by factor (1 - R/C) per operation
  • When equal quantities of two mixtures are combined, resultant concentration is arithmetic mean of both
  • In successive mixing, final ratio = Product of individual operation ratios
  • Standard milk percentage in problems: Pure milk = 100%, typical mixtures = 80%, 75%, 60%
  • Replacement formula for n operations: Final = Initial × (1 - removed/total)^n
  • Cost price mixing follows same alligation rules as percentage mixing
  • Two mixtures of same concentration when mixed give same concentration
Practice MCQs

Mixture & Alligation — Practice Questions

11graded MCQs · easy to hard · full solution & trap analysis

All MCQs →
Practice 1easy

A container has 60 litres of a solution with acid and water in ratio 7:5. How many litres of water must be added to make the ratio 1:1?

Practice 2easy

Two containers have alcohol solutions of 20% and 50% respectively. In what ratio should they be mixed to get a 35% alcohol solution?

Practice 3easy

A shopkeeper mixes two types of rice costing ₹40 per kg and ₹60 per kg in the ratio 3:2. What is the cost price of the mixture per kg?

Practice 4easy

A goldsmith mixes gold and copper in the ratio 4:1 by weight. If the total weight of the alloy is 50 kg, how much copper is present in the alloy?

Practice 5easy

A merchant has two types of tea costing ₹80 per kg and ₹120 per kg. He mixes them to get a blend costing ₹100 per kg. In what ratio should they be mixed?

Practice 6medium

A container has 120 litres of a mixture of alcohol and water in the ratio 7:5. How much pure alcohol must be added so that the ratio becomes 3:2?

Practice 7medium

A shopkeeper mixes two types of rice costing ₹40 per kg and ₹60 per kg. If the mixture is sold at ₹55 per kg and no profit or loss is made, in what ratio were the two types mixed?

Practice 8hard

Three containers have milk and water in ratios 1:1, 2:1, and 3:2 respectively. Equal volumes from each are mixed. What is the percentage of milk in the final mixture?

Practice 9hard

A merchant mixes two types of rice costing ₹40/kg and ₹60/kg in a certain ratio. The mixture is sold at ₹55/kg, earning 10% profit. In what ratio were the two types mixed?

Practice 10hard

A container has 120 litres of a mixture with alcohol and water in ratio 7:5. How much pure alcohol must be added so that the ratio becomes 3:2?

Practice 11hard

Two containers have milk and water in ratios 3:2 and 5:4 respectively. If equal volumes are mixed from both containers, the resulting mixture has milk and water in what ratio?

60-Second Revision — Mixture & Alligation

  • Formula: Ratio = (Mean - Cheaper) : (Dearer - Mean) - remember the cross pattern
  • Replacement: New concentration = Old × (1 - fraction removed)^number of times
  • Trap: Ratio direction is opposite to intuition - more cheaper when mean is closer to cheaper
  • Quick check: Final ratio quantities should be inversely related to their distance from mean
  • For successive operations, multiply the (1 - removed fraction) repeatedly
  • Cross method saves time: Draw cross, put mean center, differences give ratio directly
  • Always verify answer makes sense: percentages between 0-100, positive quantities
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