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RRB NTPC Mixture & Alligation

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This page covers RRB NTPC Mixture & Alligation with complete concept notes, 32 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

32graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 32

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Practice 1easy

Two alloys have gold and silver in the ratio 7:5 and 5:3 respectively. Equal quantities of both alloys are melted together. What is the ratio of gold to silver in the resulting alloy?

Practice 2easy

A mixture contains sugar and salt in the ratio 4:3. If 21 kg of salt is present, how much sugar is in the mixture?

Practice 3easy

Two alloys contain copper and tin in the ratios 3:2 and 4:1 respectively. Equal weights of both alloys are melted together. What is the percentage of copper in the final alloy?

Practice 4easy

A chemist mixes solution A (30% acid) and solution B (50% acid) in the ratio 2:3. What is the percentage of acid in the final mixture?

Practice 5easy

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 per kg of the mixture?

Practice 6easy

A shopkeeper mixes two types of rice costing ₹40 per kg and ₹60 per kg. If he wants to create a mixture costing ₹50 per kg, in what ratio should he mix the two types?

Practice 7easy

In what ratio must sugar costing ₹15 per kg be mixed with sugar costing ₹24 per kg to get a mixture costing ₹18 per kg?

Practice 8easy

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 9easy

Two containers have milk and water in the ratio 3:2 and 5:3 respectively. If 20 litres from the first container and 30 litres from the second container are mixed together, what is the ratio of milk to water in the resulting mixture?

Practice 10easy

A mixture contains alcohol and water in the ratio 4:1. If 10 litres of water is added to the mixture, the ratio becomes 4:3. What was the original quantity of alcohol in the mixture?

Practice 11easy

A container has 60 litres of a mixture of milk and water in the ratio 7:5. How much milk should be added to make the ratio 3:2?

Practice 12easy

A merchant has two types of sugar: Type A at ₹30 per kg and Type B at ₹50 per kg. He mixes them to get 80 kg of mixture at ₹40 per kg. How many kilograms of Type A sugar did he use?

Practice 13medium

A shopkeeper mixes two types of rice costing ₹40 per kg and ₹60 per kg. In what ratio should they be mixed to get a mixture costing ₹50 per kg?

Practice 14medium

A container has 120 litres of a mixture with alcohol and water in ratio 7:5. How much pure alcohol must be added to make the ratio 3:1?

Practice 15medium

A chemist has two solutions of acid with concentrations 30% and 50%. In what ratio should they be mixed to obtain 100 litres of 42% acid solution?

Practice 16medium

A container has 120 litres of a mixture of petrol and diesel in the ratio 7:5. How many litres of diesel should be added to make the ratio 1:1?

Practice 17medium

A chemist has three solutions with acid concentrations of 20%, 30%, and 50% respectively. He mixes the first two in equal quantities and then adds the third solution in such a way that the final mixture has 40% acid concentration. In what ratio should the third solution be added to the mixture of the first two?

Practice 18medium

A merchant blends two varieties of tea costing ₹180 per kg and ₹240 per kg in the ratio 5:3. What is the cost per kg of the blended tea?

Practice 19medium

A goldsmith has two gold alloys: one with 80% purity and another with 60% purity. He mixes them in the ratio 2:3 to create a new alloy. What is the purity percentage of the resulting alloy?

Practice 20medium

A shopkeeper mixes two types of rice costing ₹40 per kg and ₹60 per kg in the ratio 3:2. At what price per kg should he sell the mixture to make a 25% profit?

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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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