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Mixture and Alligation

Practice MCQs

Alligation. It is the rule that enables us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture of a desired price. A process or rule for the solution of problems concerning the compounding or mixing of ingredients differing in price or quality.

Quantitative Aptitude Mixture and Alligation Competitive Exams

Mixture and Alligation is an important quantitative aptitude topic used to solve problems involving mixing two or more items with different prices, concentrations, speeds, qualities, or percentages. It is especially useful in questions on milk-water mixtures, profit-based mixing, average price, alcohol-water mixtures, and replacement problems.

What are Mixture and Alligation?

A mixture is formed when two or more ingredients are combined together. The ingredients may have different costs, strengths, concentrations, or qualities.

Alligation is a shortcut method used to find the ratio in which two ingredients should be mixed to obtain a desired average value, mean price, or concentration.

Quick idea: If two items with different values are mixed to get a mean value, use alligation to find their mixing ratio quickly.
Term Meaning Example
Cheaper Quantity Ingredient with lower value or price Rice at ₹40/kg
Dearer Quantity Ingredient with higher value or price Rice at ₹60/kg
Mean Value Desired average value of the mixture Mixture at ₹48/kg
Alligation Ratio Ratio in which two ingredients are mixed Cheaper : Dearer

“Alligation is a shortcut for weighted average problems.”

Aptitude Tip
Key points
  • Alligation is used for two-value mixing problems.
  • The mean value must lie between the two given values.
  • Cheaper quantity has the lower value.
  • Dearer quantity has the higher value.
  • Alligation ratio is found by cross differences.
  • Replacement problems use repeated fraction logic.
mixture alligation ratio average

Visual Understanding

These diagrams show the basic alligation structure and common mixture-replacement concept.

Alligation Cross Method
Cheaper 40 Dearer 60 Mean 48 60 - 48 = 12 48 - 40 = 8 Ratio = 12 : 8 = 3 : 2
\[ \text{Cheaper : Dearer} = (D-M):(M-C) \]

Cross subtract the mean from the two values to get the mixing ratio.

Repeated Replacement
Initial mixture Milk Remove and replace Water added Water
\[ \text{Quantity Left} = Q\left(1-\frac{x}{Q}\right)^n \]

Used when some mixture is removed and replaced repeatedly.

Important Formulas and Rules

Alligation Rule
\[ C:D = (d-m):(m-c) \]

Here \(c\) is cheaper value, \(d\) is dearer value, and \(m\) is mean value.

Weighted Average
\[ m = \frac{q_1v_1 + q_2v_2}{q_1+q_2} \]

Used when quantities and values are directly given.

Replacement Formula
\[ R = Q\left(1-\frac{x}{Q}\right)^n \]

\(R\) is original quantity left after \(n\) replacements.

Pure Quantity
\[ \text{Pure Quantity} = \frac{p}{100}\times Q \]

Used when concentration percentage is given.

Rule: In alligation, the mean value must lie between the cheaper value and the dearer value. Otherwise, such a mixture is not possible.

Common Types of Questions

Price Mixing

Items with different prices are mixed to get an average price.

  • Rice mixture
  • Tea mixture
  • Sugar mixture
  • Average cost
Concentration Mixing

Liquids with different strengths are mixed.

  • Alcohol-water
  • Acid-water
  • Milk-water
  • Percentage strength
Replacement Problems

Some mixture is removed and replaced with another liquid.

  • Milk removed
  • Water added
  • Repeated operation
  • Use replacement formula
Profit-Based Mixture

Mixture is sold at profit or loss.

  • Find cost price
  • Find selling price
  • Use profit percentage
  • Then apply alligation
Exam approach: If two values and one mean value are given, use alligation. If quantities are directly given, use weighted average.
Alligation Method

Suppose rice costing ₹40/kg and ₹60/kg are mixed to get a mixture worth ₹48/kg.

\[ \text{Cheaper value} = 40 \]
\[ \text{Dearer value} = 60 \]
\[ \text{Mean value} = 48 \]

Use cross difference:

\[ \text{Cheaper : Dearer} = (60-48):(48-40) \]
\[ \text{Cheaper : Dearer} = 12:8 = 3:2 \]

Therefore, rice costing ₹40/kg and ₹60/kg should be mixed in the ratio \(3:2\).

Weighted Average Method

If quantities are already given, use weighted average directly.

Example: \(3\) kg rice at ₹40/kg is mixed with \(2\) kg rice at ₹60/kg.

\[ \text{Mean Price} = \frac{3\times40 + 2\times60}{3+2} \]
\[ \text{Mean Price} = \frac{120+120}{5} = 48 \]

This confirms that the ratio \(3:2\) gives an average price of ₹48/kg.

Step-by-Step Solving Method

Step Action Example
Step 1 Identify cheaper value, dearer value, and mean value. Cheaper = 40, Dearer = 60, Mean = 48
Step 2 Check whether mean lies between the two values. 48 lies between 40 and 60.
Step 3 Apply cross difference. \((60-48):(48-40)\)
Step 4 Simplify the ratio. \(12:8 = 3:2\)
Step 5 Write final answer in correct order. Cheaper : Dearer = \(3:2\)
Important: The ratio obtained from alligation is always the ratio of quantities, not the ratio of prices.

Solved Examples

Question Method Answer
In what ratio should rice costing ₹40/kg and ₹60/kg be mixed to get a mixture costing ₹48/kg?
\[ \text{Ratio} = (60-48):(48-40) \]
\[ \text{Ratio} = 12:8 = 3:2 \]
3 : 2
Tea costing ₹80/kg is mixed with tea costing ₹120/kg to get a mixture costing ₹95/kg. Find the ratio.
\[ \text{Ratio} = (120-95):(95-80) \]
\[ \text{Ratio} = 25:15 = 5:3 \]
5 : 3
Milk and water are mixed in the ratio \(4:1\). Find the percentage of milk in the mixture. Total parts:
\[ 4+1=5 \]
Milk fraction:
\[ \frac{4}{5}\times100=80 \]
80%
A mixture contains 30 litres of milk and 10 litres of water. Find the ratio of milk to water.
\[ \text{Milk : Water} = 30:10 \]
\[ 30:10 = 3:1 \]
3 : 1
How many litres of water should be added to 40 litres of milk to make milk and water ratio \(4:1\)? Let water added be \(x\).
\[ \frac{40}{x} = \frac{4}{1} \]
\[ 4x = 40 \]
\[ x = 10 \]
10 litres
A vessel has 60 litres of milk. 15 litres is removed and replaced with water. How much milk remains?
\[ R = Q\left(1-\frac{x}{Q}\right) \]
\[ R = 60\left(1-\frac{15}{60}\right) \]
\[ R = 60 \times \frac{45}{60}=45 \]
45 litres
A solution of 20% acid is mixed with a solution of 50% acid to get a 30% solution. Find the ratio.
\[ \text{Ratio} = (50-30):(30-20) \]
\[ \text{Ratio} = 20:10 = 2:1 \]
2 : 1
Rice costing ₹30/kg and ₹50/kg are mixed in the ratio \(2:3\). Find the average price.
\[ \text{Average Price} = \frac{2\times30 + 3\times50}{2+3} \]
\[ \text{Average Price} = \frac{60+150}{5}=42 \]
₹42/kg

Note: In concentration questions, treat percentages like values and apply alligation normally.

Common Traps and Shortcuts

Common Traps
  • Writing the ratio in reverse order.
  • Forgetting that alligation gives quantity ratio, not price ratio.
  • Using alligation when the mean value is outside the given values.
  • Confusing concentration percentage with quantity.
  • Using simple subtraction in repeated replacement problems.
  • Ignoring total parts while converting ratio to percentage.
Useful Shortcuts
  • Cheaper : Dearer = Dearer difference : Cheaper difference.
  • Use alligation for two-value average problems.
  • Use weighted average when quantities are directly given.
  • Use replacement formula for repeated removal and replacement.
  • For ratio to percentage, divide required part by total parts.
  • Always check final ratio order before answering.
Exam approach: If the question says “in what ratio should two things be mixed,” alligation is usually the fastest method.

Practice

A) Multiple Choice Questions
  1. In what ratio should rice costing ₹30/kg and ₹50/kg be mixed to get rice worth ₹38/kg?
    2 : 3 3 : 2 4 : 3 3 : 4
  2. Milk and water are in the ratio \(3:2\). What percentage of the mixture is milk?
    40% 50% 60% 75%
  3. A 20% solution and a 50% solution are mixed to get a 35% solution. Find the ratio.
    1 : 1 1 : 2 2 : 1 3 : 2
  4. A mixture contains 24 litres milk and 8 litres water. Find milk : water.
    2 : 1 3 : 1 4 : 1 1 : 3
  5. Rice costing ₹40/kg and ₹70/kg are mixed in the ratio \(2:1\). Find average price.
    ₹45/kg ₹50/kg ₹55/kg ₹60/kg
B) Solve the Higher-Order Problems
  1. In what ratio should tea costing ₹90/kg and ₹150/kg be mixed to obtain tea worth ₹110/kg? (Hint: Use alligation cross difference.)
  2. A vessel contains 80 litres of milk. 20 litres is removed and replaced with water. How much milk remains? (Hint: Use one-time replacement formula.)
  3. A mixture contains milk and water in the ratio \(5:3\). If the total mixture is 64 litres, find the quantity of milk. (Hint: Milk part is 5 out of 8 parts.)
  4. A 40% alcohol solution is mixed with a 70% alcohol solution to get a 50% solution. Find the ratio. (Hint: Treat percentages as values.)
  5. Rice costing ₹25/kg and ₹45/kg are mixed in the ratio \(3:2\). Find the average price. (Hint: Use weighted average.)
C) Match the Concept with the Correct Meaning
Concept Correct Meaning
Alligation Shortcut method to find mixing ratio
Mean Value Average value of the final mixture
Cheaper Quantity Ingredient with lower value
Dearer Quantity Ingredient with higher value
Weighted Average Average based on quantities and values
Replacement Removing part of mixture and adding another liquid
Aptitude Reminder

Mixture and alligation questions are based on weighted averages. Alligation is the fastest method when two values and one mean value are given. Replacement questions require fraction-based repeated reduction of the original quantity.

Task: Create five questions using price mixing, milk-water ratio, concentration mixing, replacement, and weighted average.

Show Suggested Answers
Multiple Choice
  1. 3 : 2
    \[ \text{Ratio} = (50-38):(38-30) \]
    \[ \text{Ratio} = 12:8 = 3:2 \]
  2. 60%
    Milk part \(=3\), total parts \(=3+2=5\).
    \[ \text{Milk Percentage} = \frac{3}{5}\times100=60 \]
  3. 1 : 1
    \[ \text{Ratio} = (50-35):(35-20) \]
    \[ \text{Ratio} = 15:15 = 1:1 \]
  4. 3 : 1
    \[ 24:8 = 3:1 \]
  5. ₹50/kg
    \[ \text{Average Price} = \frac{2\times40 + 1\times70}{2+1} \]
    \[ \text{Average Price} = \frac{80+70}{3}=50 \]
Higher-Order Problems
  1. Tea costing ₹90/kg and ₹150/kg, mean ₹110/kg:
    \[ \text{Ratio} = (150-110):(110-90) \]
    \[ \text{Ratio} = 40:20 = 2:1 \]
    Answer = 2 : 1.
  2. Vessel contains \(80\) litres milk, \(20\) litres replaced:
    \[ R = 80\left(1-\frac{20}{80}\right) \]
    \[ R = 80\times\frac{60}{80}=60 \]
    Answer = 60 litres.
  3. Milk : Water \(=5:3\), total \(=64\) litres:
    \[ \text{Milk} = \frac{5}{8}\times64=40 \]
    Answer = 40 litres.
  4. 40% and 70% alcohol solutions are mixed to get 50%:
    \[ \text{Ratio} = (70-50):(50-40) \]
    \[ \text{Ratio} = 20:10=2:1 \]
    Answer = 2 : 1.
  5. Rice costing ₹25/kg and ₹45/kg mixed in ratio \(3:2\):
    \[ \text{Average Price} = \frac{3\times25 + 2\times45}{3+2} \]
    \[ \text{Average Price} = \frac{75+90}{5}=33 \]
    Answer = ₹33/kg.
Concept Matching
  1. Alligation → Shortcut method to find mixing ratio
  2. Mean Value → Average value of the final mixture
  3. Cheaper Quantity → Ingredient with lower value
  4. Dearer Quantity → Ingredient with higher value
  5. Weighted Average → Average based on quantities and values
  6. Replacement → Removing part of mixture and adding another liquid
Clue Explanation

Alligation works because the final mean value is a weighted average of the two ingredients. Cross differences give the required quantity ratio quickly.

Exam tips
  • Use alligation when two values and one mean are given.
  • Mean value must lie between the two values.
  • Write the ratio in the correct order.
  • Use weighted average when quantities are directly given.
  • Use replacement formula for repeated removal and addition.
  • For ratio to percentage, divide by total parts.
Practice MCQs