Mixtures and Alligation – Complete Notes for Competitive Exams
1. Introduction to Mixtures
A mixture is formed when two or more substances are combined without any chemical reaction. Each substance in a mixture retains its own properties.
Examples:
- Mixing water and milk
- Mixing rice of different prices
- Mixing sugar and salt
2. Types of Mixtures
(a) Simple Mixture
When substances are mixed directly in a given ratio.
(b) Compound Mixture
When a mixture is mixed again with another mixture.
3. Mean Value (Average Price)
When different items are mixed, the mean value is the average price or concentration of the mixture.
Formula:
Mean Value=Total QuantityTotal Cost
4. Alligation Method
Alligation is a mathematical rule used to find the ratio in which two or more ingredients at different prices or strengths must be mixed to obtain a mixture of desired price or strength.
Used for:
- Price-based problems
- Percentage-based problems
- Strength/concentration problems
5. Alligation Rule (Basic Formula)
If:
- Cheaper price = C
- Dearer price = D
- Mean price = M
Then the ratio:Cheaper : Dearer=(D−M):(M−C)
7. Mixture with Replacement
Formula:
If a container has volume V, and x liters are replaced n times:Final Quantity of Original Liquid=V(1−Vx)n
8. Percentage in Mixtures
If two liquids with different percentages are mixed, alligation can be applied directly using percentage values.
9. Important Points for Exams
- Always check mean value lies between cheaper and dearer values
- Alligation works only for two ingredients at a time
- For more than two ingredients, combine step by step
- Replacement problems mostly use exponential formula
10. Applications in Competitive Exams
- Milk and water problems
- Alcohol and water problems
- Rice/wheat mixing
- Profit and loss combined with mixtures
Top 25 Practice Questions – Mixtures and Alligation
Q1.
In what ratio must rice at ₹20/kg be mixed with rice at ₹30/kg to get a mixture worth ₹24/kg?
Q2.
Milk costing ₹40 per liter is mixed with milk costing ₹50 per liter. Find the ratio if mean price is ₹44.
Q3.
In what ratio should sugar at ₹36/kg be mixed with sugar at ₹54/kg to get mixture at ₹42/kg?
Q4.
How many liters of water must be mixed with 10 liters of milk costing ₹60/liter to get mixture costing ₹48/liter?
Q5.
A mixture contains alcohol and water in the ratio 3:2. If 5 liters of mixture is replaced by water, what is the new ratio?
Q6.
A vessel contains 20 liters of milk. 5 liters are taken out and replaced by water. This process is repeated twice. How much milk is left finally?
Q7.
In what ratio must tea costing ₹80/kg and ₹100/kg be mixed to get mixture worth ₹92/kg?
Q8.
A 40-liter mixture contains milk and water in ratio 7:3. How much water must be added to make ratio 7:5?
Q9.
Rice worth ₹25/kg is mixed with 10 kg of rice worth ₹30/kg. What should be the quantity of first rice to get mixture worth ₹28/kg?
Q10.
A shopkeeper mixes two varieties of pulses costing ₹60/kg and ₹75/kg in ratio 2:3. Find the mean price.
Q11.
In what ratio must alcohol of 20% and 50% be mixed to get 30% alcohol?
Q12.
A mixture of 60 liters contains alcohol and water in ratio 1:2. How much alcohol must be added to make ratio 1:1?
Q13.
If 8 liters are replaced three times from a 40-liter vessel, what fraction of original liquid remains?
Q14.
Two mixtures of milk and water are in ratios 3:1 and 4:1. In what ratio should they be mixed to get ratio 5:2?
Q15.
A mixture of tea contains 40% milk. How much water should be added to 30 liters of mixture to reduce milk content to 25%?
Q16.
A container has 50 liters of a mixture. 10 liters are replaced by pure milk. If this is done twice, find final quantity of original mixture.
Q17.
In what ratio should wheat at ₹18/kg and ₹27/kg be mixed to get mixture worth ₹24/kg?
Q18.
A 20-liter mixture has milk and water in ratio 4:1. How much mixture should be removed and replaced by water to make ratio 1:1?
Q19.
Alcohol and water are mixed in ratio 5:3. If 16 liters of mixture contains alcohol, find quantity of alcohol.
Q20.
A solution contains 25% salt. How much water must be added to 20 liters to reduce salt concentration to 20%?
Q21.
Find the mean price when items costing ₹12, ₹18, and ₹24 are mixed in equal quantities.
Q22.
A mixture of two liquids is in ratio 2:5. If 14 liters of mixture is taken, how much of each liquid is present?
Q23.
In what ratio should sugar solutions of 30% and 60% be mixed to get 45% solution?
Q24.
A mixture of rice costs ₹36/kg. If rice costing ₹30/kg is mixed with rice costing ₹42/kg in equal quantities, verify the mean price.
Q25.
If a vessel contains 60 liters and 12 liters are replaced four times, find remaining fraction of original liquid.
Answer
Answers – Mixtures and Alligation
Q1. Ratio = 3 : 2
Q2. Ratio = 3 : 1
Q3. Ratio = 2 : 1
Q4. Water added = 2.5 liters
Q5. New ratio = 4 : 3
Q6. Milk left = 11.25 liters
Q7. Ratio = 2 : 3
Q8. Water to be added = 8 liters
Q9. Quantity = 20 kg
Q10. Mean price = ₹69 per kg
Q11. Ratio = 2 : 1
Q12. Alcohol to be added = 10 liters
Q13. Fraction left = 8/25
Q14. Ratio = 3 : 5
Q15. Water added = 18 liters
Q16. Original mixture left = 32 liters
Q17. Ratio = 1 : 2
Q18. Mixture replaced = 10 liters
Q19. Alcohol = 10 liters
Q20. Water added = 5 liters
Q21. Mean price = ₹18
Q22. First liquid = 4 liters, Second liquid = 10 liters
Q23. Ratio = 1 : 1
Q24. Mean price = ₹36 per kg
Q25. Fraction of original liquid left = 16/125