Practice Questions
Concept Explanation
A mixture combines two or more ingredients at different rates. The average (mean) rate of the final mix depends on the quantities and their individual prices or values.
Key Formulas
- Mean price = (C1 × Q1 + C2 × Q2) / (Q1 + Q2)
- Total cost = Sum of (quantity × rate) for each ingredient
- Total quantity = Q1 + Q2 + …
Example
Mix 10 kg at Rs. 30/kg with 10 kg at Rs. 50/kg. Total cost = 300 + 500 = Rs. 600 for 20 kg. Mean price = 600/20 = Rs. 40/kg.
Common Trap
Do not take a simple average of prices unless quantities are equal. Weight by quantity always.
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Q1 EasyA mixture of 150 litres contains milk and water in the ratio 3:2. How many litres of water should be added so that the ratio becomes 2:3?ExplanationIn a mixture, quantities follow the given ratio. Milk = (3/5) × 150 = 90 litres and water = (2/5) × 150 = 60 litres. After adding water, milk stays 90 litres and the required ratio milk:water = 2:3 gives 90 : W = 2 : 3, so W = (90 × 3)/2 = 135 litres. Water to be added = 135 − 60 = 75 litres. Hence the answer is 75 litres.
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Q2 EasyA mixture contains milk and water in the ratio 5:1. On adding 5 litres of water, the ratio of milk to water becomes 5:2. The quantity of milk in the original mixture is:ExplanationLet milk = 5x litres and water = x litres. After adding 5 litres of water, milk:water = 5x : (x + 5) = 5 : 2. Cross-multiplying: 2 × 5x = 5(x + 5) ⇒ 10x = 5x + 25 ⇒ 5x = 25 ⇒ x = 5. Original milk = 5x = 25 litres. Hence the answer is 25 litres.
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Q3 EasyIn what ratio must a grocer mix two varieties of pulses costing Rs. 15 and Rs. 20 per kg respectively so as to get a mixture worth Rs. 16.50 per kg?ExplanationBy the alligation rule, cheaper : dearer = (dearer − mean) : (mean − cheaper) = (20 − 16.50) : (16.50 − 15) = 3.5 : 1.5 = 7 : 3. Hence the required ratio is 7 : 3.
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Q4 Easy40 litres of milk and water mixture contains 10% water. How much water must be added to make water 20% of the new mixture?ExplanationWater initially = 10% of 40 = 4 litres; milk = 36 litres (unchanged). Let x litres of water be added. Then (4 + x)/(40 + x) = 20/100 = 1/5. So 5(4 + x) = 40 + x ⇒ 20 + 5x = 40 + x ⇒ 4x = 20 ⇒ x = 5 litres. Hence the answer is 5 litres.
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Q5 Easy50 litres of a mixture has milk:water = 4:1. After adding 10 litres of water, the quantity of milk is:ExplanationMilk = (4/5) × 50 = 40 litres. Adding water does not change the milk quantity, so milk remains 40 litres. Hence the answer is 40 litres.
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Q6 Easy45 litres of milk is mixed with 15 litres of water. What percent of the mixture is milk?ExplanationTotal mixture = 45 + 15 = 60 litres. Percentage of milk = (45/60) × 100 = 75%. Hence the answer is 75%.
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Q7 MediumHow many kg of pure salt must be added to 30 kg of 2% solution of salt and water to increase it to 10% solution?ExplanationSalt in 30 kg of 2% solution = 0.02 × 30 = 0.6 kg. Let x kg of pure salt be added. Then (0.6 + x)/(30 + x) = 10/100 = 1/10. So 10(0.6 + x) = 30 + x ⇒ 6 + 10x = 30 + x ⇒ 9x = 24 ⇒ x = 24/9 = 8/3 kg. Hence the answer is 8/3 kg.
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Q8 MediumIn what ratio must water be mixed with milk to gain 16⅔% by selling the mixture at cost price?ExplanationGain of 16⅔% = 50/3% = 1/6 means SP = (7/6) × CP of the mixture. Selling at the cost price of pure milk implies CP of mixture = (6/7) × CP of milk. Taking cost of milk as 1 and water as free, mean cost = milk/(milk + water) = 6/7, so milk : water = 6 : 1, i.e. water : milk = 1 : 6. Hence the answer is 1 : 6.
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Q9 MediumFind the ratio in which rice at Rs. 7.20 a kg be mixed with rice at Rs. 5.70 a kg to produce a mixture worth Rs. 6.30 a kg.ExplanationBy alligation, quantity of Rs. 7.20 rice : quantity of Rs. 5.70 rice = (6.30 − 5.70) : (7.20 − 6.30) = 0.60 : 0.90 = 2 : 3. Hence the ratio is 2 : 3.
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Q10 Medium20 litres of a mixture contains milk and water in the ratio 5:3. After replacing 4 litres of the mixture with pure milk, what is the new milk quantity?ExplanationOriginal milk = (5/8) × 20 = 12.5 litres. In 4 litres drawn, milk removed = (5/8) × 4 = 2.5 litres. Milk left = 12.5 − 2.5 = 10 litres; then 4 litres pure milk are added, so milk = 10 + 4 = 14 litres. Hence the answer is 14 litres.
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Q11 Medium60 kg of an alloy has copper and zinc in the ratio 3:2. How much copper must be added so that the ratio becomes 2:1?ExplanationCopper = (3/5) × 60 = 36 kg; zinc = 24 kg (unchanged). Let x kg copper be added. Then (36 + x) : 24 = 2 : 1 ⇒ 36 + x = 48 ⇒ x = 12 kg. Hence the answer is 12 kg.
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Q12 Medium12 litres of pure spirit is mixed with 4 litres of water. If the cost of spirit is Rs. 60 per litre and water is free, the cost per litre of the mixture is:ExplanationMean price = total cost / total quantity. Total cost = 12 × 60 + 4 × 0 = Rs. 720. Total quantity = 16 litres. Mean price = 720/16 = Rs. 45 per litre. Hence the answer is Rs. 45.
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Q13 Hard80 kg of mixture contains sand and cement in the ratio 5:3. How much sand should be added to make the ratio 2:1?ExplanationSand = (5/8) × 80 = 50 kg; cement = 30 kg. Let x kg sand be added. Then (50 + x) : 30 = 2 : 1 ⇒ 50 + x = 60 ⇒ x = 10 kg. Hence the answer is 10 kg.
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Q14 HardTwo solutions of 20% and 50% acid are mixed to get 30 litres of 30% acid solution. How many litres of the 20% solution are used?ExplanationBy alligation on acid strength: 20% : 50% for mean 30% gives ratio (50 − 30) : (30 − 20) = 20 : 10 = 2 : 1. Quantity of 20% solution = (2/3) × 30 = 20 litres. Hence the answer is 20 litres.
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DS1 Easy
How many litres of water must be added to change the milk–water ratio?
Statement IA 150-litre mixture has milk and water in the ratio 3:2.
Statement IIAfter adding water, the required milk:water ratio is 2:3.
ExplanationMilk = (3/5) × 150 = 90 litres. Statement I alone does not give the target ratio. Statement II alone lacks the initial quantities. Together: 90 : W = 2 : 3 ⇒ W = 135, so water to add = 135 − 60 = 75 litres. Both are needed. Answer: C. -
DS2 Easy
What quantity of pure salt must be added?
Statement IThere are 30 kg of a 2% salt solution.
Statement IIThe target concentration after adding pure salt is 10%.
ExplanationSalt initially = 0.6 kg. Statement I alone lacks the target %. Statement II alone lacks the initial mass and %. Together: (0.6 + x)/(30 + x) = 1/10 ⇒ x = 8/3 kg. Both are needed. Answer: C.
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DS3 Easy
For the mixing ratio of two pulse varieties. What is the ratio?
Statement IThe two varieties cost Rs. 15/kg and Rs. 20/kg and the mixture is worth Rs. 16.50/kg.
Statement IIOnly the cheaper variety costs Rs. 15/kg.
ExplanationBy alligation, Statement I alone gives (20 − 16.50) : (16.50 − 15) = 7 : 3. Statement II alone is insufficient. Answer: A. -
DS4 Medium
An HR analytics quiz asks for the milk quantity in the original mixture. What is that quantity?
Statement IMilk and water are in the ratio 5:1, and after adding 5 litres of water the ratio becomes 5:2.
Statement IIOnly the final ratio after adding water is 5:2 (initial ratio unknown).
ExplanationFrom Statement I: 5x/(x + 5) = 5/2 ⇒ x = 5, so milk = 25 litres. Statement II alone is insufficient. Answer: A.