Practice Questions
Concept Explanation
Milk and water problems are classic mixture questions. Milk is the valuable component; water is the filler. Use ratios or alligation to find quantities or concentrations.
Key Formulas
- Milk : water ratio defines concentration: milk fraction = milk/(milk + water)
- Adding water dilutes: new milk fraction = old milk / (old total + water added)
- Alligation applies when mixing two concentrations of milk
Example
Milk : water = 5 : 3 in 32 L. Milk = 5/8 × 32 = 20 L, water = 12 L.
Common Trap
When water is added, milk quantity stays the same; only total volume increases.
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Q1 EasyMilk : water = 5 : 3 in 40 L. Quantity of milk is:ExplanationMilk = 5/8 × 40 = 25 L. Hence the answer is 25 L.
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Q2 Easy36 L of pure milk: 6 L is removed and replaced by water. Milk percentage now is:ExplanationMilk left = 36 − 6 = 30 L in 36 L ⇒ 30/36 = 5/6 ≈ 83.33%. Hence the answer is 83.33%.
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Q3 EasyMilk and water are mixed in ratio 4 : 1. If the cost of pure milk is Rs. 20/L and water is free, cost of mixture per litre is:ExplanationIn 5 L mixture, milk = 4 L costing Rs. 80. Cost per litre = 80/5 = Rs. 16/L. Alligation: milk 20, water 0, ratio 4 : 1 ⇒ mean = (4×20 + 1×0)/5 = 16. Hence the answer is Rs. 16/L.
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Q4 EasyThere is 60 L of milk. 6 L water is added. New milk : water ratio is:ExplanationMilk = 60 L, water = 6 L, total = 66 L. Ratio milk : water = 60 : 6 = 10 : 1. Hence the answer is 10 : 1.
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Q5 MediumA dishonest milkman professes to sell his milk at cost price but he mixes it with water and thereby gains 25%. The percentage of water in the mixture is:ExplanationLet CP of 1 L pure milk = Rs. 1. He sells 1 L mixture at CP Rs. 1 and gains 25% ⇒ CP of mixture = 1/1.25 = Rs. 0.80. Thus 1 L mixture has 0.80 L milk and 0.20 L water. Water % = 20%. Alligation: milk 1, water 0, mean 0.8 ⇒ milk : water = 0.8 : 0.2 = 4 : 1 ⇒ water = 20%. Hence the answer is 20%.
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Q6 MediumTwo mixtures have milk:water 5:3 and 3:5. Equal quantities mixed. Ratio of milk:water in new mixture:ExplanationTake 8 L of each. First: milk 5 L, water 3 L. Second: milk 3 L, water 5 L. Combined milk = 8 L, water = 8 L ⇒ milk : water = 1 : 1. Hence the answer is 1 : 1.
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Q7 MediumPure milk costs Rs. 16 per litre. After adding water, milkman sells at Rs. 15/litre and gains 25%. Ratio milk:water:ExplanationSP = Rs. 15 with 25% gain ⇒ CP of mixture = 15/1.25 = Rs. 12/L. Alligation: milk Rs. 16, water Rs. 0, mean Rs. 12. Milk : water = (12 − 0) : (16 − 12) = 12 : 4 = 3 : 1. Hence the answer is 3 : 1.
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Q8 MediumRatio of milk and water in a mixture is 7:3. If 4 litres of water is added, ratio becomes 7:4. Quantity of milk:ExplanationLet milk = 7x, water = 3x. Then 7x : (3x + 4) = 7 : 4 ⇒ 7x/(3x + 4) = 7/4 ⇒ 28x = 21x + 28 ⇒ 7x = 28 ⇒ x = 4. Milk = 28 L. Hence the answer is 28 L.
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Q9 MediumA can of 60 litres has milk and water 4:1. How much water to add to make milk:water 3:2?ExplanationMilk = 48 L, water = 12 L. Let water added = w. Then 48 : (12 + w) = 3 : 2 ⇒ 48/3 = (12 + w)/2 ⇒ 32 = 12 + w ⇒ w = 20. Hence the answer is 20 L.
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Q10 MediumA mixture contains milk and water in the ratio 9 : 4. On adding 4 L water, ratio becomes 9 : 5. Quantity of milk is:ExplanationLet milk = 9x, water = 4x. Then 9x : (4x + 4) = 9 : 5 ⇒ 9x/(4x + 4) = 9/5 ⇒ 5 × 9x = 9(4x + 4) ⇒ 45x = 36x + 36 ⇒ 9x = 36 ⇒ x = 4. Milk = 36 L. Hence the answer is 36 L.
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Q11 HardA milk vendor has 2 cans of milk. The first contains 25% water and the rest milk. The second contains 50% water. How much milk should he mix from each of the containers so as to get 12 litres of milk such that the ratio of water to milk is 3:5?ExplanationTarget water : milk = 3 : 5 ⇒ milk % = 62.5%. Can 1 milk % = 75%; Can 2 milk % = 50%. Alligation cross: Can1 : Can2 = (62.5 − 50) : (75 − 62.5) = 12.5 : 12.5 = 1 : 1. Take 6 L from each to make 12 L. Hence the answer is 6 L, 6 L.
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Q12 HardIn 50 L mixture, milk : water = 3 : 2. How much milk must be added to make milk : water = 2 : 1?ExplanationInitially milk = 30 L, water = 20 L. Water stays 20 L. Let milk added = m. Then (30 + m) : 20 = 2 : 1 ⇒ 30 + m = 40 ⇒ m = 10. Hence the answer is 10 L.
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Q13 HardA vessel has 80 L milk. 8 L is replaced by water twice. Milk left is:ExplanationMilk left = 80 × (1 − 8/80)^2 = 80 × (0.9)^2 = 80 × 0.81 = 64.8 L. Hence the answer is 64.8 L.
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Q14 HardMilk : water = 7 : 5. If 6 L milk is added, ratio becomes 5 : 3. Original quantity of the mixture is:ExplanationLet milk = 7x and water = 5x. After adding 6 L milk: (7x + 6) : 5x = 5 : 3 ⇒ 3(7x + 6) = 25x ⇒ 21x + 18 = 25x ⇒ 18 = 4x ⇒ x = 4.5. Original mixture = 12x = 54 L. Check: milk becomes 31.5 + 6 = 37.5 L, water = 22.5 L ⇒ 37.5 : 22.5 = 5 : 3. Hence the answer is 54 L.
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DS1 Easy
How much water should be added to the 60 L can?
Statement IPresent milk : water = 4 : 1.
Statement IIRequired milk : water = 3 : 2.
ExplanationTogether: milk 48 L, water 12 L ⇒ add 20 L. Neither alone enough. -
DS2 Easy
Is the milk : water ratio in the new mixture 1 : 1?
Statement ITwo mixtures with milk : water = 5 : 3 and 3 : 5 are mixed in equal quantities.
Statement IIEqual volumes of milk-only liquids at unknown strengths are mixed.
ExplanationStatement I alone: equal quantities of 5 : 3 and 3 : 5 give milk 8 and water 8 ⇒ ratio 1 : 1 (Yes). Statement II alone has no concentrations, so insufficient. Hence I alone is sufficient.
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DS3 Medium
What percentage of water is in the mixture sold by the milkman?
Statement IHe mixes water with milk and sells at cost price, gaining 25%.
Statement IICP of mixture is 4/5 of CP of pure milk.
ExplanationI alone: water = 20%. II alone: milk fraction = 4/5 ⇒ water = 20%. Either alone is sufficient. -
DS4 Medium
What is the quantity of milk in the vessel?
Statement IMilk : water = 7 : 3.
Statement IIAfter adding 4 L water the ratio becomes 7 : 4.
ExplanationBoth needed: 7x : (3x + 4) = 7 : 4 ⇒ milk = 28 L. Neither alone finds milk.