Practice Questions
Concept Explanation
Most mixture questions involve exactly two items at different rates. Find either the mean price, a missing quantity, or the mixing ratio.
Key Formulas
- Mean m = (C1Q1 + C2Q2) / (Q1 + Q2)
- Q2 = Q1(C1 − m) / (m − C2) from alligation
- Total cost = C1Q1 + C2Q2
Example
Mix 8 kg at Rs. 25/kg with 12 kg at Rs. 40/kg. Mean = (200 + 480)/20 = Rs. 34/kg.
Common Trap
When finding a missing quantity, set up the weighted mean equation rather than guessing.
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Q1 Easy8 kg of tea at Rs. 50/kg is mixed with 12 kg of tea at Rs. 70/kg. Mean price of the mixture is:ExplanationMean = (8 × 50 + 12 × 70) / (8 + 12) = (400 + 840) / 20 = 1240 / 20 = Rs. 62/kg. Hence the answer is Rs. 62/kg.
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Q2 Easy15 L of oil at Rs. 40/L is mixed with 25 L of oil at Rs. 56/L. Mean price per litre is:ExplanationMean = (15 × 40 + 25 × 56) / 40 = (600 + 1400) / 40 = 2000 / 40 = Rs. 50/L. Hence the answer is Rs. 50/L.
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Q3 EasyTwo grades of sugar at Rs. 28/kg and Rs. 40/kg are mixed in the ratio 3 : 1. Mean price is:ExplanationMean = (3 × 28 + 1 × 40) / 4 = (84 + 40) / 4 = 124 / 4 = Rs. 31/kg. Hence the answer is Rs. 31/kg.
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Q4 Easy40 kg of flour at Rs. 25/kg is mixed with 60 kg of flour at Rs. 35/kg. Total cost of the mixture is:ExplanationTotal cost = 40 × 25 + 60 × 35 = 1000 + 2100 = Rs. 3100. Hence the answer is Rs. 3100.
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Q5 EasyA mixture is made by mixing 9 kg of pulses at Rs. 45/kg with 6 kg at Rs. 60/kg. Mean price is:ExplanationMean = (9 × 45 + 6 × 60) / 15 = (405 + 360) / 15 = 765 / 15 = Rs. 51/kg. Hence the answer is Rs. 51/kg.
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Q6 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 and water = 3x. After adding 4 L water: 7x : (3x + 4) = 7 : 4. So 7x / (3x + 4) = 7/4 ⇒ 4 × 7x = 7(3x + 4) ⇒ 28x = 21x + 28 ⇒ 7x = 28 ⇒ x = 4. Milk = 7 × 4 = 28 L. Hence the answer is 28 L.
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Q7 MediumA can of 60 litres has milk and water 4:1. How much water to add to make milk:water 3:2?ExplanationMilk = 4/5 × 60 = 48 L; water = 12 L. Milk stays 48 L. Let w litres of water be added: 48 : (12 + w) = 3 : 2 ⇒ 48/3 = (12 + w)/2 ⇒ 16 = (12 + w)/2 ⇒ 32 = 12 + w ⇒ w = 20. Hence the answer is 20 L.
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Q8 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 Rs. 1 with 25% gain ⇒ CP of 1 L mixture = 1/1.25 = Rs. 0.80. So 1 L mixture contains 0.80 L milk and 0.20 L water. Water % = 20%. (Alligation: milk CP 1, water 0, mean 0.8 ⇒ milk : water = 0.8 : 0.2 = 4 : 1 ⇒ water = 1/5 = 20%.) Hence the answer is 20%.
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Q9 MediumHow many kg of rice at Rs. 36/kg should be mixed with 20 kg of rice at Rs. 24/kg so that the mixture is worth Rs. 30/kg?ExplanationAlligation: cheaper Rs. 24, dearer Rs. 36, mean Rs. 30. Dearer : cheaper = (30 − 24) : (36 − 30) = 6 : 6 = 1 : 1. So dearer quantity = cheaper quantity = 20 kg. Check: (20 × 24 + 20 × 36)/40 = 1200/40 = 30. Hence the answer is 20 kg.
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Q10 MediumMilk at Rs. 20/L and water (free) are mixed to get a mixture costing Rs. 16/L. Ratio of milk to water is:ExplanationAlligation: milk Rs. 20, water Rs. 0, mean Rs. 16. Milk : water = (16 − 0) : (20 − 16) = 16 : 4 = 4 : 1. Hence the answer is 4 : 1.
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Q11 MediumIn what ratio must a grocer mix two varieties of pulses costing Rs. 15/kg and Rs. 20/kg respectively so as to get a mixture worth Rs. 16.50/kg?ExplanationAlligation cross: cheaper : dearer = (20 − 16.50) : (16.50 − 15) = 3.50 : 1.50 = 7 : 3. Hence the answer is 7 : 3.
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Q12 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?ExplanationRequired water : milk = 3 : 5 ⇒ milk fraction = 5/8 = 62.5%. Can 1 has 75% milk; Can 2 has 50% milk. Alligation on milk %: Can1 : Can2 = (62.5 − 50) : (75 − 62.5) = 12.5 : 12.5 = 1 : 1. From 12 L total, take 6 L from each can. Hence the answer is 6 L, 6 L.
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Q13 HardA shopkeeper mixes 30 kg of rice at Rs. 18/kg with x kg at Rs. 30/kg to get a mixture worth Rs. 22/kg. Find x.ExplanationAlligation cross: cheaper Rs. 18, dearer Rs. 30, mean Rs. 22. Cheaper : dearer = (30 − 22) : (22 − 18) = 8 : 4 = 2 : 1. Given cheaper = 30 kg, so 30 / x = 2 / 1 ⇒ x = 15 kg. Check: (30 × 18 + 15 × 30) / 45 = (540 + 450) / 45 = 990 / 45 = Rs. 22/kg. Hence the answer is 15 kg.
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Q14 HardTwo liquids costing Rs. 10/L and Rs. 16/L are mixed to get 36 L of mixture worth Rs. 13/L. Quantity of the cheaper liquid is:ExplanationAlligation: cheaper : dearer = (16 − 13) : (13 − 10) = 3 : 3 = 1 : 1. Cheaper = half of 36 L = 18 L. Hence the answer is 18 L.
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DS1 Easy
What is the mean price of the mixture?
Statement I8 kg at Rs. 50/kg and 12 kg at Rs. 70/kg are mixed.
Statement IITotal 20 kg costs Rs. 1240.
ExplanationI: mean = 1240/20 = Rs. 62/kg. II: mean = 1240/20 = Rs. 62/kg. Either alone is sufficient. -
DS2 Medium
How much milk is in the mixture?
Statement IMilk : water = 7 : 3.
Statement IIAfter adding 4 L water the ratio becomes 7 : 4.
ExplanationFrom both: 7x : (3x + 4) = 7 : 4 ⇒ milk = 28 L. Neither statement alone gives milk quantity. Hence both together are needed.
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DS3 Medium
How much water must be added to the 60 L can?
Statement IMilk : water in the can is 4 : 1.
Statement IIFinal required milk : water is 3 : 2.
ExplanationI alone: milk = 48 L, water = 12 L, but target unknown. II alone: target ratio without initial quantities. Together: add 20 L water. Hence both needed. -
DS4 Hard
How much should be taken from each can to get 12 L with water : milk = 3 : 5?
Statement ICan 1 has 25% water; Can 2 has 50% water.
Statement IIAlligation on milk % gives Can1 : Can2 = 1 : 1.
ExplanationStem already requires 12 L at water : milk = 3 : 5 (milk 62.5%). Statement I alone: alligation on 75% and 50% milk to 62.5% gives 1 : 1 ⇒ 6 L each. Statement II alone with the 12 L stem also gives 6 L each. Hence either alone is sufficient.