Practice Questions
Concept Explanation
When two ingredients form a mixture, their quantities are inversely related to how far each price is from the mean price. The ratio of ingredients can be found directly or via alligation.
Key Formulas
- Ratio Q1 : Q2 = (C2 − m) : (m − C1) where m is mean price
- Total parts = sum of ratio parts; each quantity = (its part / total parts) × total quantity
Example
Mean Rs. 24/kg from Rs. 20/kg and Rs. 32/kg. Ratio = (32 − 24) : (24 − 20) = 8 : 4 = 2 : 1.
Common Trap
Cheaper ingredient corresponds to (dearer − mean), not (mean − dearer).
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Q1 EasyIn what ratio must rice at Rs. 9.30 per kg be mixed with rice at Rs. 10.80 per kg so that the mixture be worth Rs. 10 per kg?ExplanationBy alligation, cheaper : dearer = (10.80 − 10) : (10 − 9.30) = 0.80 : 0.70 = 8 : 7. Hence the ratio is 8 : 7.
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Q2 EasyMilk and water in ratio 5:3; total 40 litres. How much milk is present?ExplanationMilk = (5/8) × 40 = 25 litres. Hence the answer is 25 litres.
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Q3 EasyRs. 12/kg and Rs. 18/kg items are mixed for mean Rs. 15/kg. The ratio cheaper:dearer is:ExplanationBy alligation, cheaper : dearer = (18 − 15) : (15 − 12) = 3 : 3 = 1 : 1. Hence the ratio is 1 : 1.
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Q4 EasyTea at Rs. 80/kg and Rs. 120/kg mixed to get Rs. 100/kg. Ratio of cheaper to dearer is:ExplanationBy alligation: (120 − 100) : (100 − 80) = 20 : 20 = 1 : 1. Hence the ratio is 1 : 1.
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Q5 EasyAlcohol:water = 7:3 in 40 litres. How much alcohol is there?ExplanationAlcohol = (7/10) × 40 = 28 litres. Hence the answer is 28 litres.
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Q6 Easy60 kg mixture in ratio 2:3 of two spices. Quantity of the larger part is:ExplanationLarger part = (3/5) × 60 = 36 kg. Hence the answer is 36 kg.
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Q7 MediumMilk and water are mixed in a vessel A in the ratio 4:1 and in vessel B in the ratio 3:2. In what ratio should quantities be taken from the two vessels so as to form a mixture in which milk to water is 7:3?ExplanationMilk fraction in A = 4/5 = 0.8, in B = 3/5 = 0.6, and required = 7/10 = 0.7. By alligation: A : B = (0.7 − 0.6) : (0.8 − 0.7) = 0.1 : 0.1 = 1 : 1. Hence quantities should be taken in the ratio 1 : 1.
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Q8 MediumA merchant has 1000 kg of sugar, part of which he sells at 8% profit and the rest at 18% profit. He gains 14% on the whole. The quantity sold at 18% profit is:ExplanationTreat profits like prices and apply alligation. Ratio of quantity sold at 8% to quantity sold at 18% = (18 − 14) : (14 − 8) = 4 : 6 = 2 : 3. Quantity at 18% profit = (3/5) × 1000 = 600 kg. Hence the answer is 600 kg.
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Q9 MediumThe ratio of milk and water in 80 litres of adulterated milk is 7:3. If water is added to make the ratio 7:9, the quantity of water added is:ExplanationMilk = (7/10) × 80 = 56 litres; water = 24 litres. Milk stays 56 litres. For milk:water = 7:9, water W satisfies 56 : W = 7 : 9 ⇒ W = (56 × 9)/7 = 72 litres. Water added = 72 − 24 = 48 litres. Hence the answer is 48 litres.
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Q10 MediumMix ingredients at Rs. 20/kg and Rs. 35/kg to get Rs. 26/kg. Cheaper:dearer ratio is:ExplanationBy alligation: (35 − 26) : (26 − 20) = 9 : 6 = 3 : 2. Hence cheaper : dearer = 3 : 2.
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Q11 MediumRs. 40/kg and Rs. 70/kg goods mixed for Rs. 52/kg. Ratio cheaper:dearer is:ExplanationBy alligation: (70 − 52) : (52 − 40) = 18 : 12 = 3 : 2. Hence the ratio is 3 : 2.
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Q12 HardTwo vessels A and B contain milk and water mixed in the ratio 8:5 and 5:2 respectively. The ratio in which these two mixtures be mixed to get a new mixture containing milk and water in the ratio 9:4 is:ExplanationMilk fraction in A = 8/13, in B = 5/7, and required = 9/13. By alligation on milk fractions: quantity from A : quantity from B = (5/7 − 9/13) : (9/13 − 8/13). Compute 5/7 = 65/91 and 9/13 = 63/91, so (65/91 − 63/91) : (63/91 − 56/91) = 2/91 : 7/91 = 2 : 7. Hence the required mixing ratio is 2 : 7.
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Q13 HardA mixture of 125 gallons of wine and water contains 20% water. How much water must be added to the mixture in order to make the water 25% of the resulting mixture?ExplanationWater initially = 20% of 125 = 25 gallons; wine = 100 gallons (unchanged). Let x gallons of water be added. Then (25 + x)/(125 + x) = 25/100 = 1/4. So 4(25 + x) = 125 + x ⇒ 100 + 4x = 125 + x ⇒ 3x = 25 ⇒ x = 25/3 = 8 1/3 gallons. Hence the answer is 8 1/3 gallons.
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Q14 HardTwo alloys contain copper and zinc in the ratios 3:2 and 1:4. In what ratio should they be mixed so that the resulting alloy has copper and zinc in equal parts?ExplanationCopper fraction in first alloy = 3/5, in second = 1/5, and required = 1/2. By alligation: first : second = (1/2 − 1/5) : (3/5 − 1/2) = (3/10) : (1/10) = 3 : 1. Hence the alloys must be mixed in the ratio 3 : 1.
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DS1 Easy
An HR analytics quiz asks how much sugar was sold at 18% profit. What is that quantity?
Statement ITotal sugar is 1000 kg; parts are sold at 8% and 18% profit with overall gain 14%.
Statement IIOnly the overall gain is 14%.
ExplanationStatement I alone: alligation gives 8%:18% = 2:3, so at 18% = 600 kg. Statement II alone is insufficient. Answer: A. -
DS2 Easy
For the mixing ratio of two rices. What is the ratio?
Statement IPrices are Rs. 9.30/kg and Rs. 10.80/kg and the mixture is worth Rs. 10/kg.
Statement IIOnly one rice costs Rs. 9.30/kg.
ExplanationStatement I alone: alligation gives 8:7. Statement II alone is insufficient. Answer: A.
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DS3 Medium
In what ratio should quantities be taken from vessels A and B?
Statement IMilk:water is 4:1 in A and 3:2 in B, and the required milk:water is 7:3.
Statement IIOnly vessel A has milk fraction 4/5.
ExplanationStatement I alone: alligation on 4/5, 3/5 and 7/10 gives ratio 1:1. Statement II alone is insufficient. Answer: A. -
DS4 Hard
In what ratio should mixtures from A and B be combined?
Statement IVessel A has milk:water = 8:5 and vessel B has milk:water = 5:2; the target milk:water is 9:4.
Statement IIOnly the target milk fraction is 9/13.
ExplanationStatement I alone is the classic alligation problem and gives A:B = 2:7. Statement II alone is insufficient. Answer: A.