Practice Questions
Concept Explanation
The alligation cross finds the ratio in which two ingredients at different prices must be mixed to obtain a desired mean price.
Key Formulas
- (Cheaper quantity) / (Dearer quantity) = (d − m) / (m − c)
- Where c = cheaper price, d = dearer price, m = mean price
- Draw the cross: link each price to the difference between mean and the other price
Example
Cheaper Rs. 20/kg, dearer Rs. 40/kg, mean Rs. 28/kg. Ratio = (40 − 28) : (28 − 20) = 12 : 8 = 3 : 2.
Common Trap
Cheaper quantity pairs with (dearer − mean), not (mean − cheaper) on the same side of the cross.
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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 may be worth Rs. 10 per kg?ExplanationAlligation cross: cheaper Rs. 9.30, dearer Rs. 10.80, mean Rs. 10. Cheaper quantity : Dearer quantity = (10.80 − 10) : (10 − 9.30) = 0.80 : 0.70 = 8 : 7. Hence the answer is 8 : 7.
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Q2 EasyFind 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.ExplanationDearer = Rs. 7.20, cheaper = Rs. 5.70, mean = Rs. 6.30. Quantity of rice at Rs. 7.20 : Quantity at Rs. 5.70 = (mean − cheaper) : (dearer − mean) = (6.30 − 5.70) : (7.20 − 6.30) = 0.60 : 0.90 = 2 : 3. Hence the answer is 2 : 3.
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Q3 EasyAlligation: wines Rs. 40 and Rs. 60 mixed to get Rs. 48. Ratio:ExplanationCheaper Rs. 40, dearer Rs. 60, mean Rs. 48. Alligation cross: cheaper : dearer = (60 − 48) : (48 − 40) = 12 : 8 = 3 : 2. Hence the answer is 3 : 2.
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Q4 EasySugar at Rs. 20/kg and Rs. 35/kg must be mixed to get Rs. 26/kg. Ratio cheaper : dearer is:ExplanationAlligation cross: cheaper : dearer = (35 − 26) : (26 − 20) = 9 : 6 = 3 : 2. Hence the answer is 3 : 2.
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Q5 EasyWheat at Rs. 12/kg and Rs. 18/kg mixed for Rs. 15/kg. Ratio cheaper : dearer is:ExplanationAlligation cross: cheaper : dearer = (18 − 15) : (15 − 12) = 3 : 3 = 1 : 1. Hence the answer is 1 : 1.
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Q6 EasyTwo varieties of rice at Rs. 36/kg and Rs. 48/kg. In what ratio should they be mixed to get Rs. 42/kg?ExplanationAlligation cross: cheaper : dearer = (48 − 42) : (42 − 36) = 6 : 6 = 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 of mixture = Rs. 15/L with 25% gain ⇒ CP of mixture = 15 / 1.25 = Rs. 12/L. Pure milk costs Rs. 16/L; water costs Rs. 0/L. Alligation cross: milk : water = (12 − 0) : (16 − 12) = 12 : 4 = 3 : 1. Hence the answer is 3 : 1.
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Q8 MediumThree types of teas A,B,C costing 70,80,90 mixed in ratio 2:3:5. Cost of mixture:ExplanationMean cost = (2 × 70 + 3 × 80 + 5 × 90) / (2 + 3 + 5) = (140 + 240 + 450) / 10 = 830 / 10 = Rs. 83. Hence the answer is Rs. 83.
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Q9 MediumTwo mixtures have milk:water 5:3 and 3:5. Equal quantities mixed. Ratio of milk:water in new mixture:ExplanationTake equal quantity 8 L of each (parts sum to 8). First: milk = 5 L, water = 3 L. Second: milk = 3 L, water = 5 L. Combined milk = 8 L, water = 8 L. Ratio milk : water = 8 : 8 = 1 : 1. (Alligation on milk %: 62.5% and 37.5% to mean 50% also gives 1 : 1.) Hence the answer is 1 : 1.
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Q10 MediumOil at Rs. 45/L and Rs. 75/L mixed to get Rs. 57/L. Ratio cheaper : dearer is:ExplanationAlligation cross: cheaper : dearer = (75 − 57) : (57 − 45) = 18 : 12 = 3 : 2. Hence the answer is 3 : 2.
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Q11 MediumIn what ratio must coffee at Rs. 50/kg be mixed with coffee at Rs. 80/kg so as to get a mixture worth Rs. 62/kg?ExplanationAlligation cross: cheaper : dearer = (80 − 62) : (62 − 50) = 18 : 12 = 3 : 2. Hence the answer is 3 : 2.
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Q12 MediumGhee at Rs. 100/kg and Rs. 140/kg for mean Rs. 124/kg. Ratio cheaper : dearer is:ExplanationAlligation cross: cheaper : dearer = (140 − 124) : (124 − 100) = 16 : 24 = 2 : 3. Hence the answer is 2 : 3.
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Q13 HardSpirit at Rs. 30/L and Rs. 50/L mixed in the ratio given by alligation for mean Rs. 42/L. Cheaper share in 40 L is:ExplanationAlligation: cheaper : dearer = (50 − 42) : (42 − 30) = 8 : 12 = 2 : 3. Cheaper share = 2/5 × 40 = 16 L. Hence the answer is 16 L.
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Q14 HardSalt at Rs. 8/kg and Rs. 14/kg mixed to get Rs. 11/kg. If total mixture is 30 kg, quantity of dearer salt is:ExplanationAlligation: cheaper : dearer = (14 − 11) : (11 − 8) = 3 : 3 = 1 : 1. Dearer = half of 30 kg = 15 kg. Hence the answer is 15 kg.
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DS1 Easy
What is the mixing ratio of cheaper to dearer rice?
Statement IRice at Rs. 9.30/kg and Rs. 10.80/kg are mixed to get mean Rs. 10/kg.
Statement IIAlligation gives cheaper : dearer = (10.80 − 10) : (10 − 9.30).
ExplanationStatement I alone: alligation gives 0.80 : 0.70 = 8 : 7. Statement II alone states the same cross differences, which also yield 8 : 7. Hence either alone is sufficient. -
DS2 Easy
Is the alligation ratio of two teas 1 : 1?
Statement ITeas cost Rs. 70/kg and Rs. 90/kg and the mean price is Rs. 80/kg.
Statement IITeas cost Rs. 75/kg and Rs. 85/kg and the mean price is Rs. 80/kg.
ExplanationI: (90 − 80) : (80 − 70) = 10 : 10 = 1 : 1. II: (85 − 80) : (80 − 75) = 5 : 5 = 1 : 1. Either alone answers Yes.
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DS3 Medium
What is the ratio in which wines at Rs. 40/L and Rs. 60/L must be mixed?
Statement IThe mean price of the mixture is Rs. 48/L.
Statement IIThe cheaper wine is mixed in larger quantity than the dearer.
ExplanationStatement I alone: alligation gives cheaper : dearer = (60 − 48) : (48 − 40) = 3 : 2. Statement II only compares quantities qualitatively and does not fix the ratio. Hence I alone is sufficient. -
DS4 Medium
What is the cost of the mixture of three teas?
Statement ITeas A, B, C cost Rs. 70, 80, 90 per kg and are mixed in ratio 2 : 3 : 5.
Statement IIThe weighted mean of costs with weights 2, 3, 5 is Rs. 83/kg.
ExplanationI alone: mean = (140 + 240 + 450)/10 = Rs. 83. II alone directly gives Rs. 83. Hence either alone is sufficient.