PBI Rules and Formulae: PBI INSTITUTE Alligation and Mixture 1. 2. 3. 4. 1. Alligation : It is the rule that enables us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture of a desired price. Mean Price : The cost price of a unit quantity of the mixture is called the mean price. Suppose a container contains x of liquid from which y units are taken out and replaced by water. After n operations, the quantity of pure liquid = units. Rule of Alligation: If two ingredients are mixed, then Quantity of cheaper Quantity of dearer We present as under : C.P. of a unit quantity of cheaper (c) C.P. of dearer - Mean Price Mean price - C.P. of cheaper C.P. of a unit quanitity of dearer (d) Mean price (Cheaper quantity) : (Dearer quantity) = (d-m):(m-c). EXERCISE In 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 ? Solution By the rule of alligation: C.P. of 1 kg rice of 1st kind (in paise) C.P. of 1 kg rice of 2nd kind (in paise)
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PBI 930 Mean price pajse 1000 80 PBI INSTITUTE 1 80 2. Required ratio = 80 : 70 = 8 : 7. In what ratio must be a grocer mix two varities of tea worth Rs. 60 a kg and Rs. 65 a Kg so that by selling the mixture at Rs. 68.20 a Kg he may gain 10% ? Solution S.P. of 1 kg of the mixture = C.P. of 1 kg of the mixture = Rs. 68.20, Gain = 10 0/0 Rs. (100 / 110 x 68.20) = Rs. 62. By the rule of alligation: C.P. of 1 kg tea of 1st kind Rs. 60 3 Mean price Rs. 62 C.P. of 1 kg tea of 2nd kind Rs 65 2 3. Required ratio = 3 : 2. In 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? Solution By the rule of alligation: Cost of 1 kg pulses of 1st Cost of 1 kg pulses of 2nd kind Rs. 15 3.50 Mean price Rs. 16.50 kind R . 20 .50 4. Required ratio = 3.50 : 1.50 = 35 : 15 = 7 : 3. 8 litres are drawn from a cask full of wine and is then filled with water. This operation is performed three more times. The ratio of the quantity of wine now left in cask to that of the water is 16 : 65. How much wine the cask hold originally ?
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PBI A. C. Solution 18 litres 32 litres B. D. PBI INSTITUTE 24 litres 42 litres Let the quantity of the wine in the cask originally be x litres then, quantity of wine left in cask after 4 operations = [x(l- 8/x)4] litres. Therefore x(l- 8/x)4 / x = 16/81 (1- 8/x)4 (2/3)2 (x -8/ x) = 2/3 - 24 = x = 24. 5. A jar full of whiskey contains 40% alcohol. A part of this whisky is replaced by another containing 19% alcohols and now the percentage of alcohol was found to be 26%. The quantity of whisky replaced of whisky replaced is A. 1/3 c. 2/5 Solution By the rule of alligation: Strength of first jar Mean price 26 % 7 B. 2/3 D. 3/5 Strength of second jar 14 So, ratio of 1st and 2nd quantities = 7 : 14 = 1 : 2. Required quantity replaced = 2/3 6. A 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 A. 400 kg C. 600 kg Solution By the rule of alligation: Profit of first part B. 560 kg D. 640 kg 8% 4 Mean profit 14 % Profit of second part 6 So, ratio of 1st and 2nd parts = 4: 2 : 3. Quantity of 2nd kind = (3/5 x 1000)kg = 600 kg. 7. How many kilograms of sugar costing Rs. 9 per kg must be mixed with 27 kg of sugar costing Rs. 7 per Kg so that there may be a gain of 10 % by selling the mixture at Rs. 9.24 per Kg ? A. 36 Kg C. 54 Kg B. 42 Kg D. 63 Kg
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PBI Solution By the rule of alligation: C.P. of 1 kg sugar of 1st kind 1.40 PBI INSTITUTE C.P. of 1 kg sugar of 2nd kind Mean price Rs. 8.40 .60 Ratio of quantities of 1st and 2nd kind = 14 : 6=7 : 3. Let x kg of sugar of 1st kind be mixed with 27 kg of 2nd kind. Then, 7: 3 = x: 27 or x = (7 x 27 / 3) = 63 kg. 8. Solution A vessel is filled with liquid, 3 parts of which are water and 5 parts of syrup. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup? A. 1/3 c. 1/5 B. 1/4 D. 1/7 Suppose the vessel initially contains 8 litres of liquid. Let x littres of this liquid be replaced with water. Quantity of water in new mixture = (3 - 3x/8 + x) litres. Quantity of syrup in new mixture = (5 - 5x/8) litres. (3 - + x) = (5 - = 5x + 24 = 40 - 5x lox = 16 x = 8/5 So, part of the mixture replaced = (8/5 x 1/8) — - 1/5. 9. The cost of Type 1 rice is Rs. 15 per kg and Type 2 rice is Rs.20 per kg. If both Type 1 and Type 2 are mixed in the ratio of 2 : 3, then the price per kg of the mixed variety of rice is A. Rs. 19.50 C. Rs. 18 Solution B. Rs. 19 D. Rs. 18.50 Let the price of the mixed variety be Rs. x per kg. By the rule of alligation, we have . Cost of 1 kg of type 1 rice Cost of 1 kg of type 2 rice Rs.15 (20-x) Mean price Rs.x Rs.20 ( -15) = 2/3 60 - = - 30 5x = 90 18. so, price of the mixture is Rs. 18 per kg. 10. In what ratio must water be mixed with milk costing Rs. 12 per litre to obtain a mixture worth of Rs.8 per litre?
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PBI INSTITUTE C. Solution PBI 4: : 90=2: 3 3 D. By the rule of alligation: C.P. of 1 litre of water Mean price Rs.B 4 C.P. of 1 litre of milk 8 Ratio of water to milk = 4: 1 : 2 In what ratio must tea at Rs.62 per Kg be mixed with tea at Rs. 72 per Kg so that the 11. mixture must be worth Rs. 64.50 per Kg? Solution By the rule of alligation: Cost of 1 kg tea of 1st kind 6200 p 750 Mean price 6450 p Cost of 1 kg tea of 2nd kind 72 op 250 Required ratio = 750 : 250 = 3 : 1 12. Find 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. Solution By the rule of alligation: Cost of 1 kg rice of 1st kind 720 p Mean price 630 p 60 Required ratio — _ 60 Cost of 1 kg rice of 2nd kind 570 p
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PBI PBI INSTITUTE 13. A container contains 40 litres of milk.From this container 4 litres of milk was taken out and replaced by water. This process was repeated further two times. How much milk is now contained by the container. A. 26.34 litres C. 28 litres Solution Amount of milk left after 3 operations litres 40 X —x—x— 10 10 10 B. 27.36 liters D. 29.16 litres - 29.16 litres. 14. Tea worth Rs. 126 per kg are mixed with a third variety in the ratio 1: 1 : 2. If the mixture is worth Rs. 153 per kg, the price of the third variety per kg will be A. Rs. 169.50 C. Rs. 175.50 Solution B. Rs.1700 D. Rs. 180 Since first second varieties are mixed in equal proportions, so their average price - Rs.(126+135/2) = Rs.130.50 So, the mixture is formed by mixing two varieties, one at Rs. 130.50 per kg and the other at say, Rs. x per kg in the ratio 2 : 2, i.e., 1 : 1. We have to find x. Cost of 1 kg tea of 1st kind Rs.130.50 (x-153) Cost of 1 kg tea of 2nd kind Rs.x Mean price Rs. 153 22.50 x-153/22.50 = 1 x - 153 = 22.50 Hence, price of the third variety = Rs.175.50 per kg. 15. A 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? A. 8 litres B. 61itres, 6 litres C. 51itres, 7 litres D. 71itres, 4 litres Solution Let the cost of 1 litre milk be Re 1 Milk in 1 litre mix. in 1st can = 3/4 litre, C.P. of 1 litre mix. in 1st can Re. 3/4 Milk in 1 litre mix. in 2nd can = 1/2 litre, C.P. of 1 litre mix. in 2nd can Re. 1/2 Milk in 1 litre of final mix. = 5/8 litre, mean price = Re. 5/8. By the rule of alligation, we have: Cost of 1 kg mixture of 1st kind Cost of 1 kg mixture of 2nd kind
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PBI IN PBI INSTITUTE 12 Mean price Ratio of two mixtures — - 1/8 : 1/8 = 1:1. So, quantity of mixture taken from each can = (1/2 X 12) = 6 litres. 16. Two vessels A and B contain spirit and water in the ratio 5 : 2 and 7 : 6 respectively. Find the ratio in which these mixture be mixed to obtain a new mixture in vessel C containing spirit and water in the ration 8 : 5 ? Solution Let the C.P. of spirit be Re. Spirit in 1 litre mix. of A = Spirit in 1 litre mix. of B = Spirit in 1 litre mix. of C = 1 litre. 5/7 litre, C.P. of 1 litre mix. in A = Re. 5/7 7/13 litre, C.P. of 1 litre mix. in B = Re. 7/13 8/13 litre, Mean price = Re. 8/13. By the rule of alligation, we have: Cost of 1 litre mixture in A (5/7) (1/13) Cost of 1 litre mixture in B (7 13) Mean price 8/13 Required ratio = 1/13 : 9/91 = 7:9. A can contains a mixture of two liquids A and B in the ratio 17. 7 : 5. When 9 litres of mixture are drawn off and the can is filled with B, the ratio of A and B becomes 7 : 9. How many litres of liquid A was contained by the can initially? A. 10 c. 21 Solution B. 20 D. 25 Suppose the can initially contains 7x and 5x litres of mixtures A and B respectively Quantity of A in mixture left = - 7/12 x 9) litres = - 21/4) litres. Quantity of B in mixture left = (5x - 5/12 x 9) litres = (5x - 15/4) litres. - 21/4) / [(5x - 15/4)+9] = 7/9 = > - + 21 = 7/9 - 189 = 147 = 336 3. So, the can contained 21 litres of A.
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