Alligation or Mixture: Formulas, Shortcuts and Solved Examples — MCQs with Answers (Set 2)

Let the container start with x = 125 litres of pure milk. Each time, 25 litres of the mixture are drawn off and replaced with water. Fraction drawn off each time = 25/125 = 0.2. So the fraction of pure milk remaining after each operation = 1 – 0.2 = 0.8. This is repeated 2 times. After operation 1, pure milk = 125 x 0.8 = 100 litres. After operation 2, pure milk = 100 x 0.8 = 80 litres. By the repeated-dilution formula, pure milk left = 125 x (0.8)^2 = 80 litres.

Let CP1 = Rs 6 per kg for pulses, and CP2 = Rs 21 per kg for rice. The mixture is sold at Rs 25 per kg with a gain of 25%. So the mixture’s cost price = 100 x 25 / (100+25) = Rs 20 per kg. By alligation, ratio of pulses to rice = (CP2 – mixture CP) : (mixture CP – CP1) = (21-20) : (20-6) = 1 : 14. Dividing by the HCF 1 gives the simplest ratio 1 : 14. With 70 kg of rice, let the pulses quantity be x kg. Then x : 70 = 1 : 14. So x = (1/14) x 70 = 5 kg. Check: 5 kg of pulses plus 70 kg of rice costs Rs 1500 for 75 kg, which is Rs 20 per kg, matching the mixture’s cost price and the 25% gain.

Let CP1 = Rs 7 per kg for cashews, and CP2 = Rs 19 per kg for almonds. The mixture is sold at Rs 18 per kg with a gain of 50%. So the mixture’s cost price = 100 x 18 / (100+50) = Rs 12 per kg. By alligation, ratio of cashews to almonds = (CP2 – mixture CP) : (mixture CP – CP1) = (19-12) : (12-7) = 7 : 5. Dividing by the HCF 1 gives the simplest ratio 7 : 5. With 20 kg of almonds, let the cashews quantity be x kg. Then x : 20 = 7 : 5. So x = (7/5) x 20 = 28 kg. Check: 28 kg of cashews plus 20 kg of almonds costs Rs 576 for 48 kg, which is Rs 12 per kg, matching the mixture’s cost price and the 50% gain.

Let CP1 = Rs 11 per kg for pulses, and CP2 = Rs 28 per kg for rice. The mixture is sold at Rs 30 per kg with a gain of 20%. So the mixture’s cost price = 100 x 30 / (100+20) = Rs 25 per kg. By alligation, ratio of pulses to rice = (CP2 – mixture CP) : (mixture CP – CP1) = (28-25) : (25-11) = 3 : 14. Dividing by the HCF 1 gives the simplest ratio 3 : 14. With 98 kg of rice, let the pulses quantity be x kg. Then x : 98 = 3 : 14. So x = (3/14) x 98 = 21 kg. Check: 21 kg of pulses plus 98 kg of rice costs Rs 2975 for 119 kg, which is Rs 25 per kg, matching the mixture’s cost price and the 20% gain.

Let CP1 = Rs 7 per kg for coffee powder, and CP2 = Rs 25 per kg for sugar. The mixture is sold at Rs 22 per kg with a gain of 10%. So the mixture’s cost price = 100 x 22 / (100+10) = Rs 20 per kg. By alligation, ratio of coffee powder to sugar = (CP2 – mixture CP) : (mixture CP – CP1) = (25-20) : (20-7) = 5 : 13. Dividing by the HCF 1 gives the simplest ratio 5 : 13. With 91 kg of sugar, let the coffee powder quantity be x kg. Then x : 91 = 5 : 13. So x = (5/13) x 91 = 35 kg. Check: 35 kg of coffee powder plus 91 kg of sugar costs Rs 2520 for 126 kg, which is Rs 20 per kg, matching the mixture’s cost price and the 10% gain.

Let CP1 = Rs 11 per kg for sugar, and CP2 = Rs 25 per kg for pulses. The mixture is sold at Rs 33 per kg with a gain of 50%. So the mixture’s cost price = 100 x 33 / (100+50) = Rs 22 per kg. By alligation, ratio of sugar to pulses = (CP2 – mixture CP) : (mixture CP – CP1) = (25-22) : (22-11) = 3 : 11. Dividing by the HCF 1 gives the simplest ratio 3 : 11. With 77 kg of pulses, let the sugar quantity be x kg. Then x : 77 = 3 : 11. So x = (3/11) x 77 = 21 kg. Check: 21 kg of sugar plus 77 kg of pulses costs Rs 2156 for 98 kg, which is Rs 22 per kg, matching the mixture’s cost price and the 50% gain.

Let CP1 = Rs 6 per kg for rice, and CP2 = Rs 11 per kg for sugar. The mixture is sold at Rs 11 per kg with a gain of 10%. So the mixture’s cost price = 100 x 11 / (100+10) = Rs 10 per kg. By alligation, ratio of rice to sugar = (CP2 – mixture CP) : (mixture CP – CP1) = (11-10) : (10-6) = 1 : 4. Dividing by the HCF 1 gives the simplest ratio 1 : 4. With 40 kg of sugar, let the rice quantity be x kg. Then x : 40 = 1 : 4. So x = (1/4) x 40 = 10 kg. Check: 10 kg of rice plus 40 kg of sugar costs Rs 500 for 50 kg, which is Rs 10 per kg, matching the mixture’s cost price and the 10% gain.

Let CP1 = Rs 15 per kg for cashews, and CP2 = Rs 35 per kg for sugar. The mixture is sold at Rs 45 per kg with a gain of 50%. So the mixture’s cost price = 100 x 45 / (100+50) = Rs 30 per kg. By alligation, ratio of cashews to sugar = (CP2 – mixture CP) : (mixture CP – CP1) = (35-30) : (30-15) = 5 : 15. Dividing by the HCF 5 gives the simplest ratio 1 : 3. With 15 kg of sugar, let the cashews quantity be x kg. Then x : 15 = 1 : 3. So x = (1/3) x 15 = 5 kg. Check: 5 kg of cashews plus 15 kg of sugar costs Rs 600 for 20 kg, which is Rs 30 per kg, matching the mixture’s cost price and the 50% gain.

Let CP1 = Rs 9 per kg for wheat, and CP2 = Rs 23 per kg for pulses. The mixture is sold at Rs 24 per kg with a gain of 20%. So the mixture’s cost price = 100 x 24 / (100+20) = Rs 20 per kg. By alligation, ratio of wheat to pulses = (CP2 – mixture CP) : (mixture CP – CP1) = (23-20) : (20-9) = 3 : 11. Dividing by the HCF 1 gives the simplest ratio 3 : 11. With 33 kg of pulses, let the wheat quantity be x kg. Then x : 33 = 3 : 11. So x = (3/11) x 33 = 9 kg. Check: 9 kg of wheat plus 33 kg of pulses costs Rs 840 for 42 kg, which is Rs 20 per kg, matching the mixture’s cost price and the 20% gain.

Let CP1 = Rs 12 per kg for coffee powder, and CP2 = Rs 19 per kg for tea. The mixture is sold at Rs 21 per kg with a gain of 50%. So the mixture’s cost price = 100 x 21 / (100+50) = Rs 14 per kg. By alligation, ratio of coffee powder to tea = (CP2 – mixture CP) : (mixture CP – CP1) = (19-14) : (14-12) = 5 : 2. Dividing by the HCF 1 gives the simplest ratio 5 : 2. With 12 kg of tea, let the coffee powder quantity be x kg. Then x : 12 = 5 : 2. So x = (5/2) x 12 = 30 kg. Check: 30 kg of coffee powder plus 12 kg of tea costs Rs 588 for 42 kg, which is Rs 14 per kg, matching the mixture’s cost price and the 50% gain.

Let CP1 = Rs 8 per kg for sugar, and CP2 = Rs 20 per kg for coffee powder. The mixture is sold at Rs 20 per kg with a gain of 25%. So the mixture’s cost price = 100 x 20 / (100+25) = Rs 16 per kg. By alligation, ratio of sugar to coffee powder = (CP2 – mixture CP) : (mixture CP – CP1) = (20-16) : (16-8) = 4 : 8. Dividing by the HCF 4 gives the simplest ratio 1 : 2. With 4 kg of coffee powder, let the sugar quantity be x kg. Then x : 4 = 1 : 2. So x = (1/2) x 4 = 2 kg. Check: 2 kg of sugar plus 4 kg of coffee powder costs Rs 96 for 6 kg, which is Rs 16 per kg, matching the mixture’s cost price and the 25% gain.

Let CP1 = Rs 14 per kg for rice, and CP2 = Rs 28 per kg for tea. The mixture is sold at Rs 24 per kg with a gain of 50%. So the mixture’s cost price = 100 x 24 / (100+50) = Rs 16 per kg. By alligation, ratio of rice to tea = (CP2 – mixture CP) : (mixture CP – CP1) = (28-16) : (16-14) = 12 : 2. Dividing by the HCF 2 gives the simplest ratio 6 : 1. With 6 kg of tea, let the rice quantity be x kg. Then x : 6 = 6 : 1. So x = (6/1) x 6 = 36 kg. Check: 36 kg of rice plus 6 kg of tea costs Rs 672 for 42 kg, which is Rs 16 per kg, matching the mixture’s cost price and the 50% gain.

Let CP1 = Rs 10 per kg for rice, and CP2 = Rs 19 per kg for almonds. The mixture is sold at Rs 18 per kg with a gain of 20%. So the mixture’s cost price = 100 x 18 / (100+20) = Rs 15 per kg. By alligation, ratio of rice to almonds = (CP2 – mixture CP) : (mixture CP – CP1) = (19-15) : (15-10) = 4 : 5. Dividing by the HCF 1 gives the simplest ratio 4 : 5. With 45 kg of almonds, let the rice quantity be x kg. Then x : 45 = 4 : 5. So x = (4/5) x 45 = 36 kg. Check: 36 kg of rice plus 45 kg of almonds costs Rs 1215 for 81 kg, which is Rs 15 per kg, matching the mixture’s cost price and the 20% gain.

Let the cost price of milk = Rs 1 per litre, and treat water as free (Rs 0 per litre). The mixture is sold at Rs 1 per litre, the same as milk’s cost price, giving a gain of 50%. So the mixture’s own cost price = 100/(100+50) per litre. By alligation, ratio of water to milk = (milk CP – mixture CP) : (mixture CP – water CP) = (1 – 100/(100+50)) : (100/(100+50) – 0) = 50 : 100. Dividing by the HCF 50 gives the simplest ratio 1 : 2. Check: mixing 1 part water with 2 parts milk (Rs 1 per litre each part of milk) costs Rs 2 for 3 parts of mixture. Selling this mixture at Rs 1 per litre gives a gain of 50% on that cost, as required.

Let the cost price of milk = Rs 1 per litre, and treat water as free (Rs 0 per litre). The mixture is sold at Rs 1 per litre, the same as milk’s cost price, giving a gain of 60%. So the mixture’s own cost price = 100/(100+60) per litre. By alligation, ratio of water to milk = (milk CP – mixture CP) : (mixture CP – water CP) = (1 – 100/(100+60)) : (100/(100+60) – 0) = 60 : 100. Dividing by the HCF 20 gives the simplest ratio 3 : 5. Check: mixing 3 parts water with 5 parts milk (Rs 1 per litre each part of milk) costs Rs 5 for 8 parts of mixture. Selling this mixture at Rs 1 per litre gives a gain of 60% on that cost, as required.

Let the cost price of milk = Rs 1 per litre, and treat water as free (Rs 0 per litre). The mixture is sold at Rs 1 per litre, the same as milk’s cost price, giving a gain of 10%. So the mixture’s own cost price = 100/(100+10) per litre. By alligation, ratio of water to milk = (milk CP – mixture CP) : (mixture CP – water CP) = (1 – 100/(100+10)) : (100/(100+10) – 0) = 10 : 100. Dividing by the HCF 10 gives the simplest ratio 1 : 10. Check: mixing 1 part water with 10 parts milk (Rs 1 per litre each part of milk) costs Rs 10 for 11 parts of mixture. Selling this mixture at Rs 1 per litre gives a gain of 10% on that cost, as required.

Let the cost price of milk = Rs 1 per litre, and treat water as free (Rs 0 per litre). The mixture is sold at Rs 1 per litre, the same as milk’s cost price, giving a gain of 75%. So the mixture’s own cost price = 100/(100+75) per litre. By alligation, ratio of water to milk = (milk CP – mixture CP) : (mixture CP – water CP) = (1 – 100/(100+75)) : (100/(100+75) – 0) = 75 : 100. Dividing by the HCF 25 gives the simplest ratio 3 : 4. Check: mixing 3 parts water with 4 parts milk (Rs 1 per litre each part of milk) costs Rs 4 for 7 parts of mixture. Selling this mixture at Rs 1 per litre gives a gain of 75% on that cost, as required.

Let the cost price of milk = Rs 1 per litre, and treat water as free (Rs 0 per litre). The mixture is sold at Rs 1 per litre, the same as milk’s cost price, giving a gain of 150%. So the mixture’s own cost price = 100/(100+150) per litre. By alligation, ratio of water to milk = (milk CP – mixture CP) : (mixture CP – water CP) = (1 – 100/(100+150)) : (100/(100+150) – 0) = 150 : 100. Dividing by the HCF 50 gives the simplest ratio 3 : 2. Check: mixing 3 parts water with 2 parts milk (Rs 1 per litre each part of milk) costs Rs 2 for 5 parts of mixture. Selling this mixture at Rs 1 per litre gives a gain of 150% on that cost, as required.

Let the cost price of milk = Rs 1 per litre, and treat water as free (Rs 0 per litre). The mixture is sold at Rs 1 per litre, the same as milk’s cost price, giving a gain of 80%. So the mixture’s own cost price = 100/(100+80) per litre. By alligation, ratio of water to milk = (milk CP – mixture CP) : (mixture CP – water CP) = (1 – 100/(100+80)) : (100/(100+80) – 0) = 80 : 100. Dividing by the HCF 20 gives the simplest ratio 4 : 5. Check: mixing 4 parts water with 5 parts milk (Rs 1 per litre each part of milk) costs Rs 5 for 9 parts of mixture. Selling this mixture at Rs 1 per litre gives a gain of 80% on that cost, as required.

Let the cost price of milk = Rs 1 per litre, and treat water as free (Rs 0 per litre). The mixture is sold at Rs 1 per litre, the same as milk’s cost price, giving a gain of 125%. So the mixture’s own cost price = 100/(100+125) per litre. By alligation, ratio of water to milk = (milk CP – mixture CP) : (mixture CP – water CP) = (1 – 100/(100+125)) : (100/(100+125) – 0) = 125 : 100. Dividing by the HCF 25 gives the simplest ratio 5 : 4. Check: mixing 5 parts water with 4 parts milk (Rs 1 per litre each part of milk) costs Rs 4 for 9 parts of mixture. Selling this mixture at Rs 1 per litre gives a gain of 125% on that cost, as required.

Let the cost price of milk = Rs 1 per litre, and treat water as free (Rs 0 per litre). The mixture is sold at Rs 1 per litre, the same as milk’s cost price, giving a gain of 100%. So the mixture’s own cost price = 100/(100+100) per litre. By alligation, ratio of water to milk = (milk CP – mixture CP) : (mixture CP – water CP) = (1 – 100/(100+100)) : (100/(100+100) – 0) = 100 : 100. Dividing by the HCF 100 gives the simplest ratio 1 : 1. Check: mixing 1 part water with 1 part milk (Rs 1 per litre each part of milk) costs Rs 1 for 2 parts of mixture. Selling this mixture at Rs 1 per litre gives a gain of 100% on that cost, as required.

Let the cost price of milk = Rs 1 per litre, and treat water as free (Rs 0 per litre). The mixture is sold at Rs 1 per litre, the same as milk’s cost price, giving a gain of 20%. So the mixture’s own cost price = 100/(100+20) per litre. By alligation, ratio of water to milk = (milk CP – mixture CP) : (mixture CP – water CP) = (1 – 100/(100+20)) : (100/(100+20) – 0) = 20 : 100. Dividing by the HCF 20 gives the simplest ratio 1 : 5. Check: mixing 1 part water with 5 parts milk (Rs 1 per litre each part of milk) costs Rs 5 for 6 parts of mixture. Selling this mixture at Rs 1 per litre gives a gain of 20% on that cost, as required.

Let the cost price of milk = Rs 1 per litre, and treat water as free (Rs 0 per litre). The mixture is sold at Rs 1 per litre, the same as milk’s cost price, giving a gain of 30%. So the mixture’s own cost price = 100/(100+30) per litre. By alligation, ratio of water to milk = (milk CP – mixture CP) : (mixture CP – water CP) = (1 – 100/(100+30)) : (100/(100+30) – 0) = 30 : 100. Dividing by the HCF 10 gives the simplest ratio 3 : 10. Check: mixing 3 parts water with 10 parts milk (Rs 1 per litre each part of milk) costs Rs 10 for 13 parts of mixture. Selling this mixture at Rs 1 per litre gives a gain of 30% on that cost, as required.

Let the cost price of milk = Rs 1 per litre, and treat water as free (Rs 0 per litre). The mixture is sold at Rs 1 per litre, the same as milk’s cost price, giving a gain of 15%. So the mixture’s own cost price = 100/(100+15) per litre. By alligation, ratio of water to milk = (milk CP – mixture CP) : (mixture CP – water CP) = (1 – 100/(100+15)) : (100/(100+15) – 0) = 15 : 100. Dividing by the HCF 5 gives the simplest ratio 3 : 20. Check: mixing 3 parts water with 20 parts milk (Rs 1 per litre each part of milk) costs Rs 20 for 23 parts of mixture. Selling this mixture at Rs 1 per litre gives a gain of 15% on that cost, as required.

Let the cost price of milk = Rs 1 per litre, and treat water as free (Rs 0 per litre). The mixture is sold at Rs 1 per litre, the same as milk’s cost price, giving a gain of 120%. So the mixture’s own cost price = 100/(100+120) per litre. By alligation, ratio of water to milk = (milk CP – mixture CP) : (mixture CP – water CP) = (1 – 100/(100+120)) : (100/(100+120) – 0) = 120 : 100. Dividing by the HCF 20 gives the simplest ratio 6 : 5. Check: mixing 6 parts water with 5 parts milk (Rs 1 per litre each part of milk) costs Rs 5 for 11 parts of mixture. Selling this mixture at Rs 1 per litre gives a gain of 120% on that cost, as required.