Alligation or Mixture problems
- 1. 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:
Options- A. 400 kg
- B. 560 kg
- C. 600 kg
- D. 640 kg Discuss
Correct Answer: 600 kg
Explanation:
By the rule of alligation, we have:
| Profit on 1st part Profit on 2nd part | ||
| 8% | Mean Profit 14% |
18% |
| 4 | 6 | |
Ration of 1st and 2nd parts = 4 : 6 = 2 : 3
| ∴ Quantity of 2nd kind = | ❨ | 3 | x 1000 | ❩kg | = 600 kg. |
| 5 |
- 2. 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:
Options- A. Rs. 18
- B. Rs. 18.50
- C. Rs. 19
- D. Rs. 19.50 Discuss
Correct Answer: Rs. 18
Explanation:
Let the price of the mixed variety be Rs. x per kg.
By rule of alligation, we have:
| Cost of 1 kg of Type 1 rice Cost of 1 kg of Type 2 rice | ||
| Rs. 15 | Mean Price Rs. x |
Rs. 20 |
| (20 - x) | (x - 15) | |
| ∴ | (20 - x) | = | 2 |
| (x - 15) | 3 |
⟹ 60 - 3x = 2x - 30
⟹ 5x = 90
⟹ x = 18.
- 3. 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?
Options- A. 4 litres, 8 litres
- B. 6 litres, 6 litres
- C. 5 litres, 7 litres
- D. 7 litres, 5 litres Discuss
Correct Answer: 6 litres, 6 litres
Explanation:
Let the cost of 1 litre milk be Re. 1
| Milk in 1 litre mix. in 1st can = | 3 | litre, C.P. of 1 litre mix. in 1st can Re. | 3 |
| 4 | 4 |
| Milk in 1 litre mix. in 2nd can = | 1 | litre, C.P. of 1 litre mix. in 2nd can Re. | 1 |
| 2 | 2 |
| Milk in 1 litre of final mix. = | 5 | litre, Mean price = Re. | 5 |
| 8 | 8 |
By the rule of alligation, we have:
| C.P. of 1 litre mixture in 1st can C.P. of 1 litre mixture in 2nd can | ||||||||
|
Mean Price
|
|
||||||
|
|
|||||||
| ∴ Ratio of two mixtures = | 1 | : | 1 | = 1 : 1. |
| 8 | 8 |
| So, quantity of mixture taken from each can = | ❨ | 1 | x 12 | ❩ | = 6 litres. |
| 2 |
- 4. 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 kg?
Options- A. 3 : 7
- B. 5 : 7
- C. 7 : 3
- D. 7 : 5 Discuss
Correct Answer: 7 : 3
Explanation:
By the rule of alligation:
| Cost of 1 kg pulses of 1st kind Cost of 1 kg pulses of 2nd kind | ||
| Rs. 15 | Mean Price Rs. 16.50 |
Rs. 20 |
| 3.50 | 1.50 | |
∴ Required rate = 3.50 : 1.50 = 7 : 3.
- 5. Tea worth Rs. 126 per kg and Rs. 135 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:
Options- A. Rs. 169.50
- B. Rs. 170
- C. Rs. 175.50
- D. Rs. 180 Discuss
Correct Answer: Rs. 175.50
Explanation:
Since first and second varieties are mixed in equal proportions.
| So, their average price = Rs. | ❨ | 126 + 135 | ❩ | = Rs. 130.50 |
| 2 |
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.
By the rule of alligation, we have:
| Cost of 1 kg of 1st kind Cost of 1 kg tea of 2nd kind | ||
| Rs. 130.50 | Mean Price Rs. 153 |
Rs. x |
| (x - 153) | 22.50 | |
| ∴ | x - 153 | = 1 |
| 22.50 |
⟹ x - 153 = 22.50
⟹ x = 175.50
- 6. 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?
Options- A. 26.34 litres
- B. 27.36 litres
- C. 28 litres
- D. 29.16 litres Discuss
Correct Answer: 29.16 litres
Explanation:
| Amount of milk left after 3 operations = | [ | 40 | ❨ | 1 - | 4 | ❩ | 3 | ] litres |
| 40 |
| = | ❨ | 40 x | 9 | x | 9 | x | 9 | ❩ | = 29.16 litres. |
| 10 | 10 | 10 |
- 7. A 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:
Options- A. 4%
- B. 6¼%
- C. 20%
- D. 25% Discuss
Correct Answer: 20%
Explanation:
Let C.P. of 1 litre milk be Re. 1
Then, S.P. of 1 litre of mixture = Re. 1, Gain = 25%.
| C.P. of 1 litre mixture = Re. | ❨ | 100 | x 1 | ❩ | = | 4 |
| 125 | 5 |
By the rule of alligation, we have:
| C.P. of 1 litre of milk C.P. of 1 litre of water | |||||
| Re. 1 | Mean Price
|
0 | |||
|
|
||||
| ∴ Ratio of milk to water = | 4 | : | 1 | = 4 : 1. |
| 5 | 5 |
| Hence, percentage of water in the mixture = | ❨ | 1 | x 100 | ❩% | = 20%. |
| 5 |
- 8. 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.
Options- A. 1 : 3
- B. 2 : 3
- C. 3 : 4
- D. 4 : 5 Discuss
Correct Answer: 2 : 3
Explanation:
By the rule of alligation:
| Cost of 1 kg of 1st kind Cost of 1 kg of 2nd kind | ||
| 720 p | Mean Price 630 p |
570 p |
| 60 | 90 | |
∴ Required ratio = 60 : 90 = 2 : 3.
- 9. How many kilogram 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?
Options- A. 36 kg
- B. 42 kg
- C. 54 kg
- D. 63 kg Discuss
Correct Answer: 63 kg
Explanation:
S.P. of 1 kg of mixture = Rs. 9.24, Gain 10%.
| ∴ C.P. of 1 kg of mixture = Rs. | ❨ | 100 | x 9.24 | ❩ | = Rs. 8.40 |
| 110 |
By the rule of allilation, we have:
| C.P. of 1 kg sugar of 1st kind Cost of 1 kg sugar of 2nd kind | ||
| Rs. 9 | Mean Price Rs. 8.40 |
Rs. 7 |
| 1.40 | 0.60 | |
∴ Ratio of quantities of 1st and 2nd kind = 14 : 6 = 7 : 3.
Let x kg of sugar of 1st be mixed with 27 kg of 2nd kind.
Then, 7 : 3 = x : 27
| ⟹ x = | ❨ | 7 x 27 | ❩ | = 63 kg. |
| 3 |
- 10. In what ratio must a grocer mix two varieties 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%?
Options- A. 3 : 2
- B. 3 : 4
- C. 3 : 5
- D. 4 : 5 Discuss
Correct Answer: 3 : 2
Explanation:
S.P. of 1 kg of the mixture = Rs. 68.20, Gain = 10%.
| C.P. of 1 kg of the mixture = Rs. | ❨ | 100 | x 68.20 | ❩ | = Rs. 62. |
| 110 |
By the rule of alligation, we have:
| Cost of 1 kg tea of 1st kind. Cost of 1 kg tea of 2nd kind. | ||
| Rs. 60 | Mean Price Rs. 62 |
Rs. 65 |
| 3 | 2 | |
∴ Required ratio = 3 : 2.
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