Alligation or Mixture problems involve combining two or more ingredients with different properties (like price, quality, concentration) to get a mixture with a desired property. The principle of alligation helps in finding the ratio in which the ingredients are to be mixed to achieve a specific quality or price in the resulting mixture. These problems are common in real-world scenarios such as cooking, chemistry, and economics.

## 1. A solution of 40 liters contains milk and water in the ratio 7:3. How much water must be added to make the ratio 3:7?

a) 30 liters

b) 40 liters

c) 50 liters

d) 60 liters

### Answer:

c) 50 liters

### Explanation:

```
- Initial quantity of milk = (7/10) * 40 = 28 liters.
- Initial quantity of water = 40 - 28 = 12 liters.
- Let x liters of water be added. The new ratio = (28)/(12+x) = 3/7.
- Solving the equation, x = 50 liters.
```

## 2. Two types of wheat costing $100 and $80 per kg are mixed together in the ratio 2:3. What is the cost price of the mixed variety per kg?

a) $88

b) $90

c) $92

d) $94

### Answer:

b) $90

### Explanation:

```
- Cost of 1 kg of mixed variety = (2*$100 + 3*$80) / (2+3) = $90.
```

## 3. A 20-liter mixture of milk and water contains milk and water in the ratio 3:2. How much more milk should be added to make the ratio 2:1?

a) 5 liters

b) 10 liters

c) 15 liters

d) 20 liters

### Answer:

b) 10 liters

### Explanation:

```
- Quantity of milk = (3/5) * 20 = 12 liters.
- Quantity of water = 20 - 12 = 8 liters.
- Let x liters of milk be added. The new ratio = (12+x)/8 = 2/1.
- Solving the equation, x = 10 liters.
```

## 4. A trader has 100 kg of rice, part of which he sells at 8% profit and the rest at 18% profit. He gains 14% on the whole. What is the quantity sold at 18% profit?

a) 30 kg

b) 40 kg

c) 50 kg

d) 60 kg

### Answer:

b) 40 kg

### Explanation:

```
- Let x kg be sold at 18% profit.
- Using the alligation rule, (18 - 14) : (14 - 8) = (4):(6) = 2:3.
- Therefore, the quantity at 18% profit = (2/5) * 100 kg = 40 kg.
```

## 5. A container contains a mixture of two liquids A and B in the ratio 7:5. When 9 liters of the mixture is drawn off and the container is filled with B, the ratio of A and B becomes 7:9. How many liters of liquid A was contained by the container initially?

a) 10 liters

b) 14 liters

c) 21 liters

d) 28 liters

### Answer:

d) 28 liters

### Explanation:

```
- Let the initial total quantity be x liters.
- Quantity of A initially = (7/12) * x.
- Quantity of A left = (7/12) * x - (7/16) * 9.
- New ratio of A:B = 7:9 implies (7/12) * x - (7/16) * 9 / (5/12) * x + 9 = 7/9.
- Solving, x = 48 liters. Thus, A initially = (7/12) * 48 = 28 liters.
```

## 6. A milk vendor has 2 types of milk: one costing $20 per liter and another $25 per liter. How much of each type should be mixed to get 20 liters of milk worth $22 per liter?

a) 8 liters of $20 milk and 12 liters of $25 milk

b) 10 liters of $20 milk and 10 liters of $25 milk

c) 12 liters of $20 milk and 8 liters of $25 milk

d) 15 liters of $20 milk and 5 liters of $25 milk

### Answer:

c) 12 liters of $20 milk and 8 liters of $25 milk

### Explanation:

```
- Using alligation, the ratio of the mix is calculated as (25 - 22) : (22 - 20) = 3 : 2.
- Based on the alligation calculation, the quantities of each type of milk required are:
- Quantity of $20 milk = (3/5) * 20 = 12 liters
- Quantity of $25 milk = (2/5) * 20 = 8 liters
```

## 7. In what ratio must a grocer mix two types of pulses costing $15/kg and $20/kg, so that the mixture costs $16.50/kg?

a) 2:3

b) 3:2

c) 7:3

d) 4:1

### Answer:

c) 7:3

### Explanation:

```
- Using alligation, (20 - 16.50) : (16.50 - 15) = 3.50 : 1.50 = 7:3.
```

## 8. How many liters of water should be added to a 60-liter mixture of milk and water containing 45% water to make the water content 50% in the new mixture?

a) 3 liters

b) 6 liters

c) 9 liters

d) 12 liters

### Answer:

b) 6 liters

### Explanation:

```
- Water in the original mixture = 45% of 60 = 27 liters.
- Let x liters of water be added. New quantity of water = 27 + x liters.
- New ratio (water) = 50%, so (27 + x) / (60 + x) = 1/2. Solving for x, x = 6 liters.
```

## 9. A 40-liter mixture contains milk and water in the ratio 2:3. How much milk should be added to make it a 3:2 ratio?

a) 5 liters

b) 10 liters

c) 20 liters

d) 25 liters

### Answer:

b) 10 liters

### Explanation:

```
- Milk = (2/5) * 40 = 16 liters, Water = 24 liters.
- Let x liters of milk be added. New ratio (milk:water) = 3:2.
- So, (16 + x)/24 = 3/2. Solving, x = 10 liters.
```

## 10. A mixture of 150 kg of wine and water contains 20% water. How much more water should be added so that water may be 25% of the new mixture?

a) 10 kg

b) 15 kg

c) 20 kg

d) 25 kg

### Answer:

a) 10 kg

### Explanation:

```
- Water in the mixture = 20% of 150 = 30 kg.
- Let x kg of water be added. New % of water = 25%.
- (30 + x) / (150 + x) = 25/100. Solving, x = 10 kg.
```

## 11. A merchant has coffee of two types, one costing $50 per kg and another $80 per kg. In what ratio should he mix them so the mixture costs $65 per kg?

a) 1:2

b) 2:1

c) 3:2

d) 2:3

### Answer:

c) 3:2

### Explanation:

```
- Using alligation, (80 - 65) : (65 - 50) = 15 : 15 = 3:2.
```

## 12. A container contains 40 liters of alcohol. From this container, 4 liters of alcohol is taken out and replaced with water. This process is repeated two more times. How much alcohol is now contained in the container?

a) 28.56 liters

b) 29.16 liters

c) 30.24 liters

d) 31.04 liters

### Answer:

b) 29.16 liters

### Explanation:

```
- Alcohol left after each operation = (1 - 4/40) * previous quantity.
- After 3 operations, alcohol = 40 * (1 - 4/40)^3 ≈ 29.16 liters.
```

## 13. A solution contains 30% alcohol and the rest water. If 3 liters of water is added to the solution, the percentage of alcohol in the new solution becomes 25%. Find the initial volume of the solution.

a) 12 liters

b) 15 liters

c) 18 liters

d) 21 liters

### Answer:

b) 15 liters

### Explanation:

```
- Let the initial volume be x liters.
- Alcohol in the initial solution = 0.30 * x.
- New volume = x + 3, Alcohol percentage = 25%.
- 0.30 * x = 0.25 * (x + 3). Solving, x = 15 liters.
```

## 14. A milkman has two types of milk: one with 15% water and another with 30% water. He wants to sell milk with 25% water content. In what ratio should he mix the two?

a) 1:1

b) 2:1

c) 1:2

d) 3:2

### Answer:

c) 1:2

### Explanation:

```
- Using alligation, (30 - 25) : (25 - 15) = 5 : 10 = 1:2.
```

## 15. A mixture contains wine and water in the ratio 4:1. On adding 5 liters of water, the ratio of wine to water becomes 4:3. Find the initial quantity of the mixture.

a) 16 liters

b) 20 liters

c) 24 liters

d) 28 liters

### Answer:

b) 20 liters

### Explanation:

```
- Let the initial quantity of the mixture be x liters.
- Wine = (4/5) * x, Water = (1/5) * x.
- New ratio (wine:water) = 4:3, New water = (1/5) * x + 5.
- (4/5) * x / [(1/5) * x + 5] = 4/3. Solving, x = 20 liters.
```

## 16. How many liters of a 60% alcohol solution must be mixed with 80 liters of a 15% alcohol solution to get a 45% alcohol solution?

a) 20 liters

b) 40 liters

c) 60 liters

d) 80 liters

### Answer:

b) 40 liters

### Explanation:

```
- Using alligation, (60 - 45) : (45 - 15) = 15 : 30 = 1:2.
- Ratio of the mix = 1:2.
- Quantity of 60% solution = (1/3) * (80 liters) = 40 liters.
```

## 17. In a 150-liter mixture of milk and water, the ratio of milk to water is 4:1. How much water must be added to make the ratio of milk to water 2:1?

a) 30 liters

b) 40 liters

c) 50 liters

d) 60 liters

### Answer:

a) 30 liters

### Explanation:

```
- Milk = (4/5) * 150 = 120 liters, Water = 30 liters.
- Let x liters of water be added. New ratio (milk:water) = 2:1.
- 120 / (30 + x) = 2/1. Solving, x = 30 liters.
```

## 18. A 24-liter mixture contains lemonade and soda in the ratio 5:3. How much soda must be added to make the ratio 5:5?

a) 6 liters

b) 8 liters

c) 10 liters

d) 12 liters

### Answer:

a) 6 liters

### Explanation:

```
- Lemonade = (5/8) * 24 = 15 liters, Soda = 9 liters.
- Let x liters of soda be added. New ratio (lemonade:soda) = 5:5 = 1:1.
- 15 / (9 + x) = 1/1. Solving, x = 6 liters.
```

## 19. A chemist has one solution that is 50% alcohol and another that is 75% alcohol. How many liters of each should be mixed to get 10 liters of a 60% alcohol solution?

a) 4 liters of 50% and 6 liters of 75%

b) 5 liters of 50% and 5 liters of 75%

c) 6 liters of 50% and 4 liters of 75%

d) 7 liters of 50% and 3 liters of 75%

### Answer:

a) 4 liters of 50% and 6 liters of 75%

### Explanation:

```
- Using alligation, (75 - 60) : (60 - 50) = 15 : 10 = 3:2.
- Ratio of the mix = 3:2.
- Quantity of 50% solution = (2/5) * 10 liters = 4 liters, 75% solution = 6 liters.
```

## 20. In a 40-liter mixture of milk and water, the ratio is 7:1. How much water must be added to make the water content 25% of the mixture?

a) 7 liters

b) 9 liters

c) 11 liters

d) 13 liters

### Answer:

c) 11 liters

### Explanation:

```
- Milk = (7/8) * 40 = 35 liters, Water = 5 liters.
- Let x liters of water be added. New ratio = 35 / (5 + x).
- For water to be 25%, (5 + x) / (40 + x) = 25/100.
- Solving, x = 11 liters.
```

## 21. A mixture of 90 kg of fruit and nuts contains 70% fruit. How many kilograms of nuts should be added to make the percentage of fruit 60%?

a) 15 kg

b) 20 kg

c) 25 kg

d) 30 kg

### Answer:

a) 15 kg

### Explanation:

```
- Fruit = 70% of 90 = 63 kg, Nuts = 27 kg.
- Let x kg of nuts be added. New percentage of fruit = 63 / (90 + x).
- 63 / (90 + x) = 60/100. Solving, x = 15 kg.
```

## 22. A shopkeeper mixes two types of rice, one costing $18 per kg and the other $25 per kg, in the ratio 2:3. What is the average cost of the mixed rice per kg?

a) $20.20

b) $21.40

c) $22.60

d) $23.80

### Answer:

b) $21.40

### Explanation:

```
- Cost of 1 kg of mixed rice = (2*$18 + 3*$25) / (2+3) = $21.40.
```