50 litres of mixture of alcohol and water is in the ratio of 1:4 re spectively. If 10 litres of mixture is taken out and replaced with 10 litres of water, What is the new ratio of alcohol and water respectively?

Options

A. `1:4`
B. `2:11`
C. `1:5`
D. `4:21`

Answer

To solve the problem step by step, we will follow the process of calculating the quantities of alcohol and water in the mixture before and after the changes.

### Step 1: Determine the initial quantities of alcohol and water in the mixture.
The initial ratio of alcohol to water is 1:4. This means that for every 1 part of alcohol, there are 4 parts of water.

- Total parts = 1 (alcohol) + 4 (water) = 5 parts
- Total volume of the mixture = 50 liters

Now, we can calculate the quantities:
- Quantity of alcohol = (1/5) * 50 = 10 liters
- Quantity of water = (4/5) * 50 = 40 liters

### Step 2: Calculate the quantities of alcohol and water after removing 10 liters of the mixture.
When we take out 10 liters of the mixture, we need to maintain the same ratio of alcohol and water in the removed mixture.

- Alcohol in 10 liters removed = (1/5) * 10 = 2 liters
- Water in 10 liters removed = (4/5) * 10 = 8 liters

Now, we subtract these quantities from the initial amounts:
- Remaining alcohol = 10 liters - 2 liters = 8 liters
- Remaining water = 40 liters - 8 liters = 32 liters

### Step 3: Add 10 liters of water to the remaining mixture.
After removing 10 liters of the mixture, we replace it with 10 liters of water:

- New quantity of water = 32 liters + 10 liters = 42 liters

### Step 4: Determine the new ratio of alcohol to water.
Now we have:
- Alcohol = 8 liters
- Water = 42 liters

The new ratio of alcohol to water is:
- Ratio = Alcohol : Water = 8 : 42

To simplify this ratio, we can divide both sides by 2:
- Simplified ratio = 4 : 21

### Final Answer:
The new ratio of alcohol to water is **4:21**.

---