Two casks of 48 and 42 litres are filled with mixture of wine and water, the proportions in the two casks being respectively 13 : 7 and 18 : 17. If the contents of the two casks be mixed and 20 litres of water added to the whole what will be the proportion of wine to water in the result ?

Options

A. 13 : 12
B. 12 : 13
C. 21 : 31
D. 31 : 21

Answer

To solve the problem, we need to find the proportion of wine to water after mixing the contents of the two casks and adding 20 liters of water. Let's break it down step by step.

### Step 1: Determine the amount of wine and water in the first cask.
The first cask has a total volume of 48 liters with a wine to water ratio of 13:7.

- Total parts = 13 (wine) + 7 (water) = 20 parts.
- Amount of wine in the first cask = (13/20) * 48 = 31.2 liters.
- Amount of water in the first cask = (7/20) * 48 = 16.8 liters.

### Step 2: Determine the amount of wine and water in the second cask.
The second cask has a total volume of 42 liters with a wine to water ratio of 18:17.

- Total parts = 18 (wine) + 17 (water) = 35 parts.
- Amount of wine in the second cask = (18/35) * 42 = 21.6 liters.
- Amount of water in the second cask = (17/35) * 42 = 20.4 liters.

### Step 3: Combine the contents of both casks.
Now, we will add the amounts of wine and water from both casks.

- Total amount of wine = 31.2 liters (from the first cask) + 21.6 liters (from the second cask) = 52.8 liters.
- Total amount of water = 16.8 liters (from the first cask) + 20.4 liters (from the second cask) = 37.2 liters.

### Step 4: Add 20 liters of water to the mixture.
Now, we add 20 liters of water to the total amount of water.

- New total amount of water = 37.2 liters + 20 liters = 57.2 liters.

### Step 5: Find the final proportion of wine to water.
Now we have the total amounts of wine and water:

- Total wine = 52.8 liters.
- Total water = 57.2 liters.

To find the ratio of wine to water:

- Ratio of wine to water = 52.8 : 57.2.

To simplify the ratio, we can divide both sides by the common factor:

- Ratio = 52.8 / 52.8 : 57.2 / 52.8 = 1 : (57.2 / 52.8).

Calculating the value:

- 57.2 / 52.8 ≈ 1.0833.

Thus, the ratio of wine to water is approximately:

- 1 : 1.0833, which can be expressed as 12 : 13 when multiplied by 12.

### Final Answer:
The proportion of wine to water in the resulting mixture is **12 : 13**.

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