12 litres of a mixture has wine and water in the ratio 2: 3. How much water must be added to get wine to water ratio of 3:7 in the resultant mixture?

Options

A. 4.5litres
B. 3.5litres
C. 3litres
D. 4litres

Answer

To solve the problem step by step, we will follow the given information and apply the concept of ratios and mixtures.

### Step 1: Understand the initial mixture
We have a total mixture of 12 litres, which consists of wine and water in the ratio of 2:3.

### Step 2: Calculate the quantities of wine and water in the initial mixture
The total parts in the ratio 2:3 is:
\[ 2 + 3 = 5 \text{ parts} \]

Now, we can find the quantity of wine and water:
- Quantity of wine:
\[ \text{Wine} = \frac{2}{5} \times 12 = \frac{24}{5} = 4.8 \text{ litres} \]

- Quantity of water:
\[ \text{Water} = \frac{3}{5} \times 12 = \frac{36}{5} = 7.2 \text{ litres} \]

### Step 3: Set up the new ratio
We want to change the ratio of wine to water to 3:7. Let \( x \) be the amount of water we need to add.

After adding \( x \) litres of water, the new quantities will be:
- Wine: 4.8 litres (remains the same)
- Water: \( 7.2 + x \) litres

### Step 4: Set up the equation based on the new ratio
According to the new ratio of wine to water (3:7):
\[ \frac{\text{Wine}}{\text{Water}} = \frac{3}{7} \]

Substituting the quantities we have:
\[ \frac{4.8}{7.2 + x} = \frac{3}{7} \]

### Step 5: Cross-multiply to solve for \( x \)
Cross-multiplying gives:
\[ 4.8 \times 7 = 3 \times (7.2 + x) \]

Calculating the left side:
\[ 33.6 = 21.6 + 3x \]

### Step 6: Isolate \( x \)
Now, we can isolate \( x \):
\[ 33.6 - 21.6 = 3x \]
\[ 12 = 3x \]
\[ x = \frac{12}{3} = 4 \]

### Conclusion
The amount of water that must be added is **4 litres**.

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