729 ml of a mixture contains milk and water in the ratio 7:2. How much more water is to be added to get a new mixture containing milk and water in the ratio 7:3?

Options

A. 81 ml
B. 60 ml
C. 71 ml
D. 52 ml

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Understand the initial mixture
The initial mixture contains milk and water in the ratio of 7:2. This means for every 9 parts of the mixture, 7 parts are milk and 2 parts are water.

### Step 2: Calculate the total parts in the mixture
The total parts of the mixture can be calculated as:
\[ 7 + 2 = 9 \text{ parts} \]

### Step 3: Determine the quantity of milk and water in the initial mixture
Given that the total volume of the mixture is 729 ml, we can find the quantity of milk and water:
- Quantity of milk:
\[ \text{Milk} = \left(\frac{7}{9}\right) \times 729 = 567 \text{ ml} \]
- Quantity of water:
\[ \text{Water} = \left(\frac{2}{9}\right) \times 729 = 162 \text{ ml} \]

### Step 4: Set up the new ratio
We want to change the ratio of milk to water from 7:2 to 7:3. This means for every 7 parts of milk, there will now be 3 parts of water.

### Step 5: Calculate the new total parts in the mixture
The new total parts in the mixture will be:
\[ 7 + 3 = 10 \text{ parts} \]

### Step 6: Determine the new quantity of water needed
Since the quantity of milk remains the same (567 ml), we can set up the equation for the new ratio:
\[ \frac{567}{\text{New Water}} = \frac{7}{3} \]
Let the new quantity of water be \( W \). Then:
\[ \frac{567}{W} = \frac{7}{3} \]
Cross-multiplying gives:
\[ 7W = 3 \times 567 \]
\[ 7W = 1701 \]
\[ W = \frac{1701}{7} = 243 \text{ ml} \]

### Step 7: Calculate how much more water is needed
We initially had 162 ml of water. To find out how much more water is needed:
\[ \text{Additional Water} = W - \text{Initial Water} = 243 - 162 = 81 \text{ ml} \]

### Final Answer
Thus, the amount of water that needs to be added is **81 ml**.

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