In a solution of 45 litres of milk and water, ` 40%` is water. How many litres of milk must be added to become the ratio of milk and water ` 5 : 3` ?

Options

A. 8
B. 3
C. 7
D. 6

Answer

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

### Step 1: Determine the volume of water in the solution
Given that the total volume of the solution is 45 liters and 40% of it is water, we can calculate the volume of water as follows:

\[ \text{Volume of water} = 40\% \text{ of } 45 = \frac{40}{100} \times 45 = 18 \text{ liters} \]

### Step 2: Determine the volume of milk in the solution
Since the total volume of the solution is 45 liters and we have found that the volume of water is 18 liters, we can find the volume of milk:

\[ \text{Volume of milk} = \text{Total volume} - \text{Volume of water} = 45 - 18 = 27 \text{ liters} \]

### Step 3: Set up the equation for the new ratio
Let \( x \) be the amount of milk that needs to be added. After adding \( x \) liters of milk, the new volume of milk will be \( 27 + x \) liters, and the volume of water remains 18 liters. We want the ratio of milk to water to be \( 5:3 \).

We can set up the equation as follows:

\[ \frac{27 + x}{18} = \frac{5}{3} \]

### Step 4: Cross-multiply to solve for \( x \)
Cross-multiplying gives us:

\[ 3(27 + x) = 5 \times 18 \]

Calculating the right side:

\[ 3(27 + x) = 90 \]

### Step 5: Expand and simplify the equation
Expanding the left side:

\[ 81 + 3x = 90 \]

### Step 6: Isolate \( x \)
Now, we will isolate \( x \):

\[ 3x = 90 - 81 \]
\[ 3x = 9 \]
\[ x = \frac{9}{3} = 3 \]

### Conclusion
The amount of milk that must be added to achieve the desired ratio of \( 5:3 \) is:

\[ \text{Amount of milk to be added} = 3 \text{ liters} \]

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