A milk man has a mixture of milk in which ratio of milk and water is 5:3. He sells 40 litres of milk i.e., mixture then he adds up 15 litres of pure water. Now the ratio of milk and water is 5:4. What is the new quantity of mixture?

Options

A. 72 litres
B. 270 litres
C. 135 litres
D. Data insufficient

Answer

To solve the problem, we will break it down step by step.

### Step 1: Understand the initial ratio of milk and water
The initial ratio of milk to water is given as 5:3. This means that for every 5 parts of milk, there are 3 parts of water.

### Step 2: Determine the total parts in the mixture
The total parts in the mixture can be calculated as:
\[ 5 + 3 = 8 \text{ parts} \]

### Step 3: Let the initial quantity of the mixture be \( x \) liters
Since the ratio is 5:3, we can express the quantities of milk and water in terms of \( x \):
- Quantity of milk = \( \frac{5}{8}x \)
- Quantity of water = \( \frac{3}{8}x \)

### Step 4: Calculate the quantity of milk and water after selling 40 liters of the mixture
When the milkman sells 40 liters of the mixture, he sells both milk and water in the same ratio. The quantity of milk and water sold can be calculated as follows:
- Milk sold = \( \frac{5}{8} \times 40 = 25 \) liters
- Water sold = \( \frac{3}{8} \times 40 = 15 \) liters

### Step 5: Update the quantities of milk and water after the sale
After selling 40 liters of the mixture:
- Remaining milk = \( \frac{5}{8}x - 25 \)
- Remaining water = \( \frac{3}{8}x - 15 \)

### Step 6: Add 15 liters of pure water
After selling, the milkman adds 15 liters of pure water:
- New quantity of water = \( \left(\frac{3}{8}x - 15\right) + 15 = \frac{3}{8}x \)

### Step 7: Set up the new ratio of milk to water
Now, the new ratio of milk to water is given as 5:4. Therefore, we can set up the equation:
\[ \frac{\frac{5}{8}x - 25}{\frac{3}{8}x} = \frac{5}{4} \]

### Step 8: Cross-multiply to solve for \( x \)
Cross-multiplying gives us:
\[ 4\left(\frac{5}{8}x - 25\right) = 5\left(\frac{3}{8}x\right) \]
Expanding both sides:
\[ \frac{20}{8}x - 100 = \frac{15}{8}x \]
Now, simplify:
\[ 20x - 800 = 15x \]
\[ 20x - 15x = 800 \]
\[ 5x = 800 \]
\[ x = 160 \]

### Step 9: Calculate the new quantity of the mixture
The new quantity of the mixture after selling 40 liters and adding 15 liters of water is:
\[ \text{New quantity of mixture} = x - 40 + 15 = 160 - 40 + 15 = 135 \text{ liters} \]

### Final Answer
The new quantity of the mixture is **135 liters**.

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