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 oure 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 step by step, we will break down the information given and perform the necessary calculations.

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

Let the total parts be \(5 + 3 = 8\) parts.

Let the initial quantity of the mixture be \(x\) litres.

- Quantity of milk = \(\frac{5}{8}x\)
- Quantity of water = \(\frac{3}{8}x\)

### Step 2: Calculate the quantities after selling 40 litres of the mixture.
When the milkman sells 40 litres of the mixture, he is selling both milk and water in the same ratio of 5:3.

The quantity of milk sold in 40 litres:
- Milk sold = \(\frac{5}{8} \times 40 = 25\) litres

The quantity of water sold in 40 litres:
- Water sold = \(\frac{3}{8} \times 40 = 15\) litres

Now, we can find the remaining quantities of milk and water after the sale:
- Remaining milk = \(\frac{5}{8}x - 25\)
- Remaining water = \(\frac{3}{8}x - 15\)

### Step 3: Add 15 litres of pure water.
After selling the mixture, the milkman adds 15 litres of pure water.

New quantity of water:
- New water quantity = \(\left(\frac{3}{8}x - 15\right) + 15 = \frac{3}{8}x\)

### Step 4: Set up the new ratio of milk to water.
After adding the water, the new ratio of milk to water is given as 5:4.

This means:
\[ \frac{\text{Remaining Milk}}{\text{New Water}} = \frac{5}{4} \]

Substituting the quantities we have:
\[ \frac{\frac{5}{8}x - 25}{\frac{3}{8}x} = \frac{5}{4} \]

### Step 5: Cross-multiply to solve for \(x\).
Cross-multiplying gives:
\[ 4\left(\frac{5}{8}x - 25\right) = 5\left(\frac{3}{8}x\right) \]

Expanding both sides:
\[ \frac{20}{8}x - 100 = \frac{15}{8}x \]

### Step 6: Rearranging the equation.
Now, we can rearrange the equation:
\[ \frac{20}{8}x - \frac{15}{8}x = 100 \]

Combining like terms:
\[ \frac{5}{8}x = 100 \]

### Step 7: Solve for \(x\).
To find \(x\), multiply both sides by \(\frac{8}{5}\):
\[ x = 100 \times \frac{8}{5} = 160 \]

### Step 8: Find the new quantity of the mixture.
The new quantity of the mixture after selling 40 litres and adding 15 litres of water is:
\[ \text{New quantity of mixture} = x - 40 + 15 = 160 - 40 + 15 = 135 \text{ litres} \]

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

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