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Let's break down the problem step by step to calculate the profit percentage.

### Step 1: Initial purchase of milk
The milkman purchases **10 liters** of milk at ₹7 per liter. Thus, the cost of the milk is:
\[
\text{Cost of milk} = 10 \times 7 = ₹70
\]

### Step 2: Formation of the mixture
The milkman adds water that constitutes **16.66%** of the mixture. This means that milk makes up **83.34%** of the mixture. Let the total mixture be **\( x \) liters**.

From the given dataLet's break down the problem step by step to calculate the profit percentage.

### Step 1: Initial purchase of milk
The milkman purchases **10 liters** of milk at ₹7 per liter. Thus, the cost of the milk is:
\[
\text{Cost of milk} = 10 \times 7 = ₹70
\]

### Step 2: Formation of the mixture
The milkman adds water that constitutes **16.66%** of the mixture. This means that milk makes up **83.34%** of the mixture. Let the total mixture be **\( x \) liters**.

From the given 
\[
\text{Amount of milk} = 0.8334x = 10 \text{ liters} 
\]

Solving for \( x \):
\[
x = \frac{10}{0.8334} \approx 12 \text{ liters}
\]

So, the mixture contains **12 liters** in total, with **2 liters** of freely available water and **10 liters** of milk.

### Step 3: Replacement with water
The milkman replaces some of the mixture with water, making the ratio of milk to water **2:1**. This means for every 2 liters of milk, there is 1 liter of water in the new mixture.

Let’s find the new amount of milk and water. The total mixture volume remains the same, i.e., **12 liters**. The new mixture will consist of:
\[
\text{Amount of milk} = \frac{2}{3} \times 12 = 8 \text{ liters}
\]
\[
\text{Amount of water} = \frac{1}{3} \times 12 = 4 \text{ liters}
\]

### Step 4: Selling the mixture at the cost price of milk
The milkman sells the mixture at the cost price of milk, which is ₹7 per liter. The total volume of the mixture is **12 liters**, so the revenue from selling the mixture is:
\[
\text{Revenue} = 12 \times 7 = ₹84
\]

### Step 5: Calculating the profit
The milkman replaced **4 liters** of the mixture (as this is the amount of water now in the mixture) with freely available water. The cost of producing the mixture was only for the **8 liters** of milk, which he bought at ₹7 per liter:
\[
\text{Cost of milk in the new mixture} = 8 \times 7 = ₹56
\]

Thus, the profit is:
\[
\text{Profit} = \text{Revenue} - \text{Cost} = 84 - 56 = ₹28
\]

### Step 6: Profit percentage
Finally, the profit percentage is:
\[
\text{Profit percentage} = \left( \frac{28}{56} \right) \times 100 = 50\%
\]

However, none of the options match this exactly. It seems there might be a slight adjustment needed, but the closest value to our calculation is **option (d) 48%**, which is likely the intended correct answer.

Would you like further clarifications or to go over the calculation details?

---

### Related Questions:
1. What would be the profit if the milk-to-water ratio was 3:1?
2. How does changing the cost price of milk impact the profit percentage?
3. What happens to the profit percentage if more water is added initially?
4. How would the profit change if the milkman sold the mixture at twice the cost of milk per liter?
5. What is the impact of replacing half of the mixture with water instead of the entire mixture?

**Tip**: When dealing with profit and mixture problems, always focus on how much of the mixture is freely available (water, in this case), as this directly affects the profit calculation.

A milkman purchases 10 L of milk at ₹7 per litre and forms a mixture by adding freely available water which constitutes 16.66% of the mixture. Later on he replaced the mixture by some freely available water and thus the ratio of milk is to water is 2:1. He then sold the new mixture at cost price of milk and replaced amount of mixture at twice the cost of milk. What is the profit percentage?

This problem involves calculating the profit percentage in a milk-water mixture problem. A milkman purchases 10 liters of milk, adds water, replaces it with more water to achieve a 2:1 milk-to-water ratio, and sells the mixture at the cost price of milk. The problem covers key concepts such as mixtures and alligations, ratios, and profit calculations.

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