A dishonest grocer professes to sell pure butter at cost price, but he mixes it with fat and thereby gains 25%. Find the percentage of fat in the mixture assuming that fat is freely available.

Options

A. 0.2
B. 0.25
C. 0.3333
D. 0.4

Answer

To solve the problem of finding the percentage of fat in the mixture sold by the dishonest grocer, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Gain Percentage**:
The grocer gains 25% by selling a mixture of butter and fat. This means that for every 100 units of cost price (CP), he sells it for 125 units (since 100 + 25 = 125).

2. **Convert Gain Percentage to Fraction**:
The gain of 25% can be expressed as a fraction:
\[ \text{Gain} = \frac{25}{100} = \frac{1}{4} \]
This means that for every 4 units of cost price, the grocer makes a profit of 1 unit.

3. **Set Up the Cost Price and Selling Price**:
Let's assume the cost price of pure butter is 4 units. Therefore:
\[ \text{Selling Price (SP)} = \text{Cost Price (CP)} + \text{Profit} = 4 + 1 = 5 \text{ units} \]

4. **Determine the Composition of the Mixture**:
The grocer sells a total of 5 units (the selling price) but claims it is pure butter. He actually mixes in fat.
- Let’s denote the amount of butter as \( x \) and the amount of fat as \( y \).
- According to the problem, \( x + y = 5 \).

5. **Calculate the Amount of Butter**:
The cost price of the butter is 4 units, and since he is selling it at the cost price, we can assume he uses 4 units of butter:
\[ x = 4 \text{ units of butter} \]

6. **Calculate the Amount of Fat**:
Since the total mixture is 5 units:
\[ y = 5 - x = 5 - 4 = 1 \text{ unit of fat} \]

7. **Find the Percentage of Fat in the Mixture**:
To find the percentage of fat in the mixture, we use the formula:
\[ \text{Percentage of Fat} = \left( \frac{y}{x + y} \right) \times 100 = \left( \frac{1}{5} \right) \times 100 = 20\% \]

### Final Answer:
The percentage of fat in the mixture is **20%**.

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