If a shopkeeper purchases a certain number of items for a certain sum and sells a fraction of the said number for the same amount, then his profit is 300%. What is the fraction?

Options

A. `1/4`
B. `1/3`
C. `1/6`
D. `1/2`

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Given Information
The shopkeeper purchases a certain number of items for a certain sum and sells a fraction of those items for the same amount. The profit made from this transaction is 300%.

### Step 2: Define Variables
Let:
- \( C \) = Total cost price of the items
- \( N \) = Total number of items purchased
- \( f \) = Fraction of items sold
- \( S \) = Selling price of the fraction sold

### Step 3: Calculate Cost Price of Sold Items
If the total cost price for \( N \) items is \( C \), then the cost price for the fraction \( f \) of the items sold is:
\[ \text{Cost Price of } fN = f \cdot C \]

### Step 4: Selling Price of Sold Items
According to the problem, the shopkeeper sells the fraction \( fN \) for the same amount \( S \) as the total cost price \( C \):
\[ S = C \]

### Step 5: Calculate Profit
Profit is calculated as:
\[ \text{Profit} = \text{Selling Price} - \text{Cost Price} \]
Substituting the values we have:
\[ \text{Profit} = S - (f \cdot C) = C - (f \cdot C) \]
This simplifies to:
\[ \text{Profit} = C(1 - f) \]

### Step 6: Calculate Profit Percentage
The profit percentage is given as 300%. The formula for profit percentage is:
\[ \text{Profit Percentage} = \left(\frac{\text{Profit}}{\text{Cost Price}}\right) \times 100 \]
Substituting the values:
\[ 300 = \left(\frac{C(1 - f)}{f \cdot C}\right) \times 100 \]
This simplifies to:
\[ 300 = \frac{1 - f}{f} \times 100 \]

### Step 7: Solve for Fraction \( f \)
Rearranging the equation gives:
\[ 3 = \frac{1 - f}{f} \]
Cross-multiplying yields:
\[ 3f = 1 - f \]
Combining like terms results in:
\[ 3f + f = 1 \]
\[ 4f = 1 \]
Thus, solving for \( f \):
\[ f = \frac{1}{4} \]

### Step 8: Conclusion
The fraction of items sold by the shopkeeper is \( \frac{1}{4} \).

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