A shopkeeper calculated his profit per cent on the selling price which comes out to be `30%`. If it had been calculated as usual on the cost price, then what is the required percentage profit?

Options

A. `42(6)/7%`
B. `4(7)/27%`
C. `7(6)/42%`
D. none of these

Answer

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

### Step 1: Understand the Given Information
The shopkeeper has a profit percentage of 30% calculated on the selling price (SP).

### Step 2: Define the Relationship Between SP, CP, and Profit
We know that:
- Profit = Selling Price (SP) - Cost Price (CP)
- Profit Percentage on SP = (Profit / SP) × 100

Given that the profit percentage on SP is 30%, we can express this as:
\[ \frac{Profit}{SP} \times 100 = 30 \]
This implies:
\[ Profit = 0.30 \times SP \]

### Step 3: Express SP in Terms of CP
From the profit formula, we can rearrange it:
\[ SP = CP + Profit \]
Substituting the expression for Profit:
\[ SP = CP + 0.30 \times SP \]

### Step 4: Solve for SP
Rearranging the equation to isolate SP:
\[ SP - 0.30 \times SP = CP \]
\[ 0.70 \times SP = CP \]
Now, we can express SP in terms of CP:
\[ SP = \frac{CP}{0.70} \]
\[ SP = \frac{10}{7} \times CP \]

### Step 5: Calculate the Profit in Terms of CP
Now, substituting SP back into the profit formula:
\[ Profit = SP - CP \]
\[ Profit = \frac{10}{7} \times CP - CP \]
To combine the terms, we need a common denominator:
\[ Profit = \frac{10CP}{7} - \frac{7CP}{7} \]
\[ Profit = \frac{3CP}{7} \]

### Step 6: Calculate the Profit Percentage on CP
Now we can find the profit percentage based on the cost price:
\[ Profit \% = \left( \frac{Profit}{CP} \right) \times 100 \]
Substituting the value of Profit:
\[ Profit \% = \left( \frac{\frac{3CP}{7}}{CP} \right) \times 100 \]
The CP cancels out:
\[ Profit \% = \frac{3}{7} \times 100 \]

### Step 7: Calculate the Final Value
Now, calculate \( \frac{3}{7} \times 100 \):
\[ Profit \% = \frac{300}{7} \approx 42.86\% \]

### Final Answer
The required percentage profit when calculated on the cost price is approximately **42.86%**.

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