A person calculates `16(2)/(3)%` profit at S.P. find his profit at C.P.

Answer

To solve the problem of finding the profit at Cost Price (C.P.) when a person calculates a profit of \(16 \frac{2}{3}\%\) at Selling Price (S.P.), we can follow these steps:

### Step 1: Convert the mixed fraction to an improper fraction
The profit percentage given is \(16 \frac{2}{3}\%\). We first convert this to an improper fraction:
\[ 16 \frac{2}{3} = \frac{50}{3}\% \]

### Step 2: Understand the relationship between Profit, Cost Price, and Selling Price
The profit percentage is calculated based on the Selling Price (S.P.). The formula for profit percentage is:
\[ \text{Profit Percentage} = \left( \frac{\text{Profit}}{\text{S.P.}} \right) \times 100 \]
From this, we can express the profit in terms of S.P.:
\[ \text{Profit} = \frac{50}{3} \times \frac{\text{S.P.}}{100} \]
This simplifies to:
\[ \text{Profit} = \frac{50 \times \text{S.P.}}{300} = \frac{1}{6} \times \text{S.P.} \]

### Step 3: Relate Selling Price to Cost Price
We know that:
\[ \text{S.P.} = \text{C.P.} + \text{Profit} \]
Substituting the expression for Profit:
\[ \text{S.P.} = \text{C.P.} + \frac{1}{6} \times \text{S.P.} \]
To isolate C.P., we can rearrange this equation:
\[ \text{S.P.} - \frac{1}{6} \times \text{S.P.} = \text{C.P.} \]
This can be simplified to:
\[ \frac{5}{6} \times \text{S.P.} = \text{C.P.} \]

### Step 4: Calculate Profit Percentage at Cost Price
Now, we can find the profit percentage based on Cost Price:
\[ \text{Profit} = \frac{1}{6} \times \text{S.P.} \]
Substituting \(\text{C.P.} = \frac{5}{6} \times \text{S.P.}\):
\[ \text{Profit Percentage at C.P.} = \left( \frac{\text{Profit}}{\text{C.P.}} \right) \times 100 \]
Substituting the values:
\[ \text{Profit Percentage at C.P.} = \left( \frac{\frac{1}{6} \times \text{S.P.}}{\frac{5}{6} \times \text{S.P.}} \right) \times 100 \]
The S.P. cancels out:
\[ \text{Profit Percentage at C.P.} = \left( \frac{1}{5} \right) \times 100 = 20\% \]

### Final Answer
The profit at Cost Price is \(20\%\).