A businessman sells a commodity at 10% profit. If he had bought it at 10% less and sold it for Rs. 2 less, then he would have gained `16(2)/(3)%`. The cost price of the commodity is

Answer

To solve the problem step by step, let's denote the cost price (CP) of the commodity as \( x \).

### Step 1: Determine the Selling Price (SP) with 10% Profit
Given that the businessman sells the commodity at a 10% profit, we can express the selling price as:
\[ SP = CP + 10\% \text{ of } CP = x + 0.1x = 1.1x \]

### Step 2: Calculate the New Cost Price (CP) if Bought at 10% Less
If he had bought the commodity at 10% less, the new cost price would be:
\[ \text{New CP} = CP - 10\% \text{ of } CP = x - 0.1x = 0.9x \]

### Step 3: Determine the New Selling Price (SP) if Sold for Rs. 2 Less
If he sold it for Rs. 2 less than the original selling price, the new selling price would be:
\[ \text{New SP} = SP - 2 = 1.1x - 2 \]

### Step 4: Calculate the Profit Percentage with New CP and SP
We are given that with the new CP and SP, the profit percentage would be \( 16\frac{2}{3}\% \), which can be converted to a fraction:
\[ 16\frac{2}{3}\% = \frac{50}{3}\% \]
This means the profit can be expressed as:
\[ \text{Profit} = \text{New SP} - \text{New CP} = (1.1x - 2) - 0.9x = 0.2x - 2 \]

### Step 5: Set Up the Equation for Profit Percentage
The profit percentage is calculated as:
\[ \text{Profit Percentage} = \frac{\text{Profit}}{\text{New CP}} \times 100 \]
Substituting the values we have:
\[ \frac{0.2x - 2}{0.9x} \times 100 = \frac{50}{3} \]

### Step 6: Solve the Equation
To eliminate the fraction, we can multiply both sides by \( 0.9x \) and simplify:
\[ (0.2x - 2) \times 100 = \frac{50}{3} \times 0.9x \]
\[ 20x - 200 = 15x \]
Rearranging gives:
\[ 20x - 15x = 200 \]
\[ 5x = 200 \]
\[ x = 40 \]

### Conclusion
The cost price of the commodity is:
\[ \text{CP} = Rs. 40 \]