A person bought some caps at rate of 27 for Rs 102 and sold them at the rate of 8 for Rs 34. Find the profit or loss percent ?

Answer

To solve the problem of finding the profit or loss percentage when a person buys caps at a certain rate and sells them at another rate, we can follow these steps:

### Step 1: Calculate the Cost Price (CP) of one cap
The person bought 27 caps for Rs. 102.

To find the cost price of one cap:
\[ \text{CP of 1 cap} = \frac{102}{27} = \frac{34}{9} \text{ Rs.} \]

### Step 2: Calculate the Selling Price (SP) of one cap
The person sold 8 caps for Rs. 34.

To find the selling price of one cap:
\[ \text{SP of 1 cap} = \frac{34}{8} = \frac{17}{4} \text{ Rs.} \]

### Step 3: Convert CP and SP to a common denominator
To compare CP and SP easily, we can convert both to a common denominator.

The CP of one cap is:
\[ \text{CP} = \frac{34}{9} \text{ Rs.} \]

The SP of one cap is:
\[ \text{SP} = \frac{17}{4} \text{ Rs.} \]

Finding a common denominator (which is 36):
\[ \text{CP} = \frac{34 \times 4}{9 \times 4} = \frac{136}{36} \text{ Rs.} \]
\[ \text{SP} = \frac{17 \times 9}{4 \times 9} = \frac{153}{36} \text{ Rs.} \]

### Step 4: Calculate Profit
Now, we can find the profit by subtracting the CP from the SP:
\[ \text{Profit} = \text{SP} - \text{CP} = \frac{153}{36} - \frac{136}{36} = \frac{17}{36} \text{ Rs.} \]

### Step 5: Calculate Profit Percentage
To find the profit percentage, we use the formula:
\[ \text{Profit Percentage} = \left( \frac{\text{Profit}}{\text{CP}} \right) \times 100 \]
Substituting the values:
\[ \text{Profit Percentage} = \left( \frac{\frac{17}{36}}{\frac{136}{36}} \right) \times 100 = \left( \frac{17}{136} \right) \times 100 \]
Calculating this gives:
\[ \text{Profit Percentage} = \frac{17 \times 100}{136} \approx 12.5 \% \]

### Final Answer
The profit percentage is **12.5%**.

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