By selling a stool for `"Rs 67.50"`,  a carpenter loses `10%`. How much percent would he gain or lose by selling it for `"Rs 82.50"`?

Answer

To solve the problem step by step, we will first determine the cost price (CP) of the stool based on the information given about the selling price (SP) and the loss percentage. Then, we will calculate the gain or loss when selling the stool for a different price.

### Step 1: Understand the Loss Percentage
The carpenter sells the stool for Rs. 67.50 and incurs a loss of 10%. This means that the selling price is 90% of the cost price (CP).

### Step 2: Set Up the Equation
We can express this relationship mathematically:
\[ SP = 90\% \text{ of } CP \]
Given that the selling price (SP) is Rs. 67.50, we can write:
\[ 67.50 = 90\% \text{ of } CP \]
This can be rewritten as:
\[ 67.50 = \frac{90}{100} \times CP \]

### Step 3: Remove the Percentage
To eliminate the percentage, we can multiply both sides by 100:
\[ 67.50 \times 100 = 90 \times CP \]
This simplifies to:
\[ 6750 = 90 \times CP \]

### Step 4: Solve for Cost Price (CP)
Now, we can solve for CP:
\[ CP = \frac{6750}{90} \]
Calculating this gives:
\[ CP = 75 \]
So, the cost price of the stool is Rs. 75.

### Step 5: Determine the New Selling Price
Now, we are given a new selling price of Rs. 82.50. We will use this to find out the gain.

### Step 6: Calculate Gain
Gain (or profit) can be calculated using the formula:
\[ \text{Gain} = SP - CP \]
Substituting the values we have:
\[ \text{Gain} = 82.50 - 75 \]
Calculating this gives:
\[ \text{Gain} = 7.50 \]

### Step 7: Calculate Gain Percentage
To find the gain percentage, we use the formula:
\[ \text{Gain Percentage} = \left( \frac{\text{Gain}}{CP} \right) \times 100 \]
Substituting the values:
\[ \text{Gain Percentage} = \left( \frac{7.50}{75} \right) \times 100 \]
Calculating this gives:
\[ \text{Gain Percentage} = 10\% \]

### Final Answer
The carpenter would gain **10%** by selling the stool for Rs. 82.50.

---