To solve the problem step-by-step, we will follow the logic presented in the video transcript.
### Step 1: Understand the problem
We know that by selling an article for ₹1,134, Anu suffers a loss that is equal to the gain she would have made by selling it at a 10% profit. We need to find her profit percentage when she sells the article for ₹1,354.50.
### Step 2: Define variables
Let the cost price (CP) of the article be represented as \( x \).
### Step 3: Calculate the selling price at 10% profit
If Anu sells the article at a 10% profit, the selling price (SP) would be:
\[
SP = CP + 10\% \text{ of } CP = x + 0.1x = 1.1x
\]
### Step 4: Calculate the selling price at a loss
If Anu sells the article for ₹1,134, and this is a loss, we can express the selling price at a loss as:
\[
SP = CP - \text{Loss} = x - \text{Loss}
\]
Since the loss is equal to the gain she would have made at 10% profit, we can say:
\[
\text{Loss} = 1.1x - 1,134
\]
### Step 5: Set up the equation
From the information given, we can set up the equation:
\[
1,134 = x - (1.1x - 1,134)
\]
This simplifies to:
\[
1,134 = x - 1.1x + 1,134
\]
\[
1,134 = -0.1x + 1,134
\]
Subtracting ₹1,134 from both sides gives:
\[
0 = -0.1x
\]
This means:
\[
0.1x = 1,134 \implies x = \frac{1,134}{0.1} = 11,340
\]
### Step 6: Calculate the cost price
Now we can find the cost price (CP):
\[
CP = 11,340
\]
### Step 7: Calculate the profit when selling for ₹1,354.50
Now, if Anu sells the article for ₹1,354.50, we can find the profit:
\[
\text{Profit} = SP - CP = 1,354.50 - 11,340
\]
Calculating this gives:
\[
\text{Profit} = 1,354.50 - 11,340 = 14.50
\]
### Step 8: Calculate the profit percentage
The profit percentage can be calculated using the formula:
\[
\text{Profit Percentage} = \left( \frac{\text{Profit}}{CP} \right) \times 100
\]
Substituting the values:
\[
\text{Profit Percentage} = \left( \frac{14.50}{11,340} \right) \times 100
\]
Calculating this gives:
\[
\text{Profit Percentage} \approx 0.128 \times 100 = 12.8\%
\]
### Conclusion
The profit percentage when Anu sells the article for ₹1,354.50 is approximately **12.8%**.
