A person sells three articles first at 10% loss, second at 20% profit and the third one at 25% loss. If SP is equal find overall profit/loss%

Answer

To solve the problem step by step, we will analyze the selling price (SP) and cost price (CP) of each article and then calculate the overall profit or loss percentage.

### Step 1: Understand the Selling Prices and Loss/Profit Percentages
1. **First Article**: Sold at a 10% loss.
- Let the cost price (CP1) be \( x \).
- Selling Price (SP1) = \( CP1 - 10\% \text{ of } CP1 = x - 0.1x = 0.9x \).

2. **Second Article**: Sold at a 20% profit.
- Let the cost price (CP2) be \( y \).
- Selling Price (SP2) = \( CP2 + 20\% \text{ of } CP2 = y + 0.2y = 1.2y \).

3. **Third Article**: Sold at a 25% loss.
- Let the cost price (CP3) be \( z \).
- Selling Price (SP3) = \( CP3 - 25\% \text{ of } CP3 = z - 0.25z = 0.75z \).

### Step 2: Set Selling Prices Equal
Since the selling prices are equal, we can set them as follows:
- \( SP1 = SP2 = SP3 \)

Let’s denote the common selling price as \( SP \).

From the equations:
- \( SP = 0.9x \)
- \( SP = 1.2y \)
- \( SP = 0.75z \)

### Step 3: Express Cost Prices in Terms of Selling Price
From the equations above, we can express the cost prices in terms of the selling price \( SP \):
1. \( x = \frac{SP}{0.9} \)
2. \( y = \frac{SP}{1.2} \)
3. \( z = \frac{SP}{0.75} \)

### Step 4: Calculate Total Cost Price
Now we can find the total cost price (CP) of all three articles:
\[ CP = CP1 + CP2 + CP3 = x + y + z \]
Substituting the values:
\[ CP = \frac{SP}{0.9} + \frac{SP}{1.2} + \frac{SP}{0.75} \]

### Step 5: Find a Common Denominator
To add these fractions, we need a common denominator. The least common multiple (LCM) of 0.9, 1.2, and 0.75 is 3.6. Rewriting each term:
- \( \frac{SP}{0.9} = \frac{SP \times 4}{3.6} \)
- \( \frac{SP}{1.2} = \frac{SP \times 3}{3.6} \)
- \( \frac{SP}{0.75} = \frac{SP \times 4.8}{3.6} \)

Now, adding them:
\[ CP = \frac{4SP + 3SP + 4.8SP}{3.6} = \frac{11.8SP}{3.6} \]

### Step 6: Calculate Total Selling Price
The total selling price (SP) is:
\[ SP = SP + SP + SP = 3SP \]

### Step 7: Calculate Overall Profit or Loss
Now we can calculate the overall profit or loss:
\[ \text{Total CP} = \frac{11.8SP}{3.6} \]
\[ \text{Total SP} = 3SP \]

### Step 8: Determine Profit or Loss Amount
The profit or loss amount is:
\[ \text{Loss} = CP - SP = \frac{11.8SP}{3.6} - 3SP \]
Finding a common denominator:
\[ \text{Loss} = \frac{11.8SP - 10.8SP}{3.6} = \frac{1SP}{3.6} \]

### Step 9: Calculate Loss Percentage
To find the loss percentage:
\[ \text{Loss Percentage} = \left( \frac{\text{Loss}}{\text{Total CP}} \right) \times 100 = \left( \frac{\frac{1SP}{3.6}}{\frac{11.8SP}{3.6}} \right) \times 100 = \left( \frac{1}{11.8} \right) \times 100 \approx 8.47\% \]

### Final Result
The overall loss percentage is approximately **8.47%**.

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