There were two articles and the sum of cost prices of these article is Rs. 500. One of them was sold at a profit of 20% and another at a loss of 20%. Besides if the selling prices of both the articles were same. Find the amount of overall loss.

Answer

To solve the problem step-by-step, we will follow a structured approach:

### Step 1: Define the Cost Prices
Let the cost price of the first article be \( CP_1 \) and the cost price of the second article be \( CP_2 \). According to the problem, the sum of the cost prices is given as:
\[ CP_1 + CP_2 = 500 \]

### Step 2: Define Selling Prices
The first article is sold at a profit of 20%. Therefore, the selling price of the first article \( SP_1 \) can be calculated as:
\[ SP_1 = CP_1 + 0.2 \times CP_1 = 1.2 \times CP_1 \]

The second article is sold at a loss of 20%. Therefore, the selling price of the second article \( SP_2 \) can be calculated as:
\[ SP_2 = CP_2 - 0.2 \times CP_2 = 0.8 \times CP_2 \]

### Step 3: Set Selling Prices Equal
Since the selling prices of both articles are the same, we can set \( SP_1 \) equal to \( SP_2 \):
\[ 1.2 \times CP_1 = 0.8 \times CP_2 \]

### Step 4: Express \( CP_2 \) in terms of \( CP_1 \)
From the equation \( 1.2 \times CP_1 = 0.8 \times CP_2 \), we can express \( CP_2 \) in terms of \( CP_1 \):
\[ CP_2 = \frac{1.2}{0.8} \times CP_1 = 1.5 \times CP_1 \]

### Step 5: Substitute \( CP_2 \) in the Cost Price Equation
Now substitute \( CP_2 \) in the sum of cost prices:
\[ CP_1 + 1.5 \times CP_1 = 500 \]
\[ 2.5 \times CP_1 = 500 \]

### Step 6: Solve for \( CP_1 \)
Now, solve for \( CP_1 \):
\[ CP_1 = \frac{500}{2.5} = 200 \]

### Step 7: Calculate \( CP_2 \)
Now that we have \( CP_1 \), we can find \( CP_2 \):
\[ CP_2 = 500 - CP_1 = 500 - 200 = 300 \]

### Step 8: Calculate Selling Prices
Now we can calculate the selling prices:
\[ SP_1 = 1.2 \times CP_1 = 1.2 \times 200 = 240 \]
\[ SP_2 = 0.8 \times CP_2 = 0.8 \times 300 = 240 \]

### Step 9: Calculate Total Cost Price and Total Selling Price
The total cost price is:
\[ Total \, CP = CP_1 + CP_2 = 200 + 300 = 500 \]
The total selling price is:
\[ Total \, SP = SP_1 + SP_2 = 240 + 240 = 480 \]

### Step 10: Calculate Overall Loss
Finally, we calculate the overall loss:
\[ Overall \, Loss = Total \, CP - Total \, SP = 500 - 480 = 20 \]

Thus, the overall loss is **Rs. 20**.

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