Finding Cost Price with Profit and Loss Differences

The problem asks us to find the original cost price (CP) of an item. We are given information about the selling prices when the item is sold at a profit and at a loss. The difference between these two selling prices is provided.

Setting Up the Problem

Let's denote the cost price of the item as \(CP\).

When the item is sold at a profit of 5%, the selling price (\(SP_1\)) is calculated as:

• \(SP_1 = CP + \text{Profit}\)
• The profit is 5% of the cost price: \(5\% \text{ of } CP = \frac{5}{100} \times CP = 0.05 \times CP\)
• So, \(SP_1 = CP + 0.05 \times CP = (1 + 0.05) \times CP = 1.05 \times CP\)

When the item is sold at a loss of 5%, the selling price (\(SP_2\)) is calculated as:

• \(SP_2 = CP - \text{Loss}\)
• The loss is 5% of the cost price: \(5\% \text{ of } CP = \frac{5}{100} \times CP = 0.05 \times CP\)
• So, \(SP_2 = CP - 0.05 \times CP = (1 - 0.05) \times CP = 0.95 \times CP\)

Using the Selling Price Difference

We are given that the difference between the two selling prices (\(SP_1\) and \(SP_2\)) is Rs. 60.

Difference = \(SP_1 - SP_2 = 60\)

Substitute the expressions for \(SP_1\) and \(SP_2\) we found:

\(1.05 \times CP - 0.95 \times CP = 60\)

Solving for the Cost Price

Now, we can solve this equation for \(CP\):

Combine the terms involving \(CP\):

\((1.05 - 0.95) \times CP = 60\)

\(0.10 \times CP = 60\)

To find \(CP\), divide both sides by 0.10:

\(CP = \frac{60}{0.10}\)

To perform the division, we can write 0.10 as \(\frac{10}{100}\) or \(\frac{1}{10}\):

\(CP = 60 \div \frac{1}{10} = 60 \times 10\)

\(CP = 600\)

So, the cost price of the item is Rs. 600.

Verification

Let's check if this cost price fits the conditions:

• If CP = Rs. 600, profit of 5% = \(0.05 \times 600 = 30\). \(SP_1 = 600 + 30 = 630\).
• If CP = Rs. 600, loss of 5% = \(0.05 \times 600 = 30\). \(SP_2 = 600 - 30 = 570\).
• Difference between selling prices = \(SP_1 - SP_2 = 630 - 570 = 60\).

The calculated difference matches the given difference of Rs. 60. Thus, the cost price of Rs. 600 is correct.

Summary of Calculation

The cost price of the item is Rs. 600.