Calculating Cost Price with Loss Percentage

This problem requires us to find the original cost price of an item when the selling price and the percentage loss incurred are known. We are given that a vendor sells a coconut for Rs. 32 and incurs a 20% loss on this sale.

Understanding the Concepts

When a vendor incurs a loss, it means the selling price (SP) is less than the cost price (CP). The loss percentage is calculated based on the cost price.

• Cost Price (CP): The original price at which the vendor bought the item.
• Selling Price (SP): The price at which the vendor sells the item.
• Loss: The difference between the cost price and the selling price when SP < CP. Loss = CP - SP.
• Loss Percentage: (Loss / CP) * 100%.

Formula for Calculating Cost Price with Loss

We can relate the Selling Price, Cost Price, and Loss Percentage using the following formula:

If there is a loss of L%, the Selling Price is calculated as:

\(\text{SP} = \text{CP} \times \left(1 - \frac{\text{L}}{100}\right)\)

To find the Cost Price when SP and L% are given, we can rearrange the formula:

\(\text{CP} = \frac{\text{SP}}{\left(1 - \frac{\text{L}}{100}\right)}\)

Step-by-Step Calculation

Given:

• Selling Price (SP) = Rs. 32
• Loss Percentage (L) = 20%

We need to find the Cost Price (CP).

Using the formula \(\text{CP} = \frac{\text{SP}}{\left(1 - \frac{\text{L}}{100}\right)}\):

1. Substitute the given values into the formula: \(\text{CP} = \frac{32}{\left(1 - \frac{20}{100}\right)}\)
2. Simplify the fraction inside the parenthesis: \(\text{CP} = \frac{32}{\left(1 - 0.20\right)}\)
3. Perform the subtraction: \(\text{CP} = \frac{32}{0.80}\)
4. Calculate the final value of CP: \(\text{CP} = \frac{32}{0.8} = \frac{320}{8}\) \(\text{CP} = 40\)

So, the cost price of the coconut is Rs. 40.

Verification

Let's check if selling at Rs. 32 results in a 20% loss on a CP of Rs. 40.

• Loss = CP - SP = Rs. 40 - Rs. 32 = Rs. 8
• Loss Percentage = \(\left(\frac{\text{Loss}}{\text{CP}}\right) \times 100\% = \left(\frac{8}{40}\right) \times 100\%\)
• Loss Percentage = \(\left(\frac{1}{5}\right) \times 100\% = 20\%\)

This matches the given information, confirming our calculated cost price is correct.

Therefore, the cost price of the coconut is Rs. 40.

Revision Table: Profit and Loss Key Formulas

Additional Information: Profit and Loss Scenarios

Profit and loss calculations are fundamental in business and everyday transactions. They help determine the financial outcome of selling goods or services.

• Profit Scenario: If a vendor sells an item for more than its cost price, they make a profit. The profit percentage is calculated based on the cost price.
• Break-even Point: If a vendor sells an item at exactly its cost price (SP = CP), there is no profit or loss.
• Calculating Percentage Change: Both profit percentage and loss percentage are calculated relative to the cost price, not the selling price, unless explicitly stated otherwise. This is a common point of confusion.
• Application: These concepts are widely used in retail, wholesale, manufacturing, and financial analysis to evaluate performance and make pricing decisions.