Calculating Cost Price from Selling Price and Loss Percentage

This problem involves calculating the original cost price of an article when its selling price and the percentage loss incurred during the sale are known.

We are given the following information:

• Selling Price (SP) of the article = Rs. 616
• Loss Percentage = 30%

Our goal is to determine the Cost Price (CP) of the article.

Understanding Profit and Loss Terms

In business mathematics, several key terms are used when dealing with buying and selling:

• Cost Price (CP): The price at which an article is purchased or manufactured.
• Selling Price (SP): The price at which an article is sold.
• Loss: Occurs when the Selling Price is less than the Cost Price (SP < CP). The amount of loss is calculated as CP - SP.
• Loss Percentage: The loss expressed as a percentage of the Cost Price. The formula is $\text{Loss Percentage} = \left( \frac{\text{Loss}}{\text{CP}} \right) \times 100$.

Formula for Selling Price with Loss

When an article is sold at a loss, the Selling Price is a fraction of the Cost Price. The formula relating Selling Price, Cost Price, and Loss Percentage is:

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

This formula shows that the Selling Price is obtained by subtracting the percentage loss (as a decimal or fraction) from 1 and then multiplying by the Cost Price.

We can rearrange this formula to find the Cost Price (CP) when SP and Loss Percentage are known:

$\text{CP} = \frac{\text{SP}}{1 - \frac{\text{Loss Percentage}}{100}}$

Step-by-Step Calculation of the Cost Price

Now, let's use the given values in the formula to find the Cost Price.

Given:

• SP = Rs. 616
• Loss Percentage = 30%

Substitute these values into the formula for CP:

$\text{CP} = \frac{616}{1 - \frac{30}{100}}$

First, calculate the value inside the parenthesis in the denominator:

$1 - \frac{30}{100} = 1 - 0.30 = 0.70$

Now, substitute this back into the formula:

$\text{CP} = \frac{616}{0.70}$

To perform the division easily, we can multiply the numerator and denominator by 10 to remove the decimal:

$\text{CP} = \frac{616 \times 10}{0.70 \times 10} = \frac{6160}{7}$

Now, divide 6160 by 7:

$\text{CP} = 880$

Therefore, the Cost Price of the article is Rs. 880.

Let's summarize the given information and the result:

Verification

To verify our answer, we can calculate the selling price if the cost price is Rs. 880 and the loss is 30%.

Loss Amount = 30% of CP = $\frac{30}{100} \times 880 = 0.30 \times 880 = 264$

Selling Price = Cost Price - Loss Amount

SP = $880 - 264 = 616$

This calculated selling price matches the given selling price (Rs. 616), confirming that our calculated cost price of Rs. 880 is correct.

Revision Table: Key Profit and Loss Formulas

Additional Information: Working with Percentages in Pricing

In profit and loss calculations, percentages are always based on the Cost Price unless stated otherwise. A 30% loss means that the loss value is 30% of the original Cost Price.

If CP is considered as 100%, a 30% loss means the loss is 30% of CP. The Selling Price will then be the remaining percentage of the CP. In this case, SP will be $(100\% - 30\%) = 70\%$ of the CP.

So, we can directly write: SP = 70% of CP.

Rs. $616 = \frac{70}{100} \times \text{CP}$

$\text{CP} = 616 \times \frac{100}{70}$

$\text{CP} = 616 \times \frac{10}{7}$

$\text{CP} = \frac{6160}{7}$

$\text{CP} = 880$

This method provides an alternative way to think about these problems, directly using the percentage of CP that the Selling Price represents.