Calculating TV Selling Price with Profit and Loss Percentages

This problem involves understanding the concepts of cost price, selling price, profit percentage, and loss percentage. We are given two scenarios for selling a T.V. and the difference in the selling prices. Our goal is to find the selling price that would result in an 18% gain.

Understanding Profit and Loss Concepts

When an item is sold for more than its cost price, there is a profit. When it is sold for less than its cost price, there is a loss. Profit or loss is usually calculated as a percentage of the cost price.

• Profit % = \(\left( \frac{\text{Selling Price} - \text{Cost Price}}{\text{Cost Price}} \right) \times 100\)
• Loss % = \(\left( \frac{\text{Cost Price} - \text{Selling Price}}{\text{Cost Price}} \right) \times 100\)

Alternatively, we can express the selling price based on the cost price and percentage gain or loss:

• Selling Price (with Gain) = Cost Price \(\times \left( 1 + \frac{\text{Gain \%}}{100} \right)\)
• Selling Price (with Loss) = Cost Price \(\times \left( 1 - \frac{\text{Loss \%}}{100} \right)\)

Setting up the Problem

Let's denote the Cost Price of the T.V. as \(CP\). We are given two scenarios:

1. The T.V. is sold at an 8% gain.
2. Had it been sold for Rs. 2553 less, there would have been a 15% loss.

Scenario 1: 8% Gain

The selling price in this case, let's call it \(SP_1\), is:

\(SP_1 = CP \times \left( 1 + \frac{8}{100} \right) = CP \times (1 + 0.08) = 1.08 \times CP\)

Scenario 2: 15% Loss

The selling price in this case, let's call it \(SP_2\), would be:

\(SP_2 = CP \times \left( 1 - \frac{15}{100} \right) = CP \times (1 - 0.15) = 0.85 \times CP\)

Finding the Cost Price (CP)

We are told that if the T.V. had been sold for Rs. 2553 less than the first scenario's selling price (\(SP_1\)), the result would have been the second scenario's selling price (\(SP_2\)).

So, the difference between \(SP_1\) and \(SP_2\) is Rs. 2553.

\(SP_1 - SP_2 = 2553\)

Substitute the expressions for \(SP_1\) and \(SP_2\) in terms of \(CP\):

\(1.08 \times CP - 0.85 \times CP = 2553\)

Combine the terms with \(CP\):

\((1.08 - 0.85) \times CP = 2553\)

\(0.23 \times CP = 2553\)

Now, solve for \(CP\):

\(CP = \frac{2553}{0.23}\)

To remove the decimal, multiply the numerator and denominator by 100:

\(CP = \frac{2553 \times 100}{0.23 \times 100} = \frac{255300}{23}\)

Performing the division:

So, the Cost Price of the T.V. is Rs. 11100.

\(CP = 11100\)

Calculating Selling Price for 18% Gain

Now that we know the Cost Price, we can calculate the selling price required to achieve an 18% gain.

Let the required selling price be \(SP_3\).

\(SP_3 = CP \times \left( 1 + \frac{18}{100} \right)\)

\(SP_3 = 11100 \times \left( 1 + 0.18 \right)\)

\(SP_3 = 11100 \times 1.18\)

Calculating the product:

\(SP_3 = 11100 \times \frac{118}{100}\)

\(SP_3 = 111 \times 118\)

Multiplying 111 by 118:

So, the selling price required to gain 18% is Rs. 13098.

Summary of Selling Prices

Let's list the selling prices for clarity:

The selling price of the T.V. to gain 18% would be Rs. 13098.

Revision Table: Profit and Loss Key Terms

Additional Information: Percentage Point Difference

In this problem, the difference in selling price (Rs. 2553) corresponds to the difference between an 8% gain scenario and a 15% loss scenario relative to the Cost Price. The total percentage difference from the perspective of the Cost Price is the sum of the gain percentage and the loss percentage (since one is above CP and the other is below CP).

• Percentage above CP (Gain) = 8%
• Percentage below CP (Loss) = 15%
• Total percentage difference = 8% + 15% = 23%

This 23% of the Cost Price is equal to the Rs. 2553 difference in selling prices.

So, \(23\% \text{ of } CP = 2553\)

\(\frac{23}{100} \times CP = 2553\)

\(CP = \frac{2553 \times 100}{23} = \frac{255300}{23} = 11100\)

This confirms our calculated Cost Price using a slightly different perspective on the percentage difference. Once the Cost Price is known, calculating the selling price for any other gain or loss percentage is straightforward.