Calculating TV Selling Price for Target Gain

This problem involves calculating the cost price of a T.V. based on two different selling scenarios with varying gains and then determining the selling price required to achieve a specific higher gain.

Let's break down the problem step-by-step:

Understanding Gain Percentage

Gain percentage is calculated on the cost price (CP). If an item is sold at a gain of R%, the selling price (SP) is given by:

\( SP = CP + (\text{R\% of CP}) \)

\( SP = CP \times (1 + \frac{R}{100}) \)

Initial Scenario: 8% Gain

When the T.V. is sold at an 8% gain, the selling price (let's call it SP1) is:

\( SP1 = CP \times (1 + \frac{8}{100}) \)

\( SP1 = CP \times (1 + 0.08) \)

\( SP1 = 1.08 \times CP \)

Second Scenario: 15% Gain

We are told that if the T.V. had been sold for Rs. 714 more, the gain would have been 15%. Let the selling price in this scenario be SP2.

\( SP2 = CP \times (1 + \frac{15}{100}) \)

\( SP2 = CP \times (1 + 0.15) \)

\( SP2 = 1.15 \times CP \)

Relating the Two Selling Prices

According to the problem, SP2 is Rs. 714 more than SP1:

\( SP2 = SP1 + 714 \)

Finding the Cost Price (CP)

Now, we can substitute the expressions for SP1 and SP2 into the equation above:

\( 1.15 \times CP = 1.08 \times CP + 714 \)

Subtract \( 1.08 \times CP \) from both sides:

\( 1.15 \times CP - 1.08 \times CP = 714 \)

\( (1.15 - 1.08) \times CP = 714 \)

\( 0.07 \times CP = 714 \)

To find CP, divide 714 by 0.07:

\( CP = \frac{714}{0.07} \)

\( CP = \frac{714}{\frac{7}{100}} \)

\( CP = 714 \times \frac{100}{7} \)

\( CP = (714 \div 7) \times 100 \)

\( CP = 102 \times 100 \)

\( CP = 10200 \)

So, the cost price of the T.V. is Rs. 10,200.

Calculating Selling Price for 18% Gain

The question asks for the selling price required to gain 18%. Let's call this SP3.

\( SP3 = CP \times (1 + \frac{18}{100}) \)

\( SP3 = CP \times (1 + 0.18) \)

\( SP3 = 1.18 \times CP \)

Substitute the calculated value of CP (Rs. 10,200):

\( SP3 = 1.18 \times 10200 \)

\( SP3 = 118 \times 102 \)

Let's multiply 118 by 102:

\( SP3 = 12036 \)

Therefore, to gain 18%, the selling price of the T.V. should be Rs. 12,036.

Summary of Calculations

• Difference in gain percentage = \( 15\% - 8\% = 7\% \)
• This 7% difference in gain corresponds to the increase in selling price of Rs. 714.
• So, \( 7\% \text{ of CP} = 714 \)
• \( \frac{7}{100} \times CP = 714 \)
• \( CP = \frac{714 \times 100}{7} = 102 \times 100 = 10200 \)
• Required selling price for 18% gain = CP + 18% of CP = \( 10200 + \frac{18}{100} \times 10200 \)
• Required selling price = \( 10200 + 18 \times 102 \)
• Required selling price = \( 10200 + 1836 \)
• Required selling price = \( 12036 \)

Revision Table: Profit and Loss Basics

Additional Information: Calculating Selling Price from Cost Price and Gain %

If you know the cost price (CP) and the desired gain percentage (G%), you can directly calculate the selling price (SP) using the formula:

\( SP = CP \times (1 + \frac{G}{100}) \)

Alternatively, you can find the gain amount and add it to the CP:

Gain Amount = \( \frac{G}{100} \times CP \)

SP = CP + Gain Amount

In this problem, once we found the CP was Rs. 10,200 and needed an 18% gain, we used the direct formula:

\( SP = 10200 \times (1 + \frac{18}{100}) = 10200 \times 1.18 = 12036 \)

Understanding the relationship between the change in selling price and the change in gain percentage is key to solving problems like this efficiently.