A T.V. is sold at 8% gain. Had it been for Rs. 714 more, the gain would have been 15%. To gain 18%, the selling price of the T.V. should be:
Rs. 12,036
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:
| Operation | Calculation |
|---|---|
| \( 118 \times 100 \) | \( 11800 \) |
| \( 118 \times 2 \) | \( 236 \) |
| \( 11800 + 236 \) | \( 12036 \) |
\( SP3 = 12036 \)
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
| Term | Definition | Formula |
|---|---|---|
| Cost Price (CP) | The price at which an article is purchased. | - |
| Selling Price (SP) | The price at which an article is sold. | - |
| Gain or Profit | When SP > CP. | Gain = SP - CP |
| Loss | When SP < CP. | Loss = CP - SP |
| Gain Percentage | Gain expressed as a percentage of CP. | \( \text{Gain}\% = \frac{\text{Gain}}{\text{CP}} \times 100 \) |
| Loss Percentage | Loss expressed as a percentage of CP. | \( \text{Loss}\% = \frac{\text{Loss}}{\text{CP}} \times 100 \) |
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.
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