Hari suffered a loss of 8% by selling an article. If he had sold it for Rs. 300 more, he would have made a profit of 4%. Find his CP (in Rs.)
2500
Finding Cost Price with Profit and Loss Percentages
This problem asks us to find the original cost price (CP) of an article based on two different selling scenarios involving loss and profit percentages.
Let the Cost Price of the article be \(CP\).
Scenario 1: Selling at a Loss
Hari suffered a loss of 8% by selling the article. This means the selling price (\(SP_1\)) was 8% less than the cost price.
We can write this relationship as:
\(SP_1 = CP - \text{Loss}\)
Since Loss is 8% of CP, Loss \( = \frac{8}{100} \times CP\).
So, \(SP_1 = CP - \frac{8}{100} \times CP\)
\(SP_1 = CP \left(1 - \frac{8}{100}\right)\)
\(SP_1 = CP \left(\frac{100 - 8}{100}\right)\)
\(SP_1 = \frac{92}{100} \times CP\)
\(SP_1 = 0.92 \times CP\)
Scenario 2: Selling for More Money to Make a Profit
If he had sold the article for Rs. 300 more than \(SP_1\), he would have made a profit of 4%. Let this new selling price be \(SP_2\).
So, \(SP_2 = SP_1 + 300\).
This new selling price (\(SP_2\)) corresponds to a profit of 4% on the cost price (\(CP\)).
We can write this relationship as:
\(SP_2 = CP + \text{Profit}\)
Since Profit is 4% of CP, Profit \( = \frac{4}{100} \times CP\).
So, \(SP_2 = CP + \frac{4}{100} \times CP\)
\(SP_2 = CP \left(1 + \frac{4}{100}\right)\)
\(SP_2 = CP \left(\frac{100 + 4}{100}\right)\)
\(SP_2 = \frac{104}{100} \times CP\)
\(SP_2 = 1.04 \times CP\)
Setting up and Solving the Equation
Now we use the relationship \(SP_2 = SP_1 + 300\). We substitute the expressions for \(SP_1\) and \(SP_2\) in terms of \(CP\):
\(1.04 \times CP = 0.92 \times CP + 300\)
To find \(CP\), we rearrange the equation to group the \(CP\) terms:
\(1.04 \times CP - 0.92 \times CP = 300\)
\((1.04 - 0.92) \times CP = 300\)
\(0.12 \times CP = 300\)
Now, isolate \(CP\) by dividing both sides by 0.12:
\(CP = \frac{300}{0.12}\)
To make the division easier, we can write 0.12 as \(\frac{12}{100}\):
\(CP = \frac{300}{\frac{12}{100}}\)
\(CP = 300 \times \frac{100}{12}\)
\(CP = \frac{30000}{12}\)
Let's perform the division:
- \(30000 \div 12\)
- \(30 \div 12 = 2\) with remainder 6.
- Bring down the next 0 to make 60. \(60 \div 12 = 5\).
- Bring down the last two 0s. These go into the quotient as 00.
So, \(CP = 2500\).
The cost price of the article is Rs. 2500.
Revision Table: Profit & Loss Basics
| Concept | Formula |
|---|---|
| Loss Percentage | \(\frac{Loss}{CP} \times 100\%\) |
| Profit Percentage | \(\frac{Profit}{CP} \times 100\%\) |
| Selling Price (with Loss) | \(CP \times \left(1 - \frac{\text{Loss } \%}{100}\right)\) |
| Selling Price (with Profit) | \(CP \times \left(1 + \frac{\text{Profit } \%}{100}\right)\) |
Additional Information: Percentage Difference in Selling Price
In this problem, the selling price increased by Rs. 300. This change in selling price caused the outcome to shift from an 8% loss to a 4% profit.
The total percentage change relative to the cost price is the sum of the loss percentage and the profit percentage because we are moving from a price below CP (loss) to a price above CP (profit).
Total percentage difference \( = \text{Loss } \% + \text{Profit } \% = 8\% + 4\% = 12\%\).
This 12% difference in selling price (relative to CP) corresponds to the Rs. 300 increase in the selling price.
So, 12% of CP = Rs. 300.
\(\frac{12}{100} \times CP = 300\)
\(CP = \frac{300 \times 100}{12}\)
\(CP = \frac{30000}{12}\)
\(CP = 2500\)
This method confirms the result obtained through calculating the individual selling prices. Understanding this relationship between the change in selling price and the total change in percentage (from loss to profit or vice versa) can be a quicker way to solve such problems.
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