A shopkeeper sells an article at 15% gain. Had he sold it for Rs. 18 more; he would have gained 18%. The cost price (in Rs.) of the article is :

Question

Accepted Answer

The correct option is 1:

This problem is a classic application of Profit and Loss concepts, specifically focusing on how a change in the selling price affects the percentage gain, assuming the Cost Price (CP) remains constant. In percentage-based word problems, the Cost Price is always the reference point for calculating profit. Since the CP is the value we are looking for, we can represent it algebraically.

Let CP be the Cost Price of the article.

In the first scenario, the shopkeeper sells the article at a 15% gain. This means the initial Selling Price ($SP\_1$) is calculated as:
$SP\_1 = CP + 15% \text{ of } CP = CP + 0.15 \times CP = 1.15 \times CP$

In the second scenario, had the article been sold for Rs. 18 more, the gain would have been 18%. This means the new Selling Price ($SP\_2$) is:
$SP\_2 = CP + 18% \text{ of } CP = CP + 0.18 \times CP = 1.18 \times CP$

The question states that the difference between the two selling prices ($SP_2 - SP_1$) is exactly Rs. 18. This difference in selling price is entirely due to the difference in the profit percentage (18% - 15% = 3%). Therefore, the Rs. 18 increase in the selling price must be equal to 3% of the Cost Price.

We can set up the equation based on the difference in selling prices:
$SP\_2 - SP\_1 = 18$
$(1.18 \times CP) - (1.15 \times CP) = 18$
$0.03 \times CP = 18$

To find the Cost Price (CP), we isolate the variable:
$CP = \frac{18}{0.03}$

To simplify the calculation, we can multiply the numerator and the denominator by 100:
$CP = \frac{1800}{3}$
$CP = 600$

Thus, the Cost Price of the article is Rs. 600. This result confirms that 3% of Rs. 600 is Rs. 18, which is the exact amount that increased the profit from 15% to 18%.

The incorrect option is 2 (540):
If the Cost Price were Rs. 540, the difference of 3% profit would be $540 \times 0.03 = \text{Rs. } 16.20$. Since the required difference in the selling price is Rs. 18, Rs. 540 is too low to be the correct cost price.

The incorrect option is 3 (350):
If the Cost Price were Rs. 350, the 3% difference in profit would only correspond to $350 \times 0.03 = \text{Rs. } 10.50$. This value is significantly less than the required increase of Rs. 18, showing that Rs. 350 is not the correct cost price.

The incorrect option is 4 (380):
If the Cost Price were Rs. 380, the 3% change in profit would equal $380 \times 0.03 = \text{Rs. } 11.40$. This calculated difference does not match the actual difference of Rs. 18 provided in the problem, confirming that Rs. 380 is also incorrect.