Final answer:
Selling an article at 2/3 of its price results in a 10% loss, implying that 2/3 of the original price is 90% of the cost price. By setting up a proportion and solving for the cost price (CP), we find that the gain when selling at the original price is 45%, a value not listed in the provided choices.
Step-by-step explanation:
When an article is sold at 2/3 of its original price and results in a 10% loss, it means that 2/3 of the price is equivalent to 90% of the cost price (because 100% - 10% = 90%). To find the profit when the item is sold at the original price, we need to set up a proportion where the cost price (CP) is defined as 100%.
Let the original price (or selling price, SP) be x. At 2/3 of this price, the item sells for (2/3)x, and that is 90% of the CP, which can be represented as:
(2/3)x = 0.9 × CP
To find CP, we rearrange the formula:
CP = (2/3)x / 0.9
Now, when the article is sold at the original price x, which is 100% of the CP, the profit percentage can be calculated from the formula:
Profit % = ((SP - CP) / CP) × 100 = ((x - CP) / CP) × 100
Substitute CP from the earlier equation:
Profit % = ((x - (2/3)x / 0.9) / ((2/3)x / 0.9)) × 100
Simplify to find the profit percentage:
Profit % = ((1/3)x / (2/3)x) × 90 = 1/2 × 90 = 45%
Therefore, the gain at the original price is 45%, which is not listed in the provided options.