Seventy percent of the cost price of an article is equal to forty percent of its selling price. What is the profit or loss percentage on the article?

Introduction / Context:
This question relates cost price and selling price using a ratio expressed in percentages. Instead of directly giving profit percentage, it gives a condition that 70% of cost price equals 40% of selling price. From this relationship, you must determine whether there is profit or loss and calculate its percentage. This is a classic algebraic application of profit and loss concepts.

Given Data / Assumptions:
- Seventy percent of cost price (CP) equals forty percent of selling price (SP). - The article has a single CP and SP. - We must determine whether there is profit or loss and express it as a percentage relative to cost price.

Concept / Approach:
Let CP be the cost price and SP be the selling price. The statement "70% of CP equals 40% of SP" becomes 0.70 * CP = 0.40 * SP. From this equation we solve for SP in terms of CP. Once SP is expressed as a multiple of CP, we compare SP and CP to see if profit or loss has occurred and compute the corresponding percentage. If SP is greater than CP, there is profit; if SP is smaller, there is loss.

Step-by-Step Solution:
Step 1: Let CP be the cost price and SP be the selling price. Step 2: According to the question, 70% of CP equals 40% of SP. Step 3: Write this as 0.70 * CP = 0.40 * SP. Step 4: Solve for SP in terms of CP: SP = (0.70 / 0.40) * CP. Step 5: 0.70 / 0.40 = 7 / 4 = 1.75. Step 6: So SP = 1.75 * CP. Step 7: This means SP is 75% more than CP because 1.75 * CP = CP + 0.75 * CP. Step 8: Hence profit percentage = 75%.

Verification / Alternative check:
Take a convenient cost price to verify. Let CP = Rs. 100. Then 70% of CP = 70. According to the condition, this equals 40% of SP. So 0.40 * SP = 70 and SP = 70 / 0.40 = 175. Profit = SP - CP = 175 - 100 = 75. Profit percentage = 75 / 100 * 100 = 75%. This numerical check confirms the algebraic result.

Why Other Options Are Wrong:
63% loss and 70% loss are not consistent with SP being greater than CP. The relationship SP = 1.75 * CP clearly shows a gain, not a loss. An 80% profit would require SP to be 1.80 * CP, which is slightly larger than what we have. Only a 75% profit exactly matches the factor 1.75 derived from the given percentage equality.

Common Pitfalls:
One common error is to misinterpret the condition and set 70% of SP equal to 40% of CP, reversing the relationship. Another mistake is to treat the percentages as independent and subtract them directly, which does not work because they are applied to different quantities. Always convert the verbal statement into a precise equation connecting CP and SP, then solve systematically.

Final Answer:
The transaction results in a 75% profit on the article.