The price of an article is reduced by 7%.\nBy what percentage must the reduced price be increased so that it returns exactly to the original price?

Introduction / Context:
This question tests the concept of percentage increase and decrease, especially the important fact that a decrease of a certain percentage does not require the same percentage increase to return to the original value. Instead, the required increase is always slightly higher. This idea appears often in exam questions related to prices, salaries, and population changes.

Given Data / Assumptions:

• An article has an original price which we can call P.
• The price is cut by 7%.
• We then need to increase the new reduced price by some percentage so that the price becomes P again.
• We assume no other changes in price besides this decrease and subsequent increase.

Concept / Approach:
When a quantity is reduced by d percent, the remaining value is P * (1 - d/100). To restore it to P, we need to find a percentage increase on this reduced value so that the final result equals P. The required percentage increase can be found using the formula: required percent = (d / (100 - d)) * 100, where d is the original percentage decrease. This approach comes directly from equating the increased reduced price to the original price.

Step-by-Step Solution:
Step 1: Assume the original price P is 100 units for simplicity.Step 2: A decrease of 7% gives new price = 100 * (1 - 7/100) = 100 * 0.93 = 93 units.Step 3: Let the required percentage increase be x% on 93 units.Step 4: After this increase, the price must return to 100 units, so 93 * (1 + x/100) = 100.Step 5: Solve for x: 1 + x/100 = 100 / 93, so x/100 = (100 / 93) - 1.Step 6: Compute x: x = 100 * (100 / 93 - 1) = 100 * (7 / 93) = 700 / 93 ≈ 7.53%.

Verification / Alternative check:
We can use the standard formula directly: required percent = (d / (100 - d)) * 100. Here d = 7. So required percent = (7 / (100 - 7)) * 100 = (7 / 93) * 100 = 700 / 93 ≈ 7.53%. This matches the detailed step-by-step calculation, confirming the correctness of the result.

Why Other Options Are Wrong:
7% is incorrect because an increase of 7% on 93 units only produces 93 * 1.07 = 99.51 units, which does not reach 100.
33.77% and 63.75% are far too high; such large increases would push the value well above the original price.
14% would give 93 * 1.14 = 106.02 units, which is again higher than the original price of 100.

Common Pitfalls:
The most frequent mistake is assuming that a decrease of 7% can be canceled by an increase of 7%. This is wrong because the base for the increase is the reduced price, not the original price. Another pitfall is applying the formula incorrectly or using the wrong denominator. Students should remember that the denominator in the formula is the remaining percentage after the decrease, which is 100 - d, and not the original percentage. Writing out a simple numerical example with a base of 100 units is often the easiest way to avoid confusion.

Final Answer:
The reduced price must be increased by approximately 7.53% to restore the original value.