The price of an article is reduced by 25%.\nBy what percentage must the new reduced price be increased to restore the article to its original price?

Introduction / Context:
This question illustrates that a decrease of a certain percentage does not require the same percentage increase to return to the original value. Understanding this asymmetry is crucial for handling successive percentage change problems correctly, especially in pricing, profit loss, and salary revision contexts.

Given Data / Assumptions:

• The original price of an article is some amount P.
• The price is reduced by 25%.
• The reduced price must then be increased by some percentage to get back to P.
• We must find that required percentage increase.

Concept / Approach:
A 25% reduction means the new price is 75% of the original price, that is 0.75P. To restore the price back to P, we need to find a percentage increase x such that 0.75P * (1 + x / 100) = P. Solving this equation gives the required percentage increase. This clearly shows why the increase must be larger than 25%.

Step-by-Step Solution:
Step 1: Let the original price be P. Step 2: After a 25% reduction, new price = P - 0.25P = 0.75P. Step 3: Let the required percentage increase be x%. Step 4: After increasing by x%, price becomes 0.75P * (1 + x / 100). Step 5: To restore to original price, set 0.75P * (1 + x / 100) = P. Step 6: Divide both sides by P: 0.75 * (1 + x / 100) = 1. Step 7: Divide both sides by 0.75: 1 + x / 100 = 1 / 0.75 = 4 / 3. Step 8: So x / 100 = 4 / 3 - 1 = 1 / 3. Step 9: Multiply by 100: x = 100 / 3 ≈ 33.33%.

Verification / Alternative check:
Assume P = 100 for easy numbers. After a 25% decrease, the price becomes 75. Increase 75 by 33.33%: 33.33% of 75 is approximately 25, so 75 + 25 = 100. This returns the price back to its original value, confirming that a 33.33% increase is correct.

Why Other Options Are Wrong:
36.31%, 57%, 71.25%: These are larger than necessary and overshoot the original price when applied to 0.75P.
25%: This is a common incorrect guess assuming symmetry, but 25% of 75 is only 18.75, which does not restore the price to 100 when added.

Common Pitfalls:
The most common error is to think that a 25% decrease followed by a 25% increase returns to the starting value. This is incorrect because the increment is applied to a smaller base. Another pitfall is incorrectly setting up the equation or forgetting to divide by 0.75 when solving for the required percentage.

Final Answer:
The new price must be increased by 33.33% to restore the original price.