The price of an article is reduced by 33%.\nBy what percentage must the new reduced price be increased so that it returns exactly to the original price?

Introduction / Context:
This question examines the reverse effect of a large percentage decrease and how much percentage increase is needed to get back to the original value. It is an important concept that a 33% decrease cannot be undone by a 33% increase. Understanding the correct relationship is very useful in questions involving discounts, salaries, and stock prices.

Given Data / Assumptions:

• The original price of an article is P.
• The price is cut by 33%.
• We must find the percentage increase required on the reduced price to restore it to P.
• There are no further changes or additional charges involved.

Concept / Approach:
A decrease of d percent leaves (100 - d)% of the original value. To restore the original value from this reduced value, we need a percentage increase that solves the equation reduced * (1 + x/100) = original. There is also a direct formula: required percentage increase = (d / (100 - d)) * 100. We apply this formula with d = 33 to find the required increase. Calculating with a base of 100 units makes the logic easier to see.

Step-by-Step Solution:
Step 1: Assume the original price P is 100 units.Step 2: A decrease of 33% means the new price is 100 * (1 - 33/100) = 100 * 0.67 = 67 units.Step 3: Let the required percentage increase be x% on this reduced price.Step 4: To reach the original 100 units, we need 67 * (1 + x/100) = 100.Step 5: Rearrange: 1 + x/100 = 100 / 67.Step 6: Therefore x/100 = (100 / 67) - 1 = (100 - 67) / 67 = 33 / 67.Step 7: Multiply by 100: x = (33 / 67) * 100 ≈ 49.25 percent.

Verification / Alternative check:
Using the standard formula directly gives the same result. Required percentage increase = (d / (100 - d)) * 100, with d = 33. So x = (33 / (100 - 33)) * 100 = (33 / 67) * 100, which is approximately 49.25 percent. This agrees exactly with the step by step calculation with a base of 100 units, confirming that we need roughly a 49.25 percent increase to undo a 33 percent decrease.

Why Other Options Are Wrong:
33 percent would restore only 67 * 1.33 = 89.11 units, which does not reach the original 100 units.
24.81 percent is too small and clearly will not recover the original 100 from 67.
41.25 percent also falls short since 67 * 1.4125 is still less than 100.
66 percent would overshoot the original amount considerably, taking 67 to over 110 units.

Common Pitfalls:
The most common mistake is to assume that the same percentage that was used for the decrease will restore the original value when used as an increase. This fails because the base amount for the increase is smaller after the reduction. Another pitfall is to misapply the formula by placing 100 in the wrong position of the fraction or forgetting to multiply by 100 at the end. To avoid such errors, always work with a hypothetical base value of 100 units and carefully track how the percentage changes affect it step by step.

Final Answer:
The reduced price must be increased by approximately 49.25 percent to restore the original price.