The price of an article is reduced (cut) by 3%.\nBy what percentage must this reduced price be increased so that the article returns exactly to its original price?

Introduction / Context:
This question highlights an important concept in percentages: a decrease of a certain percentage does not require the same percentage increase to return to the original value. Instead, the required increase is slightly larger because it is applied to a smaller base. Understanding this asymmetry is crucial for solving many successive change questions in profit, loss, and pricing.

Given Data / Assumptions:

• The original price of the article is some amount P.
• The price is first cut by 3%, leaving a reduced price.
• The reduced price is then increased by some unknown percentage x% to get back to P.
• We must compute the value of x.
• All percentage changes are applied multiplicatively to the current price at each step.

Concept / Approach:
A 3% decrease means the new price is 97% of the original, or 0.97P. Let the required percentage increase on this reduced price be x%. After the increase, the new price becomes 0.97P * (1 + x/100). To restore the original price, this expression must equal P. By forming the equation 0.97P * (1 + x/100) = P and solving for x, we obtain the required percentage increase.

Step-by-Step Solution:
Step 1: Let original price be P. Step 2: After a 3% reduction, reduced price = P - 0.03P = 0.97P. Step 3: Let the required increase be x%. New price after increase = 0.97P * (1 + x/100). Step 4: To get back to original price, set 0.97P * (1 + x/100) = P. Step 5: Divide both sides by P: 0.97 * (1 + x/100) = 1. Step 6: Divide both sides by 0.97: 1 + x/100 = 1 / 0.97. Step 7: Compute 1 / 0.97 ≈ 1.030927835. Step 8: Therefore, x/100 ≈ 1.030927835 - 1 = 0.030927835. Step 9: Multiply by 100: x ≈ 3.0927835%, which rounds to about 3.09 percent.

Verification / Alternative check:
Take a simple number for P, say P = 100. After a 3% cut, the price becomes 97. Now increase 97 by 3.09%: 3.09% of 97 ≈ 0.0309 * 97 ≈ 2.9973, and 97 + 2.9973 ≈ 99.9973, which is effectively 100 when rounded. This confirms that an increase of about 3.09 percent restores the original price after a 3 percent cut.

Why Other Options Are Wrong:
3 percent: Using 3% again on the reduced price only brings it to 97 * 1.03 = 99.91, slightly below the original.

7.11 percent and 6 percent: These are far larger than needed and overshoot the original price by a noticeable margin.

2.69 percent: Too small; increasing 97 by only 2.69% yields a value below 100.

Common Pitfalls:
Many students assume that a 3% decrease followed by a 3% increase cancels out, but this is incorrect because the increase is applied to a smaller base. Others try to add or subtract percentages linearly without using multiplicative factors. Always remember to convert percentage changes into factors (like 0.97 or 1.0309) and work with equations when restoring original values.

Final Answer:
The reduced price must be increased by approximately 3.09 percent to restore the original price.