The salary of a person is first decreased by 50% and is then increased by 50%. By what percentage does the person finally lose from the original salary?

Introduction / Context:
This question explores the effect of successive percentage changes, specifically a decrease followed by an increase of the same percentage. Many people wrongly assume that a 50% decrease followed by a 50% increase brings the salary back to the original value, but that is not correct. Such problems highlight the non additive nature of percentage changes and are very common in aptitude tests involving income, salary revisions and price fluctuations.

Given Data / Assumptions:

• Initial salary of the person is some amount, say S rupees.
• First, the salary is decreased by 50%.
• Then the reduced salary is increased by 50%.
• We need to find the net percentage loss relative to the original salary S.

Concept / Approach:
When a quantity changes by a percentage, the new value is original * (1 ± rate/100). Successive changes are handled by multiplying these factors. If the salary is reduced by 50%, the factor is 1 − 50/100 = 0.5. When increased by 50% afterward, the factor is 1 + 50/100 = 1.5, but this applies to the reduced salary. Multiplying 0.5 and 1.5 gives the overall factor. The net percentage change is then found by comparing this final factor with 1 (the original value factor).

Step-by-Step Solution:
Step 1: Let original salary be S. Step 2: After a 50% decrease, new salary = S * (1 − 50/100) = S * 0.5 = 0.5S. Step 3: This reduced salary is then increased by 50%. So final salary = 0.5S * (1 + 50/100) = 0.5S * 1.5. Step 4: Multiply the factors: 0.5 * 1.5 = 0.75. Final salary = 0.75S. Step 5: Compare final salary with original salary. Loss = S − 0.75S = 0.25S. Step 6: Percentage loss = (0.25S / S) * 100% = 25%.

Verification / Alternative check:
Take a simple numerical example. Suppose the original salary is Rs 100. After a 50% decrease, salary becomes Rs 50. A 50% increase on Rs 50 gives 50 + 25 = Rs 75. So the person ends up with Rs 75 instead of Rs 100. The loss is Rs 25 on Rs 100, which is 25%. This concrete example supports the algebraic method and shows clearly that the salary does not return to its original value.

Why Other Options Are Wrong:

• 50% would mean the final salary is zero, which is not the case.
• 75% loss would give a salary of only 25% of the original, which is far too extreme.
• No loss would mistakenly assume that opposite percentage changes cancel each other, which is only true if applied to the same base value, not successively.

Common Pitfalls:
Many students simply add or subtract percentages, concluding that a 50% decrease and a 50% increase cancel out. This ignores that the second change is applied to a modified amount. Another mistake is calculating the second 50% on the original salary instead of the reduced one. Always remember that when percentages are applied one after the other, you must multiply the corresponding factors and then compare the final value to the original to get the net effect.

Final Answer:
The person suffers a net loss of 25% of the original salary.