The length of a rectangle is halved while its breadth is tripled. By what percentage does the area of the rectangle change?

Introduction / Context:
This problem checks your understanding of how proportional changes in dimensions affect the area of a rectangle. Instead of calculating with actual numbers, you work algebraically with factors. These types of percentage change questions are very common in quantitative aptitude tests because they require clear conceptual thinking rather than just formula memorisation.

Given Data / Assumptions:

- Initial length of the rectangle is L units.
- Initial breadth of the rectangle is B units.
- Length is reduced to half of its original value.
- Breadth is increased to three times its original value.
- We must find the percentage change in area, indicating increase or decrease.

Concept / Approach:
The area of a rectangle is given by Area = length * breadth. When dimensions change by certain factors, the new area becomes the product of the new length and new breadth. The ratio new area / old area gives the overall factor by which area changes. Converting this factor into a percentage change tells us whether area increases or decreases and by how much.

Step-by-Step Solution:
Step 1: Original area A_original = L * B.Step 2: New length = L / 2 (because length is halved).Step 3: New breadth = 3B (because breadth is tripled).Step 4: New area A_new = (L / 2) * (3B) = (3L * B) / 2.Step 5: Express new area as a multiple of old area: A_new / A_original = (3L * B / 2) / (L * B) = 3 / 2.Step 6: The factor 3 / 2 equals 1.5, which means the new area is 1.5 times the original area.Step 7: An increase from 1 to 1.5 is an increase of 0.5, that is 50 percent.

Verification / Alternative check:
You can also pick simple numbers. Suppose initially L = 2 units and B = 2 units, then the original area is 4 square units. After the change, new length is 1 unit and new breadth is 6 units, so new area is 6 square units. From 4 to 6 is an increase of 2 on 4, which is 2 / 4 * 100 percent = 50 percent increase. This numerical check confirms the algebraic reasoning.

Why Other Options Are Wrong:
25 percent increase or 25 percent decrease correspond to factors like 1.25 or 0.75, which are not obtained here. A 50 percent decrease would mean the area becomes half of the original, which is also not correct. No change in area would require the product of the dimension changes to be 1, but here the product factor is 1.5, not 1.

Common Pitfalls:
Students often incorrectly add or subtract percentage changes of individual dimensions instead of multiplying the factors. For example, they may think half and triple cancel out in some way. The correct method is always to treat each dimension change as a multiplicative factor and then multiply these factors to get the final effect on area. Confusing increases and decreases or forgetting to convert fractions into percentages are also common errors.

Final Answer:
Therefore, the area of the rectangle undergoes a 50% increase.