The length of a rectangle is increased by 25%. By what percentage must the breadth be decreased so that the area of the rectangle remains unchanged?

Introduction / Context:
This problem is a standard percentage and area question about rectangles. When one dimension of a rectangle changes, the other dimension must adjust in the opposite direction if the area is to remain the same. Here the length increases, so the breadth must decrease. The question asks for the exact percentage decrease required in the breadth to offset a 25 percent increase in length.

Given Data / Assumptions:

• Original length = L.
• Original breadth = B.
• Original area A = L * B.
• New length = L + 25 percent of L = 1.25L.
• Let new breadth be B'.
• New area must equal old area: 1.25L * B' = L * B.

Concept / Approach:
When dealing with relative changes and constant products, it is often easiest to work with multiplicative factors. An increase of 25 percent corresponds to multiplying by 1.25. To keep the area constant, the product of the new length factor and the new breadth factor must equal 1. Therefore, if the length is multiplied by 1.25, the breadth must be multiplied by 1 / 1.25. Once this factor is found, it can be converted into a percentage decrease from the original breadth.

Step-by-Step Solution:
Old area A = L * B. New area A' = 1.25L * B'. Set A' = A to keep area unchanged: 1.25L * B' = L * B. Divide both sides by L: 1.25 * B' = B. So B' = B / 1.25. 1.25 = 5/4, so B' = B * (4/5) = 0.8B. This means the new breadth is 80 percent of the original breadth, a 20 percent reduction.

Verification / Alternative check:
Assume some simple values to confirm. Let L = 100 units and B = 100 units, so area = 10000 units. Increase length by 25 percent: new length = 125 units. If breadth is reduced by 20 percent, new breadth = 80 units. New area = 125 * 80 = 10000 units, which is exactly the original area, verifying that a 20 percent decrease is correct.

Why Other Options Are Wrong:
25% and 30%: These changes would make the area smaller than the original. 40%: This is a very large reduction and would reduce the area significantly below the original. 15%: This reduction is too small and would result in an area larger than the original.

Common Pitfalls:
Adding or subtracting percentage changes instead of using multiplicative factors. Assuming that a 25 percent increase must be cancelled by a 25 percent decrease, which is not correct. Forgetting that the product of the two change factors must equal 1 to keep the area constant.

Final Answer:
The breadth must be decreased by 20%