In a rectangle, the length is three times its breadth. If the length and the breadth of the rectangle are increased by 30% and 100% respectively, by what percentage does the perimeter of the rectangle increase?

Introduction:
This question is about how the perimeter of a rectangle changes when its length and breadth are increased by different percentages. Such problems test a student's understanding of ratios, percentage changes, and the formula for the perimeter of a rectangle. It also checks whether students can correctly track changes through algebra rather than relying on guesswork.

Given Data / Assumptions:

• Let the original breadth be B. • Original length = 3B (because length is three times breadth). • Length is increased by 30%, so new length = 130% of original length. • Breadth is increased by 100%, so new breadth = 200% of original breadth. • We must find the percentage increase in the perimeter.

Concept / Approach:
The perimeter P of a rectangle is given by P = 2 * (length + breadth). We can express the original perimeter in terms of B, then express the new perimeter after the percentage increases, again in terms of B, and finally compute the percentage increase using the formula: percentage increase = ((new - old) / old) * 100. Using algebra keeps the solution general and avoids choosing specific numeric values unnecessarily.

Step-by-Step Solution:
Step 1: Express original dimensions. Original breadth = B. Original length = 3B. Step 2: Compute original perimeter. Original perimeter P1 = 2 * (length + breadth) = 2 * (3B + B) = 2 * 4B = 8B. Step 3: Compute new dimensions after increase. Length increased by 30%: new length L2 = 1.30 * 3B = 3.9B. Breadth increased by 100%: new breadth B2 = 2 * B = 2B. Step 4: Compute new perimeter. New perimeter P2 = 2 * (L2 + B2) = 2 * (3.9B + 2B) = 2 * 5.9B = 11.8B. Step 5: Compute percentage increase in perimeter. Increase in perimeter = P2 - P1 = 11.8B - 8B = 3.8B. Percentage increase = (increase / original) * 100 = (3.8B / 8B) * 100. Simplify: (3.8 / 8) * 100 = 0.475 * 100 = 47.5%.

Verification / Alternative check:
Pick a specific value, for example B = 10 units. Then original length = 30 units and original perimeter P1 = 2 * (30 + 10) = 80 units. New length = 1.3 * 30 = 39 units and new breadth = 2 * 10 = 20 units, so new perimeter P2 = 2 * (39 + 20) = 2 * 59 = 118 units. The increase is 118 - 80 = 38 units. Percentage increase = (38 / 80) * 100 = 47.5%, matching our algebraic result.

Why Other Options Are Wrong:
Option 40%: Underestimates the effect of doubling the breadth and is not supported by the calculations. Option 45%: Close but not exact; it may result from rounding or partial computation errors. Option 50%: Assumes a simple half increase in perimeter, which is not consistent with the different percentage changes in length and breadth. Option 55%: Overestimates the change and does not match either numerical or algebraic checks.

Common Pitfalls:
Some students incorrectly assume that the percentage change in perimeter is simply the average or sum of the percentage changes of length and breadth. Others may recompute side changes but forget that perimeter uses the sum of both sides before doubling. Careless handling of decimal multipliers (such as 1.30 and 2.00) and incorrect arithmetic when simplifying 3.8 / 8 also cause mistakes. Writing each step and using a test value helps avoid these errors.

Final Answer:
The perimeter of the rectangle increases by 47.5%.