The length and breadth of a rectangle are increased by 40% and 70% respectively. By what percentage does the area of the rectangle increase as a result of these simultaneous changes?

Introduction / Context:
This problem is a classic example of percentage increase in area when the dimensions of a rectangle change. It highlights that percentage changes in length and breadth combine multiplicatively, not additively. Such questions are frequently used to test understanding of compound percentage ideas in quantitative aptitude exams.

Given Data / Assumptions:

• Original length of the rectangle = L.
• Original breadth of the rectangle = B.
• Length is increased by 40%, so new length = 1.4 * L.
• Breadth is increased by 70%, so new breadth = 1.7 * B.
• We must find the percentage increase in the area.

Concept / Approach:
The original area A1 of the rectangle is L * B. After the increases, the new area A2 is (1.4 * L) * (1.7 * B) = 1.4 * 1.7 * L * B. So A2 = 2.38 * A1. The factor 2.38 means the new area is 2.38 times the old area. The percentage increase is computed as ((A2 - A1) / A1) * 100%. That simplifies to (2.38 - 1) * 100% = 1.38 * 100% = 138%.

Step-by-Step Solution:
Step 1: Let original area A1 = L * B. Step 2: New length = L increased by 40%, so new length = L * (1 + 40/100) = 1.4 * L. Step 3: New breadth = B increased by 70%, so new breadth = B * (1 + 70/100) = 1.7 * B. Step 4: New area A2 = (1.4 * L) * (1.7 * B) = 1.4 * 1.7 * L * B. Step 5: Multiply 1.4 by 1.7: 1.4 * 1.7 = 2.38. So A2 = 2.38 * A1. Step 6: Percentage increase = ((A2 - A1) / A1) * 100%. Substitute A2 = 2.38 * A1: ((2.38 * A1 - A1) / A1) * 100% = (1.38 * A1 / A1) * 100%. This simplifies to 1.38 * 100% = 138%.

Verification / Alternative check:
We can test with simple numbers. Assume L = 10 and B = 10, so original area A1 = 100 square units. New length = 1.4 * 10 = 14, new breadth = 1.7 * 10 = 17. New area A2 = 14 * 17 = 238 square units. The increase in area is 238 - 100 = 138 square units, which relative to the original area 100 is a 138% increase. This numerical check confirms the algebraic result.

Why Other Options Are Wrong:
Option A (118%): This might come from partly correct but incomplete multiplication of factors or from adding incorrect adjustment. Option B (110%): This is much smaller and does not align with the product 1.4 * 1.7. Option D (128%): This may result from adding 40% and 70% and then subtracting something, but it is not accurate. Option E (90%): This is clearly too low relative to the large increases in both dimensions.

Common Pitfalls:
A common mistake is to simply add the percentage changes (40% + 70% = 110%) and assume that is the percentage increase in area. This ignores the multiplicative effect of changing both dimensions. Remember that area scales with the product of the scale factors, not their sum. Another pitfall is arithmetic errors when multiplying decimals like 1.4 and 1.7. Writing them as fractions, such as 14/10 and 17/10, can sometimes help avoid mistakes.

Final Answer:
The area of the rectangle increases by 138%.