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

Introduction / Context:
This question examines how the area of a rectangle changes when both its length and breadth are increased by certain percentages. It highlights that percentage changes in two dimensions combine multiplicatively, not additively. Understanding this is important in many real life situations, such as resizing photographs, scaling blueprints, or calculating land area changes when both directions are altered.

Given Data / Assumptions:

• Original length of the rectangle is L and original breadth is B.
• Length is increased by 40 percent, so the new length is 140 percent of L.
• Breadth is increased by 70 percent, so the new breadth is 170 percent of B.
• We are asked to find the percentage increase in the area of the rectangle.
• All calculations use basic percentage and multiplication rules.

Concept / Approach:
The original area is A = L * B. After the changes, new length is 1.4L and new breadth is 1.7B. So the new area A_new = 1.4L * 1.7B. The factor by which the area changes is A_new / A = (1.4 * 1.7). The percentage increase is then (A_new / A − 1) * 100 percent. It is important to multiply the individual factors for length and breadth rather than simply adding the percentage increases, because area depends on both dimensions together.

Step-by-Step Solution:
Original area A = L * B.New length = L_new = 1.4L (since 40 percent increase means multiply by 1.4).New breadth = B_new = 1.7B (since 70 percent increase means multiply by 1.7).New area A_new = L_new * B_new = 1.4L * 1.7B = (1.4 * 1.7) * L * B.Compute 1.4 * 1.7 = 2.38, so A_new = 2.38 * A.Percentage increase in area = (2.38 − 1) * 100 percent = 1.38 * 100 percent = 138 percent.

Verification / Alternative check:
Take simple numbers to verify. Suppose L = 10 units and B = 10 units. Then original area A = 100 sq units. New length = 1.4 * 10 = 14 units and new breadth = 1.7 * 10 = 17 units. New area A_new = 14 * 17 = 238 sq units. The increase in area is 238 − 100 = 138 sq units. As a percentage of 100, this is exactly 138 percent. This numerical example supports the algebraic calculation and confirms that the percentage increase is 138 percent.

Why Other Options Are Wrong:
Option 118 percent might come from incorrectly adding 40 percent and 70 percent and then subtracting a small value, but it does not match the correct multiplicative change. Option 110 percent could be from simply adding 40 percent and 70 percent, which gives 110 percent, but this ignores the fact that area scales with the product of both sides. Option 128 percent is another incorrect combination of the numbers. Option 90 percent arises from subtracting one of the percentages instead of combining them properly. None of these match the true 2.38 area factor.

Common Pitfalls:
The most common mistake is to simply add the two percentage increases and conclude that the area increases by 40 percent + 70 percent = 110 percent. This is wrong because the new area is based on both new length and new breadth. Another pitfall is to forget to convert percentages to decimal multipliers before multiplying. Careful step by step work with multipliers rather than raw percentages prevents these errors.

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