Question

What is the change in area of a rectangular field whose length is increased by 10% and breadth is decreased by 10%?

• A. Increases by 1%
• B. Decreases by 1%
• C. Remains the same
• D. Increases by 10%

Hint:

When a quantity is increased by \(x\%\) and then decreased by \(x\%\), there is always a net decrease given by \(\frac{x^2}{100}\%\). Here, \(\frac{10^2}{100} = 1\%\) decrease.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem requires finding the overall percentage change in a rectangle's area after its length is increased by a certain percentage and its breadth is decreased by the same percentage.
Step 2: Key Formula or Approach:
Rectangle Area = \(\text{Length} \times \text{Breadth}\).
For consecutive percentage modifications \(x\) and \(y\), the net percentage change is found via the formula:
\[ \text{Net Change %} = \left( x + y + \frac{x \times y}{100} \right) % \] Step 3: Detailed Explanation:
Assume the initial length is \(L\) and the initial breadth is \(B\).
Initial Area \(A = L \times B\).
A 10% length increase makes the new length \(1.1L\).
A 10% breadth decrease makes the new breadth \(0.9B\).
New Area \(A' = 1.1L \times 0.9B = 0.99LB\).
This shows the new area is 99% of the original, signifying a 1% reduction (\(100% - 99%\)).
Alternative method using the formula:
Assign \(x = +10\) (for increase) and \(y = -10\) (for decrease).
\[ \text{Net Change %} = 10 - 10 + \frac{10 \times (-10)}{100} = 0 - \frac{100}{100} = -1% \] The negative result confirms a 1% decrease.
Step 4: Final Answer:
The correct choice is (B).