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:
This question asks for the net percentage change in the area of a rectangle when its length undergoes a percentage increase and its breadth undergoes a percentage decrease.


Step 2: Key Formula or Approach:
The area of a rectangle is \(\text{Length} \times \text{Breadth}\).
For successive percentage changes \(x\) and \(y\), the net percentage change can be calculated using the successive change formula:
\[ \text{Net Change \%} = \left( x + y + \frac{x \times y}{100} \right) \% \]

Step 3: Detailed Explanation:
Let the original length be \(L\) and the original breadth be \(B\).
Original Area \(A = L \times B\).
The length is increased by 10%, so the new length is \(1.1L\).
The breadth is decreased by 10%, so the new breadth is \(0.9B\).
New Area \(A' = 1.1L \times 0.9B = 0.99LB\).
The new area is \(0.99\) times the original area, which means it is 99% of the original area.
Therefore, the area decreases by \(100\% - 99\% = 1\%\).
Alternative approach using the formula:
Let \(x = +10\) (increase) and \(y = -10\) (decrease).
\[ \text{Net Change \%} = 10 - 10 + \frac{10 \times (-10)}{100} = 0 - \frac{100}{100} = -1\% \] The negative sign indicates a decrease of 1%.


Step 4: Final Answer:
The correct choice is (B).