Question

The length of a square is increased by $15%$ and breadth is decreased by $15%$. The area of the rectangle so formed is

• A. neither increases nor decreases
• B. decreases by $2.25%$
• C. increases by $2.25%$
• D. decreases by $22.5%$

Hint:

For two opposite percentage changes $+p%$ and $-p%$, net change $\approx -p^2/100$ (exact for products like area).

Answer 1

The Correct Option is B

Let's first understand the problem. We begin by considering a square with a side length of \( s \). The area of this square is given by:

\( A_{square} = s \times s = s^2 \)

Now, when the length of the square is increased by 15%, the new length becomes:

\( s_{new} = s + 0.15s = 1.15s \)

Similarly, when the breadth is decreased by 15%, the new breadth becomes:

\( b_{new} = s - 0.15s = 0.85s \)

Thus, the area of the rectangle so formed is:

\( A_{rectangle} = s_{new} \times b_{new} = (1.15s) \times (0.85s) = 1.15 \times 0.85 \times s^2 \)

Calculating \( 1.15 \times 0.85 \), we have:

\( 1.15 \times 0.85 = 0.9775 \)

Therefore, the area of the rectangle becomes:

\( A_{rectangle} = 0.9775s^2 \)

Comparing with the original square's area \( s^2 \):

The percentage change in area is found using:

\( \text{Percentage Change} = \left(\frac{A_{rectangle} - A_{square}}{A_{square}}\right) \times 100 = \left(\frac{0.9775s^2 - s^2}{s^2}\right) \times 100 \)

\( = (0.9775 - 1) \times 100 = -2.25\% \)

This indicates that the area of the rectangle decreases by 2.25% compared to the square.