Quantity I: The area of a circle shown in the figure is half the area of a rectangle. Find the percentage by which the length of the rectangle is greater than its breadth.

Quantity II: When a pair of opposite sides of a square is increased by 10 cm each, the area of the resulting figure increases by 400 cm². Find the percentage increase in the area.

Question

Quantity I: The area of a circle shown in the figure is half the area of a rectangle. Find the percentage by which the length of the rectangle is greater than its breadth.

Quantity II: When a pair of opposite sides of a square is increased by 10 cm each, the area of the resulting figure increases by 400 cm². Find the percentage increase in the area.

Accepted Answer

**Solution:**

1. **Quantity I:**
   - Let the length of the rectangle be *L* and the breadth be *b*.
   - The radius of the circle is given as *b/2*.
   - Area of rectangle = *L × b*.
   - Area of circle = π × (b/2)² = πb²/4.
   - Given, area of circle is half the area of rectangle:
     $$\pi \frac{b^2}{4} = \frac{1}{2} (L \times b)$$
   - Multiply both sides by 2:
     $$\pi \frac{b^2}{2} = L \times b$$
   - Divide both sides by *b*:
     $$L = \frac{\pi b}{2}$$
   - Percentage by which length is more than breadth:
     $$\frac{L - b}{b} \times 100 = \frac{\frac{\pi b}{2} - b}{b} \times 100 = \left(\frac{\pi}{2} - 1\right) \times 100$$
   - Using $\pi \approx 3.14$,
     $$\left(\frac{3.14}{2} - 1\right) \times 100 = (1.57 - 1) \times 100 = 57\%$$

2. **Quantity II:**
   - Original figure is a square with side *s*.
   - Increasing two opposite sides by 10 cm each transforms it into a rectangle of dimensions $s + 10$ and $s$.
   - Increase in area = 400 cm².
   - New area - Original area = 400
     $$ (s + 10) \times s - s^2 = 400 $$
     $$ s^2 + 10s - s^2 = 400 $$
     $$ 10s = 400 $$
     $$ s = 40 \text{ cm} $$
   - Original area = $40 \times 40 = 1600 \text{ cm}^2$
   - Percentage increase in area:
     $$ \frac{400}{1600} \times 100 = 25\% $$

3. **Comparison:**
   - Quantity I = 57\%
   - Quantity II = 25\%
   - Therefore, Quantity I > Quantity II.

**Answer:** Quantity I > Quantity II

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