The ratio of the difference between the areas of a rectangle in two scenarios—first, when the length is decreased by 4 cm, and second, when the breadth is increased by 4 cm—to the original area of the rectangle is $4 : 9$. Find the ratio of the numerical value of the perimeter to the area of the rectangle.

Question

Accepted Answer

**Solution:**

1. Let the length and breadth of the rectangle be *l* and *b* respectively.

2. According to the question, the difference between the areas in two cases is:
   - Area when breadth is increased by 4 cm: $l 	imes (b + 4)$
   - Area when length is decreased by 4 cm: $(l - 4) 	imes b$

3. The difference between these two areas is:
   $$l(b + 4) - (l - 4)b = lb + 4l - lb + 4b = 4l + 4b$$

4. The ratio of this difference to the original area $lb$ is given as $\frac{4}{9}$:
   $$\frac{4l + 4b}{lb} = \frac{4}{9}$$

5. Multiply both sides by $9lb$:
   $$9(4l + 4b) = 4 lb$$

6. Simplify:
   $$36l + 36b = 4 lb$$

7. Divide both sides by 2:
   $$18l + 18b = 2 lb$$

8. Note that perimeter $P = 2(l + b)$ and area $A = lb$.

9. Rewrite the equation as:
   $$9 	imes 2(l + b) = 2 lb$$
   $$9P = 2A$$

10. Therefore, the ratio of perimeter to area is:
    $$P : A = 2 : 9$$

**Answer:** $2 : 9$

Answer

Answer

Answer