Perimeter of a rectangle is $x$ and the circumference of a circle is 8 more than the perimeter of the rectangle. The ratio of the radius of the circle and the length of the rectangle is $1:2$ and the ratio of length and breadth of the rectangle is $7:3$. Find the length of the rectangle.

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**Solution:**

1. Let the length and breadth of the rectangle be $7a$ and $3a$ respectively (given ratio $7:3$).

2. The perimeter of the rectangle is:
   $$P = 2(7a + 3a) = 20a$$

3. The circumference of the circle is 8 more than the perimeter:
   $$C = P + 8 = 20a + 8$$

4. The ratio of the radius of the circle to the length of the rectangle is $1:2$, so:
   $$r : 7a = 1 : 2 \ r = \frac{7a}{2} = 3.5a$$

5. Circumference of the circle is:
   $$C = 2 \pi r = 2 	imes \frac{22}{7} 	imes 3.5a = 2 	imes \frac{22}{7} 	imes \frac{7}{2} a = 22a$$

6. Equate the two expressions for circumference:
   $$20a + 8 = 22a$$

7. Solve for $a$:
   $$22a - 20a = 8 \ 2a = 8 \ a = 4$$

8. Length of the rectangle:
   $$7a = 7 	imes 4 = 28 	ext{ cm}$$

**Answer:** $28$

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