The area of circle and rectangle is 11:6 and the ratio of radius of the circle to length of the rectangle is 1:6. If the breadth of the rectangle is 7 cm, then find the circumference of the circle (in cm).

The correct answer is 154.

Let the radius of the circle be $r$ and the length of the rectangle be $l$.

According to the question, the ratio of the radius of the circle to the length of the rectangle is 1:6:
$\frac{r}{l}=\frac{1}{6}$
This gives the length of the rectangle as:
$l=6r$

The breadth ($b$) of the rectangle is given as 7 cm.

Now, let's write the expressions for the areas:
1. Area of the circle ($A_{\text{circle}}$):
$A_{\text{circle}}=\pi r^2$
2. Area of the rectangle ($A_{\text{rectangle}}$):
$A_{\text{rectangle}}=l\times b=6r\times 7=42r$

The ratio of the area of the circle to the area of the rectangle is given as 11:6:
$\frac{A_{\text{circle}}}{A_{\text{rectangle}}}=\frac{11}{6}$

Substituting the area expressions into the ratio equation:
$\frac{\pi r^2}{42r}=\frac{11}{6}$

Simplifying the equation by canceling $r$ (since $r\ne 0$):
$\frac{\pi r}{42}=\frac{11}{6}$

Multiplying both sides by 42 to solve for $\pi r$:
$\pi r=\frac{11}{6}\times 42$
$\pi r=11\times 7$
$\pi r=77$

The circumference of a circle is given by the formula:
$\text{Circumference}=2\pi r$

Substituting the value of $\pi r=77$ into the circumference formula:
$\text{Circumference}=2\times 77=154\text{cm}$