An annular region has outer radius 12 cm and inner radius 8 cm. What is the approximate ratio of the annular area to the whole outer circle?

The correct option is 5:9.

To find the ratio of the annular area to the area of the whole outer circle, let us analyze the formulas for area step-by-step.

Step 1: Understand the given dimensions.
Let $R_{\text{outer}}$ be the outer radius = 12 cm.
Let $r_{\text{inner}}$ be the inner radius = 8 cm.

Step 2: Calculate the area of the outer circle.
The area of a circle with radius $R$ is given by the formula:

$A_{\text{outer}}=\pi {R_{\text{outer}}}^2$

Substituting $R_{\text{outer}}=12$ cm:

$A_{\text{outer}}=\pi \times {12}^2=144\pi$ cm²

Step 3: Calculate the area of the inner circle.
The area of the inner circle is:

$A_{\text{inner}}=\pi {r_{\text{inner}}}^2$

Substituting $r_{\text{inner}}=8$ cm:

$A_{\text{inner}}=\pi \times 8^2=64\pi$ cm²

Step 4: Calculate the area of the annular region.
The annular area is the region between the outer and inner concentric circles, calculated by subtracting the inner area from the outer area:

$A_{\text{annular}}=A_{\text{outer}}-A_{\text{inner}}$

$A_{\text{annular}}=144\pi -64\pi =80\pi$ cm²

Step 5: Determine the ratio of the annular area to the area of the outer circle.
The required ratio is:

$\text{Ratio}=\frac{A_{\text{annular}}}{A_{\text{outer}}}=\frac{80\pi }{144\pi }$

Simplifying the fraction by dividing both the numerator and denominator by $16\pi$:

$\text{Ratio}=\frac{80}{144}=\frac{5}{9}$

Thus, the ratio of the annular area to the whole outer circle is 5:9.