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The four circles have the same center and their radii are $4 cm, 5 cm, 6 cm$ and 7cm, respectively. What is the ratio of the area of the small shaded ring to the area of the large shaded ring?

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10 : 12

9 : 13

13 : 5

3 : 9

answer is B.

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Detailed Solution

We have been given,
The four circles have the same center and their radii are

4 cm, 5 cm, 6 cm

and 7cm, respectively.
Question Image

We know that area of circle is

π \times r^{2}

, where

r

is radius.
Let

r_{1} = 4 cm, r_{2} = 5 cm, r_{3} = 6 cm, and r_{4} = 7 cm .

Area of small shaded ring is

π \times ({r_{2}}^{2} - {r_{1}}^{2})

\Rightarrow π \times (5^{2} - 4^{2})

\Rightarrow π \times 9 {cm}^{2}

Area of large shaded region is

π \times ({r_{4}}^{2} - {r_{3}}^{2})

\Rightarrow π \times (7^{2} - 6^{2})

\Rightarrow π \times 13 {cm}^{2}

The ratio of smaller to larger shaded regions is

\frac{Area of small shaded ring}{Area of large shaded region}

\Rightarrow \frac{π \times 9 {cm}^{2}}{π \times 13 {cm}^{2}}

\Rightarrow \frac{9}{13}

\Rightarrow 9 : 13

Therefore, the required ratio is 9:13.
Hence, option 2 is correct.

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