Question

Lisa drew three circles to form a figure. The areas of the circles were in the

ratio 1:4:16. She then shaded some parts of the figure as shown.
What fraction of the figure was shaded?​

Answer 1

Fraction of the figure shaded =
`(13)/(16)`

Explanation:

Ratio of the areas of the given circles are 1 : 4 : 16

Then the radii of the circles will be in the ratio =
`√(1):√(4):√(16)`

= 1 : 2 : 4

If the radius of the smallest circle = x units

Then the radius of the middle circle = 2x units

and the radius of the largest circle = 4x units

Area of the smallest circle = πx²

Area of the middle circle = π(2x)² = 4πx²

Area of the largest circle = π(4x)²= 16πx²

Area of the region which is not shaded in the middle circle = πx²(4 - 1)

= 3πx²

Therefore, area of the shaded region = Area of the largest circle - Area of the region which is not shaded

= 16πx² - 3πx²

= 13πx²

Fraction of the figure which is not shaded =
`\frac{\text{Area of the shaded region}}{\text{Area of the largest circle}}`

=
`(13\pi x^(2) )/(16\pi x^(2) )`

=
`(13)/(16)`