IMC 2017 Q5
5. The diagram shows two circles with the same centre. The radius of the outer circle is twice the radius of the inner circle. The region between the inner circle and the outer circle is divided into six equal segments as shown.
What fraction of the area of the outer circle is shaded?
- A \(\dfrac{3}{7}\)
- B \(\dfrac{3}{8}\)
- C \(\dfrac{3}{9}\)
- D \(\dfrac{3}{10}\)
- E \(\dfrac{3}{11}\)
Intermediate Mathematical Challenge 2017, Question 5. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B \(\dfrac{3}{8}\)
Let the area of the inner circle be \(A\). The radius of the outer circle is twice that of the inner circle, so its area is four times that of the inner circle , that is \(4A\). Therefore the region between the two circles has area \(3A\). As three of the six equal segments are shaded, the total shaded area is \(3A \div 2\). So the required fraction is \(\frac{3}{2}A \div 4A = \frac{3}{8}\).
Solution © UK Mathematics Trust, from the IMC 2017 solutions. Longer versions with investigations: Solutions and investigations (PDF).