Bull's eye
What fractions of the largest circle are the two shaded regions?
Age
11 to 14
Challenge level
Problem
The diagram below shows circles with radii 3, 4, 5, 6 and 7 cm.
What fractions of the largest circle are each of the two shaded regions?
Image
Student Solutions
We have had correct responses to this problem from a number of
students: Andrei Lazanu (aged 12) from No. 205 School in Bucharest
(Romania), Chua Zhi Yu (aged 13) from River Valley High School in
Singapore, Michael Brooker (aged 10) educated at home, Belinda Guo
(aged 14) from Riccarton High School in Christchurch (New Zealand),
Prateek Mehrotra, Sim Jingwei (aged 12) from Raffles Girls' Primary
School in Singapore, and Fiona Watson from Stamford High School.
Well done to you all.
Everyone reasoned in a similar way:
Area of circle $=\pi r^2$
Area of the largest circle $= \pi \times 7^2 = 49 \pi cm^2$
Area of the red ring $= \pi \times 4^2 - \pi \times 3 ^ 2 = 7 \pi cm^2$
$\frac{7}{49}= \frac{1}{7}$
The red ring is $\frac{1}{7}$ of the whole circle.
Area of the green ring $= \pi \times 6^2 - \pi \times 5^2 = 11 \pi cm^2$
The green ring is $\frac{11}{49}$ of the whole circle.