IMC 2020 Q4
4. What fraction of this diagram is shaded?
- A \(\dfrac{13}{32}\)
- B \(\dfrac{1}{2}\)
- C \(\dfrac{9}{16}\)
- D \(\dfrac{5}{8}\)
- E \(\dfrac{13}{16}\)
Intermediate Mathematical Challenge 2020, Question 4. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D \(\dfrac{5}{8}\)
We choose units so the outer square has side-length 4, and hence area 16.
The unshaded area of the square consists of two congruent triangles of base 3 and height 2.
So the area of each unshaded triangle is \(\tfrac{1}{2} \times 3 \times 2 = 3\).
Hence the area of the shaded part of the square is \(16 - 2 \times 3 = 10\).
Therefore the fraction of the square which is shaded is \(\tfrac{10}{16} = \tfrac{5}{8}\).
Solution © UK Mathematics Trust, from the IMC 2020 solutions. Longer versions with investigations: Solutions and investigations (PDF).