IMC 2009 Q23
23. The diagram shows part of a tiling pattern which is made from two types of individual tiles: 8 by 6 rectangular white tiles and square black tiles. If the pattern is extended to cover an infinite plane, what fraction is coloured black?
- A \(\dfrac{1}{12}\)
- B \(\dfrac{1}{13}\)
- C \(\dfrac{1}{25}\)
- D \(\dfrac{1}{37}\)
- E \(\dfrac{1}{40}\)
Intermediate Mathematical Challenge 2009, Question 23. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C \(\dfrac{1}{25}\)
Firstly, note that the black squares have side 2 units. The pattern may be considered to be a tessellation of the shape shown on the right. So the ratio of squares to rectangles is 1:2 and hence the fraction coloured black is \(\dfrac{4}{4 + 2 \times 48} = \dfrac{4}{100} = \dfrac{1}{25}\).
Solution © UK Mathematics Trust, from the IMC 2009 solutions.