IMC 2008 Q9
9. A solid wooden cube is painted blue on the outside. The cube is then cut into 27 smaller cubes of equal size. What fraction of the total surface area of these new cubes is blue?
- A \(\dfrac{1}{6}\)
- B \(\dfrac{1}{5}\)
- C \(\dfrac{1}{4}\)
- D \(\dfrac{1}{3}\)
- E \(\dfrac{1}{2}\)
Intermediate Mathematical Challenge 2008, Question 9. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D \(\dfrac{1}{3}\)
Let the smaller cubes have side of length 1 unit. So the original cube had side of length 3 units and hence a surface area of 54 units2, all of which was painted blue. The total surface area of the 27 small cubes is 27 × 6 units2, that is 162 units2. So the required fraction is \(\dfrac{54}{162} = \dfrac{1}{3}\).
Solution © UK Mathematics Trust, from the IMC 2008 solutions.