IMC 2011 Q13
13. The three blind mice stole a piece of cheese. In the night, the first mouse ate \(\tfrac{1}{3}\) of the cheese. Later, the second mouse ate \(\tfrac{1}{3}\) of the remaining cheese. Finally, the third mouse ate \(\tfrac{1}{3}\) of what was then left of the cheese.
Between them, what fraction of the cheese did they eat?
- A \(\dfrac{16}{27}\)
- B \(\dfrac{17}{27}\)
- C \(\dfrac{2}{3}\)
- D \(\dfrac{19}{27}\)
- E \(\dfrac{20}{27}\)
Intermediate Mathematical Challenge 2011, Question 13. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D \(\dfrac{19}{27}\)
After the first mouse has eaten, \(\tfrac{2}{3}\) of the cheese remains. After the second mouse has eaten, \(\tfrac{2}{3}\) of \(\tfrac{2}{3}\), that is \(\tfrac{4}{9}\), of the cheese remains. Finally, after the third mouse has eaten, \(\tfrac{2}{3}\) of \(\tfrac{4}{9}\), that is \(\tfrac{8}{27}\), of the cheese remains. So the mice ate \(\tfrac{19}{27}\) of the cheese.
Solution © UK Mathematics Trust, from the IMC 2011 solutions. Longer versions with investigations: Solutions and investigations (PDF).