Three blind mice
Each of the three blind mice in turn ate a third of what remained of a piece of cheese. What fraction of the cheese did they eat in total?
Age
11 to 14
Challenge level
Problem
The three blind mice stole a piece of cheese. In the night, the first mouse ate $\frac{1}{3}$ of the cheese. Later, the second mouse ate $\frac{1}{3}$ of the remaining cheese. Finally, the third mouse ate $\frac{1}{3}$ of what was then left of the cheese.
What fraction of the cheese is left?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Answer: $\frac8{27}$
Using a diagram
Image
first mouse eats $\frac13$
2 pieces left
Image
split each piece into 3 - each piece $\frac19$
4 pieces left
Image
split each piece into 3 - each piece $\frac1{27}$
8 pieces left
Direct calculation
After the first mouse has eaten, $\frac{2}{3}$ of the cheese remains.
After the second mouse has eaten, $\frac{2}{3}$ of $\frac{2}{3}$, that is $\frac{4}{9}$, of the cheese remains.
Finally, after the third mouse has eaten, $\frac{2}{3}$ of $\frac{4}{9}$, that is $\frac{8}{27}$ of the cheese remains.