Breakfast Time
Four hobbits each eat one quarter of the porridge remaining in the pan. How much is left?
Age
14 to 16
Challenge level
Problem
The hobbits Frodo, Sam, Pippin and Merry have breakfast at different times. Each one takes a quarter of the porridge they find in the pan, thinking that the others have not yet eaten. What fraction of the porridge is left after all four hobbits have had their breakfast?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Answer: $\frac{81}{256}$
Each of the hobbits eats $\frac{1}{4}$ of the porridge in the pan, so leaves $\frac{3}{4}$ in place. This means that the amount of porridge is multiplied by $\frac{3}{4}$ each time one of the hobbits eats.
Since four of them eat, the amount of porridge is multiplied by $\frac{3}{4} \times \frac{3}{4} \times \frac{3}{4} \times \frac{3}{4} = \left( \frac{3}{4} \right)^4 = \frac{81}{256}$
Therefore, $\frac{81}{256}$ of the porridge remains.