(i) A little monkey had 60 peaches.

On the first day he decided to keep ${\bf \frac{3}{4}}$ of his peaches.

He gave the rest away. Then he ate one.

On the second day he decided to keep ${\bf \frac{7}{11}}$ of his peaches.

He gave the rest away. Then he ate one.

He gave the rest away. Then he ate one.

On the third day he decided to keep ${\bf \frac{5}{9}}$ of his peaches.

He gave the rest away. Then he ate one.

On the fourth day he decided to keep ${\bf \frac{2}{7}}$ of his peaches.

He gave the rest away. Then he ate one.

He gave the rest away. Then he ate one.

On the fifth day he decided to keep ${\bf \frac{2}{3}}$ of his peaches.

He gave the rest away. Then he ate one.

How many did he have left at the end?

(ii) A little monkey had 75 peaches.

Each day, he kept a fraction of his peaches, gave the rest away, and then ate one.

These are the fractions he decided to keep:

$ \frac{1}{2} \frac{1}{4} \frac{3}{4} \frac{3}{5} \frac{5}{6} \frac{11}{15}$

In which order did he use the fractions so that he was left with just one peach at the end?

(iii) Whenever the monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one.

I wonder how long he could make his peaches last for?

Here are his rules:

- Each fraction must be in its simplest form and must be less than 1.
- The denominator is never the same as the number of peaches left (for example, if there were 45 peaches left, he would not be allowed to keep $\frac{44}{45}$ of them).

Click here for a poster of this problem.

*An older version of the problem also included the question below:*

A little monkey had some peaches.

On the first day he decided to keep ${\bf \frac{1}{2}}$ of his peaches.

He gave the rest away. Then he ate one.

On the second day he decided to keep ${\bf \frac{1}{2}}$ of his peaches.

He gave the rest away. Then he ate one.

On the third day he decided to keep ${\bf \frac{1}{2}}$ of his peaches.

He gave the rest away. Then he ate one.

On the fourth day he found he had only one peach left.

How many did he have at the beginning?