### Rationals Between

What fractions can you find between the square roots of 56 and 58?

### F'arc'tion

At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and paper.

### Do Unto Caesar

At the beginning of the night three poker players; Alan, Bernie and Craig had money in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of them won $1 200. What were the assets of the players at the beginning of the evening? # Peaches Today, Peaches Tomorrow.... ##### Stage: 3 and 4 Challenge Level: (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. 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. 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${\bf 75}$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:${\bf \frac{1}{2}, \frac{1}{4}, \frac{3}{4}, \frac{3}{5}, \frac{5}{6}, \frac{11}{15}}$In which order did he use the fractions so that he was left with just one peach at the end? (iii) Peach Rationing 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). Can you start with fewer than 100 peaches and choose fractions so that there is at least one peach left after a week? What is the longest that you can make them last, starting with fewer than 100 peaches? Send us your solutions showing how many peaches you started with and the fractions you used each day. 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?