## 29. Apples

Amelie picked apples, ate a quarter of them herself, and gave the rest to her little sisters Bertha, Charlotte, and Dorothy. Bertha ate a quarter of her apples and gave the rest to her three sisters. Charlotte ate one apple and gave the rest to her three sisters. As for little Dorothy, she also ate a quarter of her share before giving the rest to her three sisters.

"Look," said Bertha to Charlotte, "I have twice as many apples left as you do."

How many apples did Amelie pick?
29. Let $a$ be the number of apples Amelie ends up with, $c$ be the number of apples Charlotte has at this stage, and $2c - y$ be the number of apples Berta has (0 being the number of apples Dorothy has). In the following table, we will represent all events in reverse order of their actual occurrence:

|  | Amelie | Berta | Charlotte | Dorothy |
| :--- | :---: | :---: | :---: | :---: |
| In the end | $a$ | $2c$ | $c$ | 0 |
| Before Dorothy distributed the apples | $a-c$ | $c$ | 0 | $4+3c$ |
| Before Charlotte distributed the apples | $a-2c$ | 0 | $1+3c$ | $4+2c$ |
| Before Berta distributed the apples, but after Amelie did | 0 | $4a-8c$ | $1+5c-a$ | $4+4c-a$ |

The last three parts should be equal. Therefore,

$$
4a-8c=1+5c-a
$$

and

$$
4a-8c=4+4c-a,
$$

from which $c=3, a=8$.

Thus, Amelie gave her sisters 8 apples each, and she herself picked 32 apples.