Vodník Kebule nakupoval v rybárně kapitána Nema, kde ceny všeho zboží byly uvedeny v celých šupinkách. Kdyby Kebule koupil 2 raky, 3 škeble a 1 štiku, zaplatil by 49 šupinek. Pokud by přikoupil ještě 5 raků, 11 škeblí a 1 štiku, platil by celkem 154 šupinek.

Kolik šupinek by platil za 1 raka, 2 škeble a 3 štiky? Určete všechny možnosti.

(K. Pazourek)

Nápověda. Vyjádřete vztahy ze zadání pomocí neznámých.

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Water Sprite Kebule was shopping at Captain Nema's fish market, where the prices of all goods were listed in whole scales. If Kebule bought 2 crayfish, 3 perch, and 1 pike, he would pay 49 scales. If he also bought 5 more crayfish, 11 more perch, and 1 more pike, he would pay a total of 154 scales.

How many scales would he pay for 1 crayfish, 2 perch, and 3 pike? Determine all possibilities.

(K. Pazourek)

Hint. Express the relationships from the problem using unknowns.
Let's denote by $r, k$, and $t$ the number of scales that one rak, one skeble, and one stika cost, respectively. The information about the first considered purchase means

$$
2 r+3 k+t=49 \text {. }
$$

For the purchased goods, Kebule would have to pay an additional $154-49=105$ scales, thus

$$
5 r+11 k+t=105 .
$$

By subtracting the first equation from the second, we get

$$
3 r+8 k=56 .
$$

The unknowns $r$ and $k$ are integers and

$$
k=7-\frac{3}{8} r
$$

therefore $r$ must be divisible by eight. The unknowns $r$ and $k$ are also positive numbers, so $r$ can be either 8 or 16.

For $r=8$ and 16, we can express $k$ according to (3) and $t$ using (1) or (2). If $t$ also turns out to be positive, we can calculate the desired price for 1 rak, 2 skebles, and 3 stikas:

| $r$ | $k$ | $t$ | $r+2 k+3 t$ |
| :---: | :---: | :---: | :---: |
| 8 | 4 | 21 | 79 |
| 16 | 1 | 14 | 60 |

Kebule could pay either 79 or 60 scales for 1 rak, 2 skebles, and 3 stikas.

Note. The divisibility of the number $r$ by eight means that $r=8 r^{\prime}$, where $r^{\prime}$ is some integer. From (3), we get $k=7-3 r^{\prime}$ and using (1) or (2), we get $t=28-7 r^{\prime}$. The desired sum is then equal to $r+2 k+3 t=98-19 r^{\prime}$ and the columns in the previous table correspond to

| $8 r^{\prime}$ | $7-3 r^{\prime}$ | $28-7 r^{\prime}$ | $98-19 r^{\prime}$ |
| :--- | :--- | :--- | :--- |

All numbers are positive precisely when $r^{\prime}$ is either 1 or 2, which agrees with the above constraints.