3. Frane has a total of $108 \mathrm{kn}$ in his piggy bank in $5 \mathrm{kn}$, $2 \mathrm{kn}$, and $1 \mathrm{kn}$ coins. The value of the $5 \mathrm{kn}$ and $2 \mathrm{kn}$ coins is the same. The number of $1 \mathrm{kn}$ coins is equal to the number of $5 \mathrm{kn}$ and $2 \mathrm{kn}$ coins combined. How many coins of each type are in the piggy bank?
First solution.

A $1 \mathrm{kn}$ coin has exactly as many as a $5 \mathrm{kn}$ coin and a $2 \mathrm{kn}$ coin together, so we can place a $1 \mathrm{kn}$ coin on each $5 \mathrm{kn}$ coin to form a "packet" of $6 \mathrm{kn}$, and place a $1 \mathrm{kn}$ coin on each $2 \mathrm{kn}$ coin to form a "packet" of $3 \mathrm{kn}$.

2 POINTS

![](https://cdn.mathpix.com/cropped/2024_05_30_ca549eb60a6f3c3b6299g-03.jpg?height=288&width=437&top_left_y=661&top_left_x=798)

The amounts in the $5 \mathrm{kn}$ and $2 \mathrm{kn}$ coins are equal, so for every two $5 \mathrm{kn}$ coins (i.e., packets of $6 \mathrm{kn}$), there are five $2 \mathrm{kn}$ coins (i.e., packets of $3 \mathrm{kn}$).

2 POINTS

![](https://cdn.mathpix.com/cropped/2024_05_30_ca549eb60a6f3c3b6299g-03.jpg?height=291&width=1379&top_left_y=1102&top_left_x=333)

Thus, all the coins can be grouped such that each group has two packets of $6 \mathrm{kn}$ and five packets of $3 \mathrm{kn}$, and it holds that $2 \cdot 6 + 5 \cdot 3 = 27 \mathrm{kn}$.

2 POINTS

The total amount is 108 kn, so we have $108 \div 27 = 4$ such groups.

1 POINT

Each group has 2 coins of 5 kn, so there are 8 such coins in total.

1 POINT

Each group has 5 coins of 2 kn, so there are 20 such coins in total.

1 POINT

Each group has 7 coins of 1 kn, so there are 28 such coins in total.

1 POINT

TOTAL 10 POINTS

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