Task B-1.7. Ivan and Nenad collect 2-kuna and 5-kuna coins in their respective piggy banks. After a year, they opened their piggy banks and counted the coins. Together, they collected 280 2-kuna coins. In Ivan's piggy bank, $60\%$ of all the coins are 2-kuna coins, and in Nenad's piggy bank, there are twice as many 2-kuna coins as 5-kuna coins. The total value of the money from both piggy banks together is 1360 kuna. What is the value of the money from Ivan's piggy bank, and what is the value from Nenad's?
## Solution.

Let's introduce the following notations:

$x_{1}$ - the number of 2 kn coins in Ivan's piggy bank
$y_{1}$ - the number of $5 \mathrm{kn}$ coins in Ivan's piggy bank

$x_{2}$ - the number of $2 \mathrm{kn}$ coins in Nenad's piggy bank

$y_{2}$ - the number of $5 \mathrm{kn}$ coins in Nenad's piggy bank

Then we have

$$
\begin{aligned}
& x_{1}+x_{2}=280 \\
& x_{1}=0.6\left(x_{1}+y_{1}\right) \\
& x_{2}=2 y_{2} \\
& 2\left(x_{1}+x_{2}\right)+5\left(y_{1}+y_{2}\right)=1360
\end{aligned}
$$

From the first two equations, we get $x_{2}=280-x_{1}, y_{1}=\frac{2}{3} x_{1}$,

so $y_{2}=\frac{1}{2} x_{2}=140-\frac{1}{2} x_{1}$,

which, when substituted into the last equation, gives

$$
2 \cdot 280+5\left(\frac{2}{3} x_{1}+140-\frac{1}{2} x_{1}\right)=360
$$

This simplifies to

$$
\begin{aligned}
& 5\left(\frac{2}{3} x_{1}+140-\frac{1}{2} x_{1}\right)=800 \\
& \frac{2}{3} x_{1}+140-\frac{1}{2} x_{1}=160 \\
& x_{1}=120
\end{aligned}
$$

Then we have

$$
\begin{aligned}
& x_{2}=280-x_{1}=160 \\
& y_{1}=\frac{2}{3} x_{1}=80 \\
& y_{2}=\frac{1}{2} x_{2}=80
\end{aligned}
$$

The value of Ivan's money is $120 \cdot 2+80 \cdot 5=640 \mathrm{kn}$, and the value of Nenad's money is $160 \cdot 2+80 \cdot 5=720 \mathrm{kn}$.

## COUNTY COMPETITION IN MATHEMATICS

## 2nd grade - high school - B variant

## February 18, 2014.

IF A STUDENT HAS A DIFFERENT APPROACH TO SOLVING THE PROBLEM, THE COMMITTEE IS OBLIGED TO GRADE AND EVALUATE THAT APPROACH APPROPRIATELY.