7. During the seasonal sale, Marko bought three items at a very favorable price: shoes, shorts, and a T-shirt. He spent half of the money he brought and an additional $10 \mathrm{kn}$ on the shoes. On the shorts, he spent half of the remaining money and an additional $20 \mathrm{kn}$. He then spent half of the remaining money and an additional $30 \mathrm{kn}$ on the T-shirt. This way, he spent all his money. How much money did Marko have at the beginning? How much did he pay for each item?
First method:

Let $x$ be the amount of money at the beginning. The entire amount is expressed in kuna.

If $a$ is half of the amount of $x$, then he spent $a+10$ on shoes, and $a-10$ remained. 1 POINT

If $b$ is half of the remaining amount of $a-10$, then he spent $b+20$ on socks, after which $b-20$ remained.

1 POINT

If $c$ is half of the remaining amount of $b-20$, then he spent $c+30$ on a T-shirt, and $c-30$ remained.

1 POINT

Since he had nothing left after adding 30 kn, we conclude that $c-30=0$, or $c=30 \mathrm{kn}$.

He paid 60 kn for the T-shirt.

1 POINT

![](https://cdn.mathpix.com/cropped/2024_05_30_cbd21e468efb4566cfb2g-7.jpg?height=429&width=1425&top_left_y=328&top_left_x=401)

Furthermore, it holds that:

$b-20=2c$

$b-20=60$

He paid 100 kn for the socks.

2 POINTS

$b=80 \mathrm{kn}$

$a-10=2b$

$a-10=160$

He paid 180 kn for the shoes.

2 POINTS

$a=170 \mathrm{kn}$

$x=2a$

$x=340 \mathrm{kn}$

At the beginning, he had 340 kn.

2 POINTS

TOTAL 10 POINTS

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