## Task 6 - 210736

A liquid is sold in small, medium, and large bottles. Each small bottle holds exactly $200 \mathrm{~g}$, each medium bottle exactly $500 \mathrm{~g}$, and each large bottle exactly $1000 \mathrm{~g}$ of the liquid. Each filled $200 \mathrm{~g}$ bottle costs $1.20 \mathrm{M}$, and each filled $500 \mathrm{~g}$ bottle costs $2.80 \mathrm{M}$. The price of an empty $500 \mathrm{~g}$ bottle is $50 \%$ higher than that of an empty $200 \mathrm{~g}$ bottle. The empty $1000 \mathrm{~g}$ bottle, in turn, is $50 \%$ more expensive than the empty $500 \mathrm{~g}$ bottle.

What amount is saved if, instead of five filled $200 \mathrm{~g}$ bottles, one filled $1000 \mathrm{~g}$ bottle is purchased?
The empty 200 g bottle costs $x$ marks. From this, it follows: The empty 500 g bottle costs 50% more, so $\frac{3}{2} \cdot x$ marks.

Furthermore, it follows:

200 g of the liquid cost $(1.20 - x)$ marks,

500 g of the liquid cost $\left(2.80 - \frac{3}{2} \cdot x\right)$ marks.

Since the price for 500 g must be $\frac{5}{2}$ of the price for 200 g, it follows that

$$
2.80 - \frac{3}{2} \cdot x = \frac{5}{2}(1.20 - x) = 3 - \frac{5}{2} x
$$

from which $x = 0.20$ is obtained.

Thus, the empty 200 g bottle costs $0.20$ marks, the empty 500 g bottle therefore costs $0.30$ marks, and finally, the empty 1000 g bottle costs $\frac{3}{2} \cdot 0.30$ marks = $0.45$ marks.

Consequently, five empty 200 g bottles cost $5 \cdot 0.20$ marks = $1$ mark. If one buys the 1000 g liquid not in these five bottles but instead in a 1000 g bottle, one saves $(1 - 0.45)$ marks = $0.55$ marks.

Solutions of the third round 1981 taken from [5]

### 4.24 22nd Olympiad 1982

### 4.24.1 First Round 1982, Class 7