## 157. Math Puzzle $6 / 78$

At a class party, in addition to the milkshake drinks, various soda bottles were also emptied. There were more than 20 and fewer than 25, and specifically five times as many Astoria bottles as Cola, half as many Lemonade bottles as Astoria, and three fewer Selters bottles than Astoria. How many bottles were consumed at this class party?
At the class party, an additional 24 bottles of soda were emptied. Let the number of Astoria bottles be $a$, $c$ the number of Cola bottles, $l$ the number of Lemonade bottles, and $s$ the number of Seltzer bottles. Then it must hold that:

$$
20<a+c+l+s<25 \quad \text { and } \quad a=5 c, \quad a=2 l, \quad s=a-3
$$

The total number of bottles is then

$$
a+\frac{a}{5}+\frac{a}{2}+a-3=\frac{27}{10} a-3
$$

For the sum to be an integer, $a$ must be divisible by 10. The solutions are:

$$
a=10, c=2, l=5, s=7 \quad ; \quad a=20, c=4, l=10, s=17
$$

The second solution is invalid because the sum exceeds 25 from here on.

Only in the first case are the conditions met.