13.324. In the first week of his vacation, the engineer spent a few rubles less than $3 / 5$ of the amount of money he had taken with him; in the second week, $1 / 4$ of the remainder plus 30 rubles; in the third week, $2 / 5$ of the new remainder plus 12 rubles; after which $6 / 35$ of the amount of money taken remained. It is also known that the amount of money left unspent at the end of the first, second, and third weeks decreased in an arithmetic progression. How much money was spent over the three weeks of the vacation?
Solution.

Let $S$ be the total amount of money taken. We fill in the table:

| Period | Spent, rubles | Balance, rubles |
| :---: | :---: | :---: |
| first week | $x$ | $S-x$ |
| second week | $\frac{S-x}{4}+30$ | $\frac{3(S-x)}{4}-30$ |
| third week | $\frac{2}{5}\left(\frac{3(S-x)}{4}-30\right)+12$ | $\frac{6}{35} S$ |

By the condition $\left\{\begin{array}{l}x<\frac{3}{5} S, \\ \frac{S-x+\frac{6}{35} \cdot S}{2}=\frac{3(S-x)}{4}-30, \\ \frac{3(S-x)}{4}-30-\frac{2}{5}\left(\frac{3(S-x)}{4}-30\right)-12=\frac{6}{35} S,\end{array}\right.$

from which $S=1400$ rubles.

Then spent $S-\frac{6}{35} S=1160$ (rubles).

Answer: 1160 rubles.