Three musicians, Janek, Mikeš, and Vávra, usually divide their shared honorarium in the ratio $4: 5: 6$, with Janek receiving the least and Vávra the most. This time, Vávra did not play well and thus waived his share. Janek suggested that they split Vávra's share equally between himself and Mikeš. However, Mikeš insisted that they should divide this share unevenly as usual, in the ratio $4: 5$. Mikeš would have received 40 Kč less according to Janek's proposal than according to his own.

Determine the total amount of the shared honorarium.

(L. Šimünek)
Given the sequential ratio $4: 5: 6$, we will expand it so that we can appropriately divide its third term according to both Janek's proposal (division in the ratio $1: 1$) and Mikš's proposal (division in the ratio $4: 5$). Therefore, we will expand the ratio so that its third term is divisible by both two and nine, expanding it by three:

$$
12: 15: 18
$$

The amount originally intended for Vávra consists of 18 equal parts, of which Janek proposes to allocate 9 parts to himself and 9 to Mikš, while Mikš proposes to give 8 parts to Janek and 10 to himself. According to Janek's proposal, Mikš would receive 1 part less than according to his own proposal. This part corresponds to 40 Kč. The total honorarium for the musicians was 45 such parts $(12 + 15 + 18 = 45)$, so in total they received

$$
45 \cdot 40 = 1800 \text{ (Kč). }
$$

Grading proposal. 2 points for the result; 4 points for the solution process.