## Task 2 - 340722

a) A prize of 1485 DM is to be divided among three participants in the ratio $2: 3: 4$.

How much does each receive?

b) In another game, one participant receives one-fifth of the prize money, which is 150 DM. The remainder is to be divided between the other two participants in the ratio $5: 7$.

How much does each of them receive?

c) In a third game, it was agreed to distribute the prize in proportion to the stakes with which the participants had joined the competition. The total of these stakes of the three participants Anke, Bertram, and Claus was 44 DM, and furthermore: If Anke had bet 6 DM more and Claus had bet twice his stake, all three would have achieved the same winning claim.

How large were the three stakes?
a) The participants are to receive 2 parts, 3 parts, and 4 parts of the prize, respectively, with all these parts being of equal size and together making up the entire prize. Since there should be $2+3+4=9$ parts, each part must be 1485 DM : $9=165$ DM. Therefore, the participants receive the following amounts in sequence:

$$
2 \cdot 165 \mathrm{DM}=330 \mathrm{DM}, \quad 3 \cdot 165 \mathrm{DM}=495 \mathrm{DM}, \quad 4 \cdot 165 \mathrm{DM}=660 \mathrm{DM}
$$

b) Since one fifth of the prize is 150 DM, the total prize is $5 \cdot 150 \mathrm{DM}=750 \mathrm{DM}$.

The two other participants mentioned in the problem text together receive 750 DM - 150 DM $=600$ DM. For the distribution of this amount in the ratio 5:7, it follows similarly to part a) that $5+7=12$ and $600: 12=50$: The two other participants receive $5 \cdot 50 \mathrm{DM}=250 \mathrm{DM}$ and $7 \cdot 50 \mathrm{DM}=350 \mathrm{DM}$, respectively.

c) If $a, b, c$ are the stakes in DM placed by Anke, Bertram, and Claus, respectively, then $a+b+c=44$ and $a+6=2 c=b$.

Substituting $a=2 c-6$ and $b=2 c$ into the first equation, we get $2 c-6+2 c+c=44$, which means $c=10$ and thus $b=20, a=14$.

Therefore, the following amounts were placed: Anke: 14 DM, Bertram: 20 DM, Claus: 10 DM.