In the new

In a new television show, a competition was announced for the funniest logo of the show. A total prize fund of 13,300 CZK was set for the three winners: the amount will be divided so that the second prize is two-thirds of the first prize and the third prize is two-thirds of the second prize.

Determine the amount that the winner of:

a) first prize,
b) second prize,
c) third prize will receive.

Final Answer:

a =  6300
b =  4200
c =  2800

Step-by-step explanation:

$$\begin{gathered} a+b+c=13300 \\ b=2/3a \\ c=2/3b \\ a+2/3a+2/3b=13300 \\ a+2/3a+2/3\cdot 2/3a=13300 \\ r=1+\frac{2}{3}+\frac{4}{9}=\frac{19}{9}=2\frac{1}{9}\doteq 2.1111 \\ a\cdot r=13300 \\ a=13300/r=13300/\frac{19}{9}=13300:\frac{19}{9}=13300\cdot \frac{9}{19}=\frac{13300\cdot 9}{19}=\frac{119700}{19}=6300 \end{gathered}$$

$b=\frac{2}{3}\cdot a=\frac{2}{3}\cdot 6300=\frac{2\cdot 6300}{3}=\frac{12600}{3}=4200$

$$\begin{gathered} c=\frac{2}{3}\cdot b=\frac{2}{3}\cdot 4200=\frac{2\cdot 4200}{3}=\frac{8400}{3}=2800 \\ \text{ Verifying Solution: } \\ t=a+b+c=6300+4200+2800=13300\text{Kc} \end{gathered}$$

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