Prizes in a competition

In a competition announced for a design of a poster, the prizes for the first three winners are prizes of a total value of 9 500 Kč. The second prize has a value of two thirds of the first prize, and the third prize has a value of two thirds of the second prize.
Which statement is false? (For each statement write the solution with procedure.)
a) The first prize has a value of 4 500 Kč (Czech crown).
b) The third prize has half the value of the first prize.
c) The second prize is 1 000 Kč more expensive than the third prize.
d) The second and third prizes have a total value greater than is the value of the first prize.
e) Twice the value of the first prize is the same as three times the value of the second prize.

Final Answer:

x = B

Step-by-step explanation:

$$\begin{gathered} a+b+c=9500 \\ b=\frac{2}{3}\cdot a \\ c=\frac{2}{3}\cdot b \\ b=2/3\cdot a \\ c=2/3\cdot b \\ a+b+c=9500 \\ 2a-3b=0 \\ 2b-3c=0 \\ Pivot:Row1\leftrightarrow Row2 \\ Row2-\frac{1}{2}\cdot Row1\to Row2 \\ 2.5b+c=9500 \\ Row3-\frac{2}{2.5}\cdot Row2\to Row3 \\ -3.8c=-7600 \\ c=\frac{-7600}{-3.8}=2000 \\ b=\frac{9500-c}{2.5}=\frac{9500-2000}{2.5}=3000 \\ a=\frac{0+3b}{2}=\frac{0+3\cdot 3000}{2}=4500 \\ a=4500 \\ b=3000 \\ c=2000 \\ x=B \end{gathered}$$

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