Coins in a money box

Karol has in his money box in total 19 coins, and only ten-crown and fifty-crown coins. In total he has in the money box 830 Kč saved.
Which statement is false? (Provide a solution procedure for the individual statements.)
a) In the money box 170 Kč (Czech crown) are missing to a thousand.
b) In the money box there are fewer ten-crowns than fifty-crowns.
c) In the money box there are 13 more fifty-crowns than ten-crowns.
d) In the money box there is an equal number of ten-crowns and fifty-crowns.
e) In the money box the ten-crowns and fifty-crowns are in the ratio 3 : 16 (in this order).

Final Answer:

x = d

Step-by-step explanation:

$$\begin{gathered} a+b=19 \\ 10a+50b=830 \\ 10\cdot a+50\cdot b=830 \\ a+b=19 \\ Pivot:Row1\leftrightarrow Row2 \\ Row2-\frac{1}{10}\cdot Row1\to Row2 \\ -4b=-64 \\ b=\frac{-64}{-4}=16 \\ a=\frac{830-50b}{10}=\frac{830-50\cdot 16}{10}=3 \\ a=3 \\ b=16 \\ {x}_1=1000-170=830\text{Kc} \\ a<b \\ x=d \end{gathered}$$

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