Four pupils

Four pupils divided euro 634 so that the second received 30% less than the first, the third 1/5 less than a fourth, and the fourth euro 95 less than the first. How much money had each of them?

Correct answer:

a =  230 euro
b =  161 euro
c =  108 euro
d =  135 euro

Step-by-step explanation:

$$\begin{gathered} a+b+c+d=634 \\ b=a-30/100\cdot a \\ c=d\cdot (1-1/5) \\ d=a-95 \\ a+b+c+d=634 \\ 70a-100b=0 \\ 5c-4d=0 \\ a-d=95 \\ Pivot:Row1\leftrightarrow Row2 \\ 70a-100b=0 \\ a+b+c+d=634 \\ 5c-4d=0 \\ a-d=95 \\ Row2-\frac{1}{70}\cdot Row1\to Row2 \\ 70a-100b=0 \\ 2.4286b+c+d=634 \\ 5c-4d=0 \\ a-d=95 \\ Row4-\frac{1}{70}\cdot Row1\to Row4 \\ 70a-100b=0 \\ 2.4286b+c+d=634 \\ 5c-4d=0 \\ 1.4286b-d=95 \\ Row4-\frac{1.42857143}{2.42857143}\cdot Row2\to Row4 \\ 70a-100b=0 \\ 2.4286b+c+d=634 \\ 5c-4d=0 \\ -0.5882c-1.5882d=-277.9412 \\ Row4-\frac{-0.58823529}{5}\cdot Row3\to Row4 \\ 70a-100b=0 \\ 2.4286b+c+d=634 \\ 5c-4d=0 \\ -2.0588d=-277.9412 \\ d=\frac{-277.94117647}{-2.05882353}=135 \\ c=\frac{0+4d}{5}=\frac{0+4\cdot 135}{5}=108 \\ b=\frac{634-c-d}{2.42857143}=\frac{634-108-135}{2.42857143}=161 \\ a=\frac{0+100b}{70}=\frac{0+100\cdot 161}{70}=230=230\text{euro} \\ a=230 \\ b=161 \\ c=108 \\ d=135 \end{gathered}$$

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