Four pupils

Four pupils divided euro 765 so that the second received 50% less than the first, the third 1/2 less than a fourth, and the fourth euro 50 less than the first. How much money had each of them?

Final Answer:

a =  280 euro
b =  140 euro
c =  115 euro
d =  230 euro

Step-by-step explanation:

$$\begin{gathered} a+b+c+d=765 \\ b=a-50/100\cdot a \\ c=d\cdot (1-1/2) \\ d=a-50 \\ a+b+c+d=765 \\ 50a-100b=0 \\ 2c-d=0 \\ a-d=50 \\ Pivot:Row1\leftrightarrow Row2 \\ Row2-\frac{1}{50}\cdot Row1\to Row2 \\ 3b+c+d=765 \\ Row4-\frac{1}{50}\cdot Row1\to Row4 \\ 2b-d=50 \\ Row4-\frac{2}{3}\cdot Row2\to Row4 \\ -0.6667c-1.6667d=-460 \\ Row4-\frac{-0.66666667}{2}\cdot Row3\to Row4 \\ -2d=-460 \\ d=\frac{-460}{-2}=230 \\ c=\frac{0+d}{2}=\frac{0+230}{2}=115 \\ b=\frac{765-c-d}{3}=\frac{765-115-230}{3}=140 \\ a=\frac{0+100b}{50}=\frac{0+100\cdot 140}{50}=280=280\text{euro} \\ a=280 \\ b=140 \\ c=115 \\ d=230 \end{gathered}$$

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