Четверо школьников вместе купили ноутбук. Первый внес половину суммы, вносимой остальными, второй-треть суммы, вносимой остальными, третий внес четверть суммы, вносимой остальными, а четвертый внес 5200 рублей. сколько рублей стоил ноутбук?
Ответы
Ответ дал:
8
Первый внёс x руб, второй y руб, третий z руб. Составим и решит систему:
![\begin{cases}x=\frac12(y+z+5200)\\y=\frac13(x+z+5200)\\z=\frac14(x+y+5200)\end{cases}\Rightarrow\begin{cases}x=\frac12(y+z+5200)\\y=\frac13(\frac12(y+z+5200)+z+5200)\\z=\frac14(\frac12(y+z+5200)+y+5200)\end{cases}\\y=\frac13(\frac12(y+z+5200)+z+5200)\\y=\frac13\left(\frac12y+\frac12z+2600+z+5200\right)\\y=\frac16y+\frac12z+2600\\\frac56y=\frac12z+2600\\y=\frac35z+3120 \begin{cases}x=\frac12(y+z+5200)\\y=\frac13(x+z+5200)\\z=\frac14(x+y+5200)\end{cases}\Rightarrow\begin{cases}x=\frac12(y+z+5200)\\y=\frac13(\frac12(y+z+5200)+z+5200)\\z=\frac14(\frac12(y+z+5200)+y+5200)\end{cases}\\y=\frac13(\frac12(y+z+5200)+z+5200)\\y=\frac13\left(\frac12y+\frac12z+2600+z+5200\right)\\y=\frac16y+\frac12z+2600\\\frac56y=\frac12z+2600\\y=\frac35z+3120](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7Dx%3D%5Cfrac12%28y%2Bz%2B5200%29%5C%5Cy%3D%5Cfrac13%28x%2Bz%2B5200%29%5C%5Cz%3D%5Cfrac14%28x%2By%2B5200%29%5Cend%7Bcases%7D%5CRightarrow%5Cbegin%7Bcases%7Dx%3D%5Cfrac12%28y%2Bz%2B5200%29%5C%5Cy%3D%5Cfrac13%28%5Cfrac12%28y%2Bz%2B5200%29%2Bz%2B5200%29%5C%5Cz%3D%5Cfrac14%28%5Cfrac12%28y%2Bz%2B5200%29%2By%2B5200%29%5Cend%7Bcases%7D%5C%5Cy%3D%5Cfrac13%28%5Cfrac12%28y%2Bz%2B5200%29%2Bz%2B5200%29%5C%5Cy%3D%5Cfrac13%5Cleft%28%5Cfrac12y%2B%5Cfrac12z%2B2600%2Bz%2B5200%5Cright%29%5C%5Cy%3D%5Cfrac16y%2B%5Cfrac12z%2B2600%5C%5C%5Cfrac56y%3D%5Cfrac12z%2B2600%5C%5Cy%3D%5Cfrac35z%2B3120)
![x=\frac12\left(\frac35z+3120+z+5200\right)\\x=\frac12\left(\frac85z+8320\right)\\x=\frac45z+4160\\\\z=\frac14\left(\frac45z+4160+\frac35z+3120+5200\right)\\z=\frac14\left(\frac75z+12480\right)\\z=\frac7{20}z+3120\\\frac{13}{20}z=3120\\z=4800 x=\frac12\left(\frac35z+3120+z+5200\right)\\x=\frac12\left(\frac85z+8320\right)\\x=\frac45z+4160\\\\z=\frac14\left(\frac45z+4160+\frac35z+3120+5200\right)\\z=\frac14\left(\frac75z+12480\right)\\z=\frac7{20}z+3120\\\frac{13}{20}z=3120\\z=4800](https://tex.z-dn.net/?f=x%3D%5Cfrac12%5Cleft%28%5Cfrac35z%2B3120%2Bz%2B5200%5Cright%29%5C%5Cx%3D%5Cfrac12%5Cleft%28%5Cfrac85z%2B8320%5Cright%29%5C%5Cx%3D%5Cfrac45z%2B4160%5C%5C%5C%5Cz%3D%5Cfrac14%5Cleft%28%5Cfrac45z%2B4160%2B%5Cfrac35z%2B3120%2B5200%5Cright%29%5C%5Cz%3D%5Cfrac14%5Cleft%28%5Cfrac75z%2B12480%5Cright%29%5C%5Cz%3D%5Cfrac7%7B20%7Dz%2B3120%5C%5C%5Cfrac%7B13%7D%7B20%7Dz%3D3120%5C%5Cz%3D4800)
![\begin{cases}x=8000\\y=6000\\z=4800\end{cases} \begin{cases}x=8000\\y=6000\\z=4800\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7Dx%3D8000%5C%5Cy%3D6000%5C%5Cz%3D4800%5Cend%7Bcases%7D)
Ноутбук стоил 8000+6000+4800+5200 = 24000 руб.
Ноутбук стоил 8000+6000+4800+5200 = 24000 руб.
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