Саша + Серёжа = 10 дней
Серёжа + Алёша = 15 дней
Саша + Алёша = 18 дней
Обозначим, х - Саша, у - Серёжа, z - Алёша
- производительность Саши;
- производительность Серёжи;
- производительность Алёши;
1 - вся работа
исходя из данных условия задачи составим систему уравнений и решим её
![\begin{cases}(x+y)\cdot10=1\\(y+z)\cdot15=1\\(x+z)\cdot18=1\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D%28x%2By%29%5Ccdot10%3D1%5C%5C%28y%2Bz%29%5Ccdot15%3D1%5C%5C%28x%2Bz%29%5Ccdot18%3D1%5Cend%7Bcases%7D)
![\begin{cases}10\cdot(x+y)=1\\15\cdot(y+z)=1\\18\cdot(x+z)=1\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D10%5Ccdot%28x%2By%29%3D1%5C%5C15%5Ccdot%28y%2Bz%29%3D1%5C%5C18%5Ccdot%28x%2Bz%29%3D1%5Cend%7Bcases%7D)
![\begin{cases}10x+10y=1\\15y+15z=1\\18x+18z=1\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D10x%2B10y%3D1%5C%5C15y%2B15z%3D1%5C%5C18x%2B18z%3D1%5Cend%7Bcases%7D)
![\left \{ {{10x=1-10y}\atop{15z=1-15y}}\right](https://tex.z-dn.net/?f=%5Cleft+%5C%7B+%7B%7B10x%3D1-10y%7D%5Catop%7B15z%3D1-15y%7D%7D%5Cright)
![\left \{ {{x=\frac{1-10y}{10}}\atop{z=\frac{1-15y}{15}}}\right](https://tex.z-dn.net/?f=%5Cleft+%5C%7B+%7B%7Bx%3D%5Cfrac%7B1-10y%7D%7B10%7D%7D%5Catop%7Bz%3D%5Cfrac%7B1-15y%7D%7B15%7D%7D%7D%5Cright)
подставляем в (3) уравнение
![18\cdot(\frac{1-10y}{10})+18\cdot(\frac{1-15y}{15})=1](https://tex.z-dn.net/?f=18%5Ccdot%28%5Cfrac%7B1-10y%7D%7B10%7D%29%2B18%5Ccdot%28%5Cfrac%7B1-15y%7D%7B15%7D%29%3D1)
![\frac{18}{10}\cdot({1-10y)+\frac{18}{15}\cdot(1-15y)=1](https://tex.z-dn.net/?f=%5Cfrac%7B18%7D%7B10%7D%5Ccdot%28%7B1-10y%29%2B%5Cfrac%7B18%7D%7B15%7D%5Ccdot%281-15y%29%3D1)
![1,8\cdot(1-10y)+1,2\cdot(1-15y)=1](https://tex.z-dn.net/?f=1%2C8%5Ccdot%281-10y%29%2B1%2C2%5Ccdot%281-15y%29%3D1)
![1,8-18y+1,2-18y=1](https://tex.z-dn.net/?f=1%2C8-18y%2B1%2C2-18y%3D1)
![3-36y=1](https://tex.z-dn.net/?f=3-36y%3D1)
![36y=3-1](https://tex.z-dn.net/?f=36y%3D3-1)
![36y=2](https://tex.z-dn.net/?f=36y%3D2)
![y=2:36](https://tex.z-dn.net/?f=y%3D2%3A36)
![y=\frac{2}{36}=\frac{1}{18}](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B2%7D%7B36%7D%3D%5Cfrac%7B1%7D%7B18%7D)
![x=\frac{1-10y}{10}=\frac{1}{10}(1-10y)=\frac{1}{10}-y=\frac{1}{10}-\frac{1}{18}=\frac{9-5}{90}=\frac{4}{90}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B1-10y%7D%7B10%7D%3D%5Cfrac%7B1%7D%7B10%7D%281-10y%29%3D%5Cfrac%7B1%7D%7B10%7D-y%3D%5Cfrac%7B1%7D%7B10%7D-%5Cfrac%7B1%7D%7B18%7D%3D%5Cfrac%7B9-5%7D%7B90%7D%3D%5Cfrac%7B4%7D%7B90%7D)
![z=\frac{1-15y}{15}=\frac{1}{15}\cdot(1-15y)=\frac{1}{15}-y=\frac{1}{15}-\frac{1}{18}=\frac{6-5}{90}=\frac{1}{90}](https://tex.z-dn.net/?f=z%3D%5Cfrac%7B1-15y%7D%7B15%7D%3D%5Cfrac%7B1%7D%7B15%7D%5Ccdot%281-15y%29%3D%5Cfrac%7B1%7D%7B15%7D-y%3D%5Cfrac%7B1%7D%7B15%7D-%5Cfrac%7B1%7D%7B18%7D%3D%5Cfrac%7B6-5%7D%7B90%7D%3D%5Cfrac%7B1%7D%7B90%7D)
- нарубит дрова Саша.
- нарубит дрова Серёжа.
- нарубит дрова Алёша.
Ответ: Саша -
дня, Серёжа - 18 дней, Алёша - 90 дней.