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Each morning Boris walks to school. At onefourth of the way, he passes the machine and tractor station; at onethird of the way, the railroad station. At the machine and tractor station its clock shows 07:30, and at the railroad station its clock shows 07:35. When does Boris leave his house, when does he reach school?
Between 07:30 and 07:35, \al{ \Delta t &= \tau_2  \tau_1\\ &= \SI{7}{h}\,\SI{35}{min}  \SI{7}{h}\,\SI{30}{min}\\ &= \SI{5}{min} = \SI{300}{s} } are passing.\\ This corresponds to \al{ \Delta\eta &= \eta_2  \eta_1\\ &= \frac{1}{3}  \frac{1}{4}\\ &= \frac{1}{12} } of the time for the whole way, i.e. the whole way takes \al{ t &= \frac{\Delta t}{\Delta \eta} = \frac{\tau_2\tau_1}{\eta_2\eta_1}\\ &= \frac{\SI{300}{s}}{\frac{1}{12}}\\ &= \SI{3600}{s} = \SI{1.0}{h}. } One third of this time is \al{ t_1 &= \eta_1\cdot t = \frac{\eta_1}{\eta_2\eta_1}\cdot \qty(\tau_2\tau_1)\\ &= \frac{1}{3}\cdot\SI{3600}{s}\\ &= \SI{1200}{s} = \SI{20}{min}. } Thus he leaves the house at \al{ \ssc{\tau}{leave} &= \tau_2  t_1 = \tau_2  \frac{\eta_2}{\eta_2\eta_1}\cdot \qty(\tau_2\tau_1)\\ &= \SI{7}{h}\,\SI{35}{min}  \SI{20}{min}\\ &= \SI{7}{h}\,\SI{15}{min} \rightarrow \text{07:15}. } From the gas station to school it takes him \al{ t_2 &= (1\eta_2) \cdot t = \frac{1\eta_2}{\eta_2\eta_1}\cdot \qty(\tau_2\tau_1)\\ &= \qty(1\frac{1}{3})\cdot \SI{3600}{s}\\ &= \SI{2400}{s} = \SI{40}{min} } and he therefore reaches school at \al{ \ssc{\tau}{reach} &= \tau_2 + t_2 = \tau_2 + \frac{1\eta_2}{\eta_2\eta_1}\cdot \qty(\tau_2\tau_1)\\ &= \SI{7}{h}\,\SI{35}{min} + \SI{40}{min} \\ &= \SI{8}{h}\,\SI{15}{min} \rightarrow \text{08:15}. } \emph{Less formal solution:} From onefourth to onethird, its onetwelfth: \begin{align} \frac13\frac14=\frac{1}{12} \end{align} For this part of the way, the boy Boris needs 5 minutes; his whole route to school is 60 minutes long. He starts at 07:15 at home, goes $\frac{3}{12}=\frac{1}{4}$ of the way, is at 07:30 at the machine and tractor station, 07:35 at the railroad station, where $\frac{2}{3}=\frac{8}{12}$ of the way are still remaining, which takes him another 40 minutes; he arreives at 08:15.
12:11, 7. Aug. 2020  diff>4, formalisiert  Patrik Weber (patrik)  Current Version 
13:31, 18. May 2017  lsg  Urs Zellweger (urs)  Compare with Current 
12:51, 18. May 2017  Initial Version.  Urs Zellweger (urs)  Compare with Current 