Speedboat and train

At 12:15, a train speeding 55 km/h left the station. Twelve minutes later, an 80 km/h speedboat set off behind the train. What time does the express train catch up? How far from the station?

Result

h = 12:53 hh:mm
s =  35.2 km

Step-by-step explanation:

$$\begin{gathered} v_1=55\text{km/h} \\ {v}_2=80\text{km/h} \\ {t}_1=12:15=12hr15min=12+\frac{15}{60}=12.25hr=12.25 \\ \Delta t=12\text{min}\to \text{h}=12:60\text{h}=0.2\text{h} \\ s=v_1\cdot (t-t_1) \\ s=v_2\cdot (t-(t_1+\Delta t)) \\ {v}_1\cdot (t-t_1)=v_2\cdot (t-(t_1+\Delta t)) \\ 55\cdot (t-12.25)=80\cdot (t-(12.25+0.2)) \\ 25t=322.25 \\ t=\frac{322.25}{25}=12.89 \\ t=\frac{1289}{100}=12.89 \\ h=t=12.89=12:53\text{hh}:\text{mm} \end{gathered}$$

$s=v_1\cdot (t-t_1)=55\cdot (12.89-12.25)=35.2\text{km}$

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