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The car traveled a distance of 120 km in 1 hour 50 minutes. The first part of this track traveled at a speed of 50 km per hour, and the second part traveled at a speed of 75 km per hour. The 10 minutes the driver needed to fill the gas are included in the given time. How big was each track?

Correct answer:

a =  10 km
b =  110 km

Step-by-step explanation:

$$\begin{gathered} s=120\text{km} \\ {t}_1=1:50=1hr50min=1+\frac{50}{60}=1.8333hr\doteq 1.8333 \\ {v}_1=50\text{km/h} \\ {v}_2=75\text{km/h} \\ {t}_2=0:10=0hr10min=0+\frac{10}{60}=0.1667hr\doteq 0.1667 \\ t=t_1-t_2=1.8333-0.1667=\frac{5}{3}\doteq 1.6667\text{h} \\ s=a+b \\ a=v_1\cdot x \\ b=v_2\cdot (t-x) \\ a=v_1\cdot x \\ s-a=v_2\cdot (t-x) \\ s-v_1\cdot x=v_2\cdot (t-x) \\ 120-50\cdot x=75\cdot (1.6666666666666-x) \\ 25x=5 \\ x=\frac{5}{25}=0.2 \\ x=\frac{1}{5}\doteq 0.2 \\ a=v_1\cdot x=50\cdot 0.2=10\text{km} \end{gathered}$$

$b=s-a=120-10=110\text{km}$

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