Two cities

The car goes from city A to city B at an average speed of 70 km/h and back at an average speed of 50 km/h. If it goes to B and back at an average speed of 60 km/h, the whole ride will take 8 minutes less. What is the distance between cities A and B?

Correct answer:

s =  140 km

Step-by-step explanation:

$$\begin{gathered} v_1=70\text{km/h} \\ {v}_2=50\text{km/h} \\ {v}_3=60\text{km/h} \\ t=8\text{min}\to \text{h}=8:60\text{h}=0.13333\text{h} \\ s=v_1\cdot a \\ s=v_2\cdot b \\ 2\cdot s=v_3\cdot (a+b-t) \\ 2\cdot s=v_3\cdot a+v_3\cdot b-v_3\cdot t \\ 2\cdot s=v_3\cdot (s/v_1)+v_3\cdot (s/v_2)-v_3\cdot t \\ 2\cdot s=60\cdot (s/70)+60\cdot (s/50)-60\cdot 0.13333333333333 \\ 20s=2800 \\ s=\frac{2800}{20}=140=140\text{km} \\ s=140 \\ \text{ Verifying Solution: } \\ a=s/v_1=140/70=2\text{h} \\ b=s/v_2=140/50=\frac{14}{5}=2.8\text{h} \\ c=a+b-t=2+2.8-0.1333=\frac{14}{3}\doteq 4.6667\text{h} \end{gathered}$$

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