# Between two bus stops

Wanda lives between two bus stops at three-eighths of their distance. He started the house today and found that whether he was running to one or the other stop, he would have arrived at the bus stop. The average bus speed is 60 km/h.

What is the average speed of Wanda running today?

Result

v =  15 km/h

#### Solution:

$s_{ 1 } = \dfrac{ 3 }{ 8 } s \ \\ s_{ 2 } = s-s_{ 1 } = \dfrac{ 5 }{ 8 } s \ \\ \ \\ v_a = 60 \ km/h \ \\ \ \\ s_{ 1 } = v \cdot \ t_{ 1 } \ \\ s_{ 2 } = v \cdot \ t_{ 2 } \ \\ \ \\ s_{ 3 } = v_a \cdot \ t_{ 1 } \ \\ s_{ 3 }+s = v_a \cdot \ t_{ 2 } \ \\ v_a \cdot \ t_{ 1 } + s = v_a \cdot \ t_{ 2 } \ \\ \ \\ v_a \cdot \ s_{ 1 }/v + s = v_a \cdot \ s_{ 2 }/v \ \\ v_a \cdot \ \dfrac{ 3 }{ 8 } \cdot \ s/v + s = v_a \cdot \ \dfrac{ 5 }{ 8 } \cdot \ s/v \ \\ v_a \cdot \ \dfrac{ 3 }{ 8 } /v + 1 = v_a \cdot \ \dfrac{ 5 }{ 8 } /v \ \\ v_a \cdot \ \dfrac{ 3 }{ 8 } + v = v_a \cdot \ \dfrac{ 5 }{ 8 } \ \\ \ \\ v = v_a \cdot \ \dfrac{ 5 }{ 8 } - v_a \cdot \ \dfrac{ 3 }{ 8 } \ \\ \ \\ v = v_a \cdot \ ( \dfrac{ 5 }{ 8 } - \dfrac{ 3 }{ 8 } ) = 60 \cdot \ ( \dfrac{ 5 }{ 8 } - \dfrac{ 3 }{ 8 } ) = 15 = 15 \ \text { km/h }$

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Matematik
I bow to the author of the solution, it is brilliant :)

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