Question

On a 120 km track, a train travels the first 30 km a uniform
speed of 30km/hr. How fast must the train travel the next
90km so as to average 60km/hr for the entire trip?

Answer 1

Answer:

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• Total distance (D) = 120 km
• Distance travelled by the train in first half (D₁) = 30 km
• Distance travelled by the train in second half (D₂) = 90 km
• Speed of the train in first half (S₁) = 30 km/h
• Speed of the train in second half (S₂) = ?
• Average Speed = 60 km/h

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$\longrightarrow\:\:\tt Average \: Speed = \dfrac{Total \: distance}{Total \: time}$

$\longrightarrow\:\:\tt Average \: Speed = \dfrac{Total \: distance}{\dfrac{Distance}{Speed}}$

$\longrightarrow\:\:\tt Average \: Speed = \dfrac{D}{\dfrac{D_1}{S_1} + \dfrac{D_2}{S_2} }$

$\longrightarrow\:\:\tt 60 = \dfrac{120}{\dfrac{30}{30} + \dfrac{90}{S_2} }$

$\longrightarrow\:\:\tt 60 = \dfrac{120}{1 + \dfrac{90}{S_2} }$

$\longrightarrow\:\:\tt 60 = \dfrac{120}{\dfrac{S_2 + 90}{S_2} }$

$\longrightarrow\:\:\tt 60 =120 \times\dfrac{S_2 }{S_2 + 90}$

$\longrightarrow\:\:\tt \dfrac{60}{120} =\dfrac{S_2 }{S_2 + 90}$

$\longrightarrow\:\:\tt \dfrac{1}{2} =\dfrac{S_2 }{S_2 + 90}$

$\longrightarrow\:\:\tt 1( S_2 + 90)=2(S_2)$

$\longrightarrow\:\:\tt S_2 + 90=2S_2$

$\longrightarrow\:\:\tt 2S_2 - S_2 = 90$

$\longrightarrow\:\: \underline{ \underline{\tt S_2 = 90 \: {km}^{ - 1}}}$

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