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Alan works 6 miles away from the apartment and rollerblades from his work place to t he apartment and a constant rate of 7 miles per hour, Soren works 8 miles away from the apartment and bikes from his workplace to the apartment at a constant rate of 12 miles per hour How much time, in hours do Alan and Soren have to travel to be the same distance from their apartment

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Answer:

[tex]\large \boxed{\text{24 min}}[/tex]

Step-by-step explanation:

        Let d = distance from apartment

Then 6 - d = distance travelled by Alan

 and 8 - d = distance travelled by Soren

They will each be on the road for the same time.

Distance = speed × time

[tex]\text{Time} = \dfrac{\text{distance}}{\text{speed}}[/tex]

[tex]\begin{array}{rcl}\dfrac{6 - d}{7} & = & \dfrac{8 - d}{12}\\72 - 12d & = & 56 - 7d\\16 & = & 5d\\d & = & \dfrac{16}{5}\\\\ & = & 3.2\\ \end{array}\\[/tex]

Calculate the travel times

Alan's distance = (6 - 3.2) mi = 2.8 mi

[tex]\text{Alan's time} = \dfrac{2.8}{7} = \text{0.4 h} = \textbf{24 min}[/tex]

Soren's distance = (8 - 3.2) mi = 4.8 mi

[tex]\text{Soren's time} = \dfrac{4.8}{12} = \text{0.4 h} = \textbf{24 min}\\\\\text{Alan and Soren must travel $\large \boxed{\textbf{24 min}}$ to be the same distance from their apartment.}[/tex]

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