Part A:
Let [tex]t[/tex] be the time (in hours) that has passed since Alan/Soren has started to head for their apartment.
Alan works 6 miles away from the apartment and moves at a rate of 7 miles per hour towards his apartment
The distance [tex]d[/tex] remaining to Alan's apartment after [tex]t[/tex] hours is:
[tex]d=6-7t[/tex]
Soren works 8 miles away and moves at a rate of 12 miles per hour. The distance [tex]d[/tex] remaining to Soren's apartment after [tex]t[/tex] hours is:
[tex]d=8-12t[/tex]
Set the equations equal to each other to find how long it takes for them to be at the same distance from their apartment:
[tex]6-7t=8-12t[/tex]
Add both sides by 12t
[tex]6+5t=8[/tex]
Subtract both sides by 6
[tex]5t=2[/tex]
Divide both sides by 5
[tex]t=2/5=0.4[/tex]
They will be at the same distance from their apartment after 0.4 hours, or 2/5 hours.
Part B:
Set d equal to 0 for each equation
Alan's equation:
[tex]0=6-7t[/tex]
Add both sides by 7t
[tex]7t=6[/tex]
Divide both sides by 7
[tex]t=6/7[/tex]
Soren's equation:
[tex]0=8-12t[/tex]
Add both sides by 12t
[tex]12t=8[/tex]
Divide both sides by 12 and simplify
[tex]t=2/3[/tex]
Since [tex]6/7>2/3[/tex], it takes less time for Soren to get to his apartment, which means Soren gets back first.
The difference in the amount of time (how much less time it took) between Soren and Alan is:
[tex]6/7-2/3=4/21[/tex]
It took Soren 4/21 hours less than Alan to get back to his apartment.
Let me know if you need any clarifications, thanks!
~ Padoru
