Linear Algebra problem

Suppose that I own a car rental company with 3 offices. One office is in Denver, one is in Boulder, and one is in Pueblo (All in Colorado, USA). My office is successful enough that I rent out all of the cars in all of the offices every day. Also, assume that every car that is rented is also returned on the same day. Now, my customers' behavior is exactly the same from one day to the next. So, I know for a fact that 50% of the cars rented in the Denver site will be returned to the Denver site. 30% of the cars rented in Denver are returned to Boulder, and the last 20% of the cars rented in Denver are returned to Pueblo. 40% of the cars rented in Boulder are returned to Denver, and the other 60% are returned to Boulder. 40% of the cars rented in Pueblo are returned to Denver, 10% returned to Boulder, and 50% returned to back to Pueblo. If I have X cars to rent, how many cars should I originally have in each site so that at the end of the day, each site will have the same number of cars it started the day with? For those who'd like a hint, request it via PM.

Originally posted by rheymans
Suppose that I own a car rental company with 3 offices. One office is in Denver, one is in Boulder, and one is in Pueblo (All in Colorado, USA). My office is successful enough that I rent out all of the cars in all of the offices every day. Also, assume that every car that is rented is also returned on the same day. Now, my customers' behavior is exactl ...[text shortened]... e same number of cars it started the day with? For those who'd like a hint, request it via PM.

I used Excel to get an iterative solution, although the matrix solution would be straightforward as well.

I got 44.4% should be in Denver, 37.8% should be in Boulder, and 17.8% should be in Pueblo. If you have 90 cars, that translates to 40, 34 and 16 cars in each lot, respectively.

Originally posted by PBE6
I used Excel to get an iterative solution, although the matrix solution would be straightforward as well.

I got 44.4% should be in Denver, 37.8% should be in Boulder, and 17.8% should be in Pueblo. If you have 90 cars, that translates to 40, 34 and 16 cars in each lot, respectively.

Nicely done! With 45 cars, you would get whole number answers as well.

al you need is twentyfive vehicals at each site.

Originally posted by shaz24
al you need is twentyfive vehicals at each site.

Nope.