100 people (or some multiple of 100) want to travel from San Diego to LA. A trip by car takes C(x) = 100 + 0.6x minutes where x is the number of people who drive. The bus from San Diego to La takes C(x) = 120 + 0.2x minutes.
Suppose each person bases her decision to drive or take the bus to minimize her own commuting time. What will the total travel time be for all 100 people?
a) 13000 minutes
b) 12000 minutes
c) 11000 minutes
d) 10000 minutes
My try: if 100 people are driving on road, one car would take 160 minutes. then for 100 people, total time would be 16000 minutes. but the answer is 13000 minutes. somebody plz help.
I find this problem to be a little bit awkward and the only way I could find to reach the answer of 13.000 minutes is the following:
Suppose that $x$ people go by car and $y$ people go by bus. Individual travelling times are:
$$t_c=100+0.6x, \ t_b=120+0.2y$$
If you take into account that $x+y=100$:
$$t_c=100+0.6x, \ t_b=140-0.2x$$
Suppose that these times are different. In that case there will be some people who can claim that their travelling time is not minimal (because there are some people with shorter transfer times). The only way to avoid that situation is the case when:
$$t_c=t_b\implies x=50$$
This gives:
$$t_c=t_b=130$$
...and the total travelling time is $130\times100=13000$ minutes.