Question

The organizers want to devise a task time policy such that the total time for the participants to reach the treasure is minimized. The policy should also ensure that not more than 70 per cent of the total traffic passes through junction B. The time taken by the participants in travelling from point S to point T under this policy will be:

• A. 7 hours
• B. 9 hours
• C. 10 hours
• D. 13 hours

Answer 1

The correct option is C 10 hours

The possible routes are -

$\begin{array}{ccc}Route& TotalTime& Junction\\ S-A-T& 14& A\\ S-B-A-T& 9& A,B\\ S-B-C-T& 7& B,C\\ S-D-C-T& 10& C,D\\ S-D-T& 13& D\end{array}$

It is given that, for each of the routes, the only way to increase the total time is to impose task time at the junctions. Let the task time at A, B, C, D be a, b, c, d hours respectively.

Now, the total time for each of the five routes will be as follows.

$\begin{array}{cc}Route& TotalTime\\ S-A-T& 14+a\\ S-B-A-T& 9+(a+b)\\ S-B-C-T& 7+(b+c)\\ S-D-C-T& 10+(c+d)\\ S-D-T& 13+d\end{array}$

There must be another route other than those involving B with the least time as at most only $70\%$ participants can use this route.
Also, the other route should take time equal to the route through B. The minimum time for the other route will be through route S-D-C-T which is 10 hours (if task time at C and D is 0).

In that case, for route S-B-C-T, the total time = 10 hours (if task time at B = 3 hours).
$\therefore$ 10 hours is the least time.