Four cars need to travel from Akala (A) to Bakala (B)

Four cars need to travel from Akala (A) to Bakala (B). Two routes are available, one via Mamur (M) and the other via Nanur (N). The roads from A to M, and from N to B, are both short and narrow. In each case, one car takes 6 minutes to cover the distance, and each additional car increases the travel time per car by 3 minutes because of congestion. (For example, if only two cars drive from A to M, each car takes 9 minutes.) On the road from A to N, one car takes 20 minutes, and each additional car increases the travel time per car by 1 minute. On the road from M to B, one car takes 20 minutes, and each additional car increases the travel time per car by 0.9 minute.

The police department orders each car to take a particular route in such a manner that it is not possible for any car to reduce its travel time by not following the order, while the other cars are following the order.

1. How many cars would be asked to take the route A-N-B, that is Akala-Nanur-Bakala route, by the police department?

2. If all the cars follow the police order, what is the difference in travel time (in minutes) between a car which takes the route A-N-B and a car that takes the route A-M-B?

  1. 1
  2. 0.1
  3. 0.2
  4. 0.9

A new one-way road is built from M to N. Each car now has three possible routes to travel from A to B: A-M-B, A-N-B and A-M-N-B. On the road from M to N, one car takes 7 minutes and each additional car increases the travel time per car by 1 minute. Assume that any car taking the A-M-N-B route travels the A-M portion at the same time as other cars taking the A-M-B route, and the N-B portion at the same time as other cars taking the A-N-B route.

3. How many cars would the police department order to take the A-M-N-B route so that it is not possible for any car to reduce its travel time by not following the order while the other cars follow the order? (Assume that the police department would never order all the cars to take the same route.)

4. If all the cars follow the police order, what is the minimum travel time (in minutes) from A to B? (Assume that the police department would never order all the cars to take the same route.)

  1. 26
  2. 32
  3. 29.9
  4. 30

Answers

  1. 2
  2. B
  3. 2
  4. B

1. As there are four cars and as the time through each route is nearly the same, two cars should go through A-M-B and the other two through A-N-B. In case three cars are directed to go through any of the routes, one of the three cars can break the police order and reduce its travel time.

2. According to the police order 2 cars each would pass through A – M – B and A – N – B. Then time taken through A – M – B = 29.9 and time taken through A – N – B = 30.0.

Difference = 0.1

3. No car should be able to reduce its travel time by not following the order and all the cars cannot take the same route. So either two or three cars should go through A-M. If two cars go through M-B, one car can break the police order and go through M-N and reach B in 9 + 7 + 12 = 28 minutes as compared to 29.9 minutes had both gone through A-M-B.

If two cars go through A-M and one is directed to go through M-N, one of the cars which was directed to go through A-N can break the police order and go through A-M-B and save time as follows:

Original time (A-N-B) = 21 + 12 = (three cars) = 33

New time = 12 (3 cars) + 20 .9 = 32.9

The police department cannot direct both cars to go through M-N as in that case all four cars would go through N-B In case three cars are directed to go through A-M, either one car can be directed through M-N or two cars can be directed through M-N.

If one car is directed through M-N, one of the two cars directed through M-B, can break the police order and go through M-N, and save time.

Original time (A-M-B) = 12 (3 cars) + 20.9 = 32.9

New time (A-M-N-B) = 12 + 8 + 12 = 32 minutes.

Therefore, two cars must be directed through M-N such that any car breaking the police order cannot reduce the travel time.

4. When all cars follow the police order the time taken would be A-M-B (1 car) = 12 + 20 = 32 minutes.

A-M-N-B (2 cars) = 12 + 8 + 12 = 32 minutes.

A-N-B (1 car) = 20 + 12 = 32 minutes.