Qs 1-4: Read the following information and answer the questions that follow.

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.

Q.1

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

TITA type.

There are actually 5 possibilities … 0/1/2/3/4 car take the A-M-B route (and rest 4/3/2/1/0 the A-N-B route)

The extreme cases of all cars travelling on same route and other route being empty are trivial cases and can be eliminated. So make a table with the other 3 possible cases and find the time taken along the two routes for the 3 cases. Check if anyone can reduce his travel time by changing his current route to the other route while others continue on the same.

2 cars.

Q.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?

A. 1

B. 0.1

C. 0.2

D. 0.9

There are actually 5 possibilities …

0/1/2/3/4 car take the A-M-B route (and rest 4/3/2/1 the A-N-B route)

The extreme cases of all cars travelling on same route and other route being empty are trivial cases and can be eliminated. So make a table with the other 3 possible cases and find the time taken along the two routes for the 3 cases. Check if anyone can reduce his travel time by changing his current route to the other route while others continue on the same.

0.1

Same as that for first question of the set.

Additional directions for questions 3 & 4: 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.

Q.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.)

TITA type.

Similar to as done in Q. 1 & 2, we need to draw a table for time taken along the 3 routes − A-M-B, A-M-N-B, A-N-B, for different cases of number of cars traveling along the routes.

Check which of the route takes the least time for just 1 car. That route can be given more number of cars. So, start with a distribution of cars being uniformly distributed along the routes, with a little more on the route taking least time for 1 car.

2 cars

Additional directions for questions 3 & 4: 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.

Q.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.)

A. 26

B. 32

C. 29.9

D. 30

Similar to as done in Q. 1 & 2, we need to draw a table for time taken along the 3 routes − A-M-B, A-M-N-B, A-N-B, for different cases of number of cars traveling along the routes.

Check which of the route takes the least time for just 1 car. That route can be given more number of cars. So, start with a distribution of cars being uniformly distributed along the routes, with a little more on the route taking least time for 1 car.

B. 32

Same as given in the previous question.