## Routes and Networks

#### Routes and Networks

Direction: A significant amount of traffic flows from point S to point T in the one-way street network shown below.
Points A, B, C and D are junctions in the network, and the arrows mark the direction of traffic flow.
The fuel cost in rupees for travelling along a street is indicated by the number adjacent to the arrow representing the street.
Motorists travelling from point S to point T would obviously take the route for which the total cost of travelling is the minimum.
If two or more routes have the same least travel cost, then motorists are indifferent between them.
Hence, the traffic gets evenly distributed among all the least cost routes The government can control the flow of traffic only by levying appropriate toll at each junction.
For example, if a motorist takes the route S - A - T (using junction A alone), then the total cost of travel would be Rs 14 (i.e. Rs 9 + Rs 5) plus the toll charged at junction A.

1. If the government wants to ensure that all routes from S to T get the same amount of traffic, then a feasible set of toll charged (in rupees) at junctions A, B, C and D respectively to achieve this goal is :

1. Let the toll charged at junctions A, B, C and D be a, b, c and d respectively.
Then 1st we will list down all the routes and corresponding cost of travel.
Here in this case all 5 routes have the same toll charge hence 14 + a = 7 + b + c = 13 + d = 9 + a + b = 10 + c + d

##### Correct Option: D

As per the given diagram , we can see that
Let the toll charged at junctions A, B, C and D be a, b, c and d respectively.
Then 1st we will list down all the routes and corresponding cost of travel.
Here in this case all 5 routes have the same toll charge hence 14 + a = 7 + b + c = 13 + d = 9 + a + b = 10 + c + d
After solving we will get a = 1, b = 5, c = 3 and d = 2

1. Eight cities A, B, C, D, E, F, G and H are connected with one-way roads R1, R2, R3, R4, R5 and R6 in the following manner:
R1 leads from A to C via B;
R2 leads from C to D and then via B to F;
R3 leads from D to A and then via E to H;
R4 leads from F to B via G;
R5 leads from G to D; and R6 leads from F to H.
The minimum number of road segments that have to be blocked in order to make all traffic form B to D impossible is

1. On the basis of above given question , we can say that
We can block B to D if A to B means R1 and B to F means R2 is blocked.

##### Correct Option: D

On the basis of above given question , we can say that
We can block B to D if A to B means R1 and B to F means R2 is blocked.
Therefore minimum 2 ways needed to be block.

1. In the adjoining figure, the lines represent one-way roads allowing travel only northwards or only westwards. Along how many distinct routes can a car reach point B from point A?

1. Here , number of vertical steps ( v ) = 3
Number of horizontal steps ( h ) = 5

##### Correct Option: B

Here , number of vertical steps ( v ) = 3
Number of horizontal steps ( h ) = 5
Then in this case total number of ways is given by h+vCh = h+vCv = 8C3 = 6 x 7 x ( 8/6 ) = 7 x 8 = 56.
Hence , 56 distinct routes can a car reach point B from point A .

1. If the government wants to ensure that all motorists travelling from S to T pay the same amount ( fuel costs and toll combined ) regardless of the route they choose and the street from B to C is under repairs ( and hence unusable ), then a feasible set of toll charged (in rupees) at junctions A, B, C and D respectively to achieve this goal is :

1. As per the given figure , we can see that
Let the toll charged at junctions A, B, C and D be a, b, c and d respectively.
Then 1st we will list down all the routes and corresponding cost of travel.
Since Route BC is under repair hence route S-B-C-T is not in use.
Rest all four have the same toll charges hence 14 + a = 9 + a + b ⇒ b = 14 - 9 = 5

##### Correct Option: E

As per the given figure , we can see that
Let the toll charged at junctions A, B, C and D be a, b, c and d respectively.
Then 1st we will list down all the routes and corresponding cost of travel.
Since Route BC is under repair hence route S-B-C-T is not in use.
Rest all four have the same toll charges hence 14 + a = 9 + a + b ⇒ b = 14 - 9 = 5
Similarly 10 + c + d = 13 + d ⇒ c = 13 - 10 = 3
Hence Options 4 is ruled out, now if we check option rest 3 options we will find out that option 2 and 3 both are correct. Option (2)/(3) Inconsistent options .

1. If the government wants to ensure that the traffic at S gets evenly distributed along streets from S to A, from S to B, and from S to D, then a feasible set of toll charged (in rupees) at junctions A, B, C, and D respectively to achieve this goal is:

1. According to question ,
If all the five routes have the same cost, then there will be an equal flow in all the five routes, i.e. 20% in each route.
But then the percentage of traffic in S - A = 20% (Only one route involving S-A)
S - B = 40% (As there are two routes involving S - B)
S - D = 40% (As there are two routes involving S - D)
But here the given condition that traffic in S - A is equal to that in S - B, which in turn is equal to S - D is not satisfied.

##### Correct Option: A

According to question ,
If all the five routes have the same cost, then there will be an equal flow in all the five routes, i.e. 20% in each route.
But then the percentage of traffic in S - A = 20% (Only one route involving S - A)
S - B = 40% (As there are two routes involving S - B)
S - D = 40% (As there are two routes involving S - D)
But here the given condition that traffic in S-A is equal to that in S - B, which in turn is equal to S - D is not satisfied.
Of the routes, that can be used the number of routes involving S - A must be the same as S - B, which in turn is same as that as S - D.
That is possible only when we block the junction C and that can be done by taking higher toll charge at C to achieve this goal c > 3.
Hence , required answer will be option A .