Set 1: Airlines Company
Q1-4. A new airlines company is planning to start operations in a country. The company has identified ten different cities which they plan to connect through their network to start with. The flight duration between any pair of cities will be less than one hour. To start operations, the company has to decide on a daily schedule.
The underlying principle that they are working on is the following:
Any person staying in any of these 10 cities should be able to make a trip to any other city in the morning and should be able to return by the evening of the same day.
1. If the underlying principle is to be satisfied in such a way that the journey between any two cities can be performed using only direct (non-stop) flights, then the minimum number of direct flights to be scheduled is:
- 45
- 90
- 180
- 135
EXPLANATION
Answer: 3
Consider, the ten cities to be A,B,C,D and so on till City J.
Let us consider Cities A and B.
As per the underlying principle any person should be able to take a round trip between the two cities in the same day.
So, from A to B and vice versa (B to A) one flight should leave from each city in the morning and similarly in the evening, two flight should leave for the person to reutrn.
A total 4 flight are to be operate between any 2 cities.
We can select 2 cities out of 10 cities in 10C2 = 45 ways.
Four flights from a pair of cities, therefore total of 45 * 4 = 180 flights.
The question is “The minimum number of direct flights to be scheduled is:”
Hence, the answer is “180 flights”.
2. Suppose three of the ten cities are to be developed as hubs. A hub is a city which is connected with every other city by direct flights each way, both in the morning as well as in the evening. The only direct flights which will be scheduled are originating and/or terminating in one of the hubs. Then the minimum number of direct flights that need to be scheduled so that the underlying principle of the airline to serve all the ten cities is met without visiting more than one hub during one trip is:
- 54
- 120
- 96
- 60
EXPLANATION
Answer: C
Consider 3 cities A,B and C to be the hub cities. So, any flight from hub city and towards hub city are direct flights.
Also, it has to satifsy the underlying principle of the airline.
For the moment, let us forget about the underlying principle.
In the first step, forget about the flights between the three hub cities.
Consider only one hub city, say A. So, it will have direct flights with 7 other non hub cities D,E,F,G,H,I and J.
As observed in the previous question, in order to satisfy the underlying principle, 4 flights should be connected to each of the 7 seven cities. So, a total of 7 * 4 = 28 flights are to be operated per hub city.
Similarly, the other two hub cities B and C will have 28 flights each to the other 7 non-hub cities.
Therefore, the total no. of flight will be 28 + 28 + 28 = 84.
But, we still have to account for the flights between the hub cities.
The three cities can be connected by direct flights.
Similarly, if we account for the flights between any two hub cities, a total of 4 flights will have to be operated between the cities such that they satisfy the underlying principle.
For 3 such parirs (AB, BC and AC), we have 4 * 3 = 12 flights.
Total number of flights to be operated = 84 + 12 = 96 flights.
Hence, the answer is “96”.
3. Suppose the 10 cities are divided into 4 distinct groups G1, G2, G3, G4 having 3, 3, 2 and 2 cities respectively and that G1 consists of cities named A, B and C. Further, suppose that direct flights are allowed only between two cities satisfying one of the following:
1. Both cities are in G1
2. Between A and any city in G2
3. Between B and any city in G3
4. Between C and any city in G4
Then the minimum number of direct flights that satisfies the underlying principle of the airline is: (TITA)
EXPLANATION
Answer: 40 flights
First consider the flights to G2.
But, only flight from and to A can be allowed for cities in G2.
So, between A and G21, 4 flights should be operated (morning/evening and to/from).
Similarly between A-G22 and A-G23.
Total flight = 4 + 4 + 4 = 12 flights
Similarly, G3 consists of two cities and can only be connected to city B.
Therefore, total flights = 4 + 4 = 8.
And, for G4 flights are connected only to city C.
Total flights = 4 + 4 = 8
Finally, we have to consider flights within G1.
As stated in the previous questions, the total number of flights between the hub cities is 4 + 4 + 4 = 12.
Minimum no. of flights to be operated = 12 + 8 + 8 + 12 = 40 flights.
Hence, the answer is “40”.
4. Suppose the 10 cities are divided into 4 distinct groups G1, G2, G3, G4 having 3, 3, 2 and 2 cities respectively and that G1 consists of cities named A, B and C. Further, suppose that direct flights are allowed only between two cities satisfying one of the following:
1. Both cities are in G1
2. Between A and any city in G2
3. Between B and any city in G3
4. Between C and any city in G4
However, due to operational difficulties at A, it was later decided that the only flights that would operate at A would be those to and from B. Cities in G2 would have to be assigned to G3 or to G4.
What would be the maximum reduction in the number of direct flights as compared to the situation before the operational difficulties arose? (TITA)
EXPLANATION
Answer: 4 flights
It is to be noted that as per new conditions, the flights between A and C is not possible, as well as flights between A and any city in G2 is not allowed.
The cities in G2 are shifted to group G3 and G4.
The new group og cities would look like this:
Consider the group of cities in G3, only flights from/to B are possible.
Total flights for G3 = 4 * 4 = 16 flights.
Similarly, for group G4,
Total flights for G4 = 4 * 3 = 12 flights.
Only flights in G1 and the flights from A are to be accounted.
The flights in G1 is only between B and C,
Total flights = 4
Since, flights between A and C are not possible, we have to consider only flights between A and B, which is 4.
Total flights operated = 16 + 12 + 4 + 4 = 36.
Maximum reduction in the number of flights = 40 – 36 = 4 flights.
Hence, the answer is “4”.
