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.
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
The Correct Option is C
Solution and Explanation
To adhere to the fundamental requirement, each pair of cities, such as A and B, must have both morning and evening flights in both directions. This ensures that individuals from A or B can travel to the other city and return on the same day. Consequently, there should be a total of four flights connecting any pair of cities.
Regarding the selection of two cities from the total of ten cities, the number of ways to achieve this can be calculated using
\(=\frac{10\times9}{2}=45\)
Hence, the minimum number of flights that must be scheduled = 45 ×4 = 180.
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
The Correct Option is C
Solution and Explanation
Consider the ten cities labeled A through J, with A, B, and C identified as hubs and the remaining seven cities designated as non-hub cities. The condition specifies that any direct flight must originate and/or terminate at a hub.
Take city D as an example, which is a non-hub city. D should be connected to each of the hubs, A, B, and C. According to the previous solution, there must be four flights between D and each of A, B, and C. Therefore, from D, there must be a total of \(4 × 3 = 12\) flights connecting to the three hubs.
Similarly, for each of the other six non-hub cities, there must be 12 flights connecting each non-hub city with the three hubs. This results in a total of \(12 × 7 = 84\) flights linking a non-hub city with a hub. Additionally, the three hubs must be connected among themselves. As per the requirement of four flights between any pair of cities, there must be a total of \(4 × 3 = 12\) flights connecting any pair of hubs.
Hence, the minimum total number of flights that should be scheduled is 84 (connecting non-hub cities with hubs) + 12 (connecting hubs) = 96.
Suppose the 10 cities are divided into 4 distinct groups 01,02,03,04 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
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
Correct Answer: 40
Solution and Explanation
Given that G1 has cities A, B, and C, and G2, G3, and G4 have 3, 2, and 2 cities, respectively. According to the given conditions, a city in G2 cannot have a direct flight to a city in G3 or G4. To travel from a city in G2 to a city in G3 or G4, all cities in G2 must be connected to A. Subsequently, a person can travel from A to B or C to reach a city in G3 or G4.
Consequently, the 3 cities in G2 must be connected to A, requiring 4 flights between each pair of cities. This results in a total of 4 × 3 = 12 flights between cities in G2 and A. Considering the 2 cities in G3, there must be 2 × 4 = 8 flights connecting cities in G3 and B. Similarly, for the 2 cities in G4, there must be 2 × 4 = 8 flights connecting cities in G4 and C. Additionally, cities in G1 (A, B, and C) must be interconnected, requiring an extra 4 × 3 = 12 flights between these three cities.
Hence, the total minimum number of direct flights that must be scheduled is 12 (G2 to A) + 8 (G3 to B) + 8 (G4 to C) + 12 (interconnecting cities in G1) = 40.
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?
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?
Correct Answer: 4
Solution and Explanation
The given information states that cities in G2 will be assigned to G3 or G4. However, this alone does not lead to a reduction in the number of flights, as cities in G2 still need to be connected to either B or C.
However, an additional piece of information is provided: there are now no flights between A and C. Consequently, the 4 flights that would have been scheduled in the previous case connecting A and C will not be scheduled.
Therefore, the maximum reduction in the number of flights can be 4.
Top Questions on Conclusion
- Faculty members in a management school can belong to one of four departments – Finance and Accounting (F&A), Marketing and Strategy (M&S), Operations and Quants (O&Q) and Behaviour and Human Resources (B&H). The numbers of faculty members in F&A, M&S, O&Q and B&H departments are 9, 7, 5 and 3 respectively. Prof. Pakrasi, Prof. Qureshi, Prof. Ramaswamy and Prof. Samuel are four members of the school's faculty who were candidates for the post of the Dean of the school. Only one of the candidates was from O&Q. Every faculty member, including the four candidates, voted for the post. In each department, all the faculty members who were not candidates voted for the same candidate. The rules for the election are listed below.
1. There cannot be more than two candidates from a single department.
2. A candidate cannot vote for himself/herself.
3. Faculty members cannot vote for a candidate from their own department.
After the election, it was observed that Prof. Pakrasi received 3 votes, Prof. Qureshi received 14 votes, Prof. Ramaswamy received 6 votes and Prof. Samuel received 1 vote. Prof. Pakrasi voted for Prof. Ramaswamy, Prof. Qureshi for Prof. Samuel, Prof. Ramaswamy for Prof. Qureshi and Prof. Samuel for Prof. Pakrasi - There are only four neighbourhoods in a city-Levmisto, Tyhrmisto,Pesmisto and Kitmisto. During the onset of a pandemic,the number of new cases of a disease in each of these neighbourhoods was recorded over a period of five days. On each day, the number of new cases recorded in any of the neighbourhoods was either 0,1,2 or 3.
The following facts are also known:
1. There was at least one new case in every neighbourhood on Day 1.
2. On each of the five days, there were more new cases in Kitmisto than in Pesmisto.
3. The number of new cases in the city in a day kept increasing during the five-day period. The number of new cases on Day 3 was exactly one more than that on Day 2.
4. The maximum number of new cases in a day in Pesmisto was 2 and this happenedonly once during the five-day period.
5. Kitmisto is the only place to have 3 new cases on Day 2.
6. The total numbers of new cases in Levmisto,Tyhrmisto,Pesmisto and Kitmisto over the five-day period were 12,12,5 and 14 respectively. - The Humanities department of a college is planning to organize eight seminars, one for each of the eight doctoral students - A, B, C, D, E, F, G and H. Four of them are from Economics, three from Sociology and one from Anthropology department. Each student is guided by one among P, Q, R, S and T. Two students are guided by each of P, R and T, while one student is guided by each of Q and S. Each student is guided by a guide belonging to their department.
Each seminar is to be scheduled in one of four consecutive 30-minute slots starting at 9:00 am, 9:30 am, 10:00 am and 10:30 am on the same day. More than one seminars can be scheduled in a slot, provided the guide is free. Only three rooms are available and hence at the most three seminars can be scheduled in a slot. Students who are guided by the same guide must be scheduled in consecutive slots.
The following additional facts are also known.
1. Seminars by students from Economics are scheduled in each of the four slots.
2. A’s is the only seminar that is scheduled at 10:00 am. A is guided by R.
3. F is an Anthropology student whose seminar is scheduled at 10:30 am.
4. The seminar of a Sociology student is scheduled at 9:00 am.
5. B and G are both Sociology students, whose seminars are scheduled in the same slot. The seminar of an Economics student, who is guided by T, is also scheduled in the same slot.
6. P, who is guiding both B and C, has students scheduled in the first two slots.
7. A and G are scheduled in two consecutive slots. - Ten musicians (A, B, C, D, E, F, G, H, I and J) are experts in at least one of the following three percussion instruments: tabla, mridangam, and ghatam. Among them, three are experts in tabla but not in mridangam or ghatam, another three are experts in mridangam but not in tabla or ghatam, and one is an expert in ghatam but not in tabla or mridangam. Further, two are experts in tabla and mridangam but not in ghatam, and one is an expert in tabla and ghatam but not in mridangam.
The following facts are known about these ten musicians.
1. Both A and B are experts in mridangam, but only one of them is also an expert in tabla.
2. D is an expert in both tabla and ghatam.
3. Both F and G are experts in tabla, but only one of them is also an expert in mridangam.
4. Neither I nor J is an expert in tabla.
5. Neither H nor I is an expert in mridangam, but only one of them is an expert in ghatam. - A large store has only three departments, Clothing, Produce, and Electronics. The following figure shows the percentages of revenue and cost from the three departments for the years 2016, 2017 and 2018. The dotted lines depict percentage levels. So for example, in 2016, 50% of store's revenue came from its Electronics department while 40% of its costs were incurred in the Produce department.
In this setup, Profit is computed as (Revenue – Cost) and Percentage Profit as Profit/Cost × 100%.
It is known that
1. The percentage profit for the store in 2016 was 100%.
2. The store’s revenue doubled from 2016 to 2017, and its cost doubled from 2016 to 2018.
3. There was no profit from the Electronics department in 2017.
4. In 2018, the revenue from the Clothing department was the same as the cost incurred in the Produce department.
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