Solution
Understanding the set :-
This is a tables set :-
- In this set - there were six teachers (Annie, Beti, Chetan, Dave, Esha, and Fakir) teaching six sections of a business school class. They jointly prepared the Midterm (MT) and Endterm (ET) question papers.
- Key features of the exams:
a) Both MT and ET were for 100 marks.
b) ET had more questions than MT.
c) Each question carried either 5, 10, or 15 marks.
d) Question order by marks:
All 5-mark questions came first,
Then all the 10-mark questions,
And then all the 15-mark questions.
e) Each teacher prepared the same number of questions in total (MT + ET).
f) A teacher’s questions were grouped together (consecutive) in both MT and ET.
So lets Use the constraints to logically deduce the full structure of the papers.
| MT | ET | Total Marks | |
|---|---|---|---|
| 5 marks | ≥ 4 | ≥ 4 | ≥ 20 marks |
| 10 marks | ≥ 3 | ≥ 3 | ≥ 30 marks |
| 15 marks | ≥ 2 | ≥ 2 | ≥ 30 marks |
We know, All six sections had a common midterm (MT) and a common end term (ET) worth 100 marks each.
And, Each of MT and ET had at least four questions that were worth 5 marks, at least three questions that were worth 10 marks, and at least two questions that were worth 15 marks
=>
| MT | ET | Total Marks | |
|---|---|---|---|
| 5 marks | ≥ 4 | ≥ 4 | ≥ 20 marks |
| 10 marks | ≥ 3 | ≥ 3 | ≥ 30 marks |
| 15 marks | ≥ 2 | ≥ 2 | ≥ 30 marks |
The total possible with considering the minimum number of questions of each type
= marks
Rest 20 marks are possible by the following cases:
{5,5,5,5} {5,5,10} {10,10} {5,15}
| MT | ET | Total Marks | |
|---|---|---|---|
| 5 marks | ≥ 4 | ≥ 4 | ≥ 20 marks |
| 10 marks | ≥ 3 | ≥ 3 | ≥ 30 marks |
| 15 marks | ≥ 2 | ≥ 2 | ≥ 30 marks |
Now, we know ET contained more questions than MT.
Thus MT cannot consider the case {5,5,5,5}
The number of questions in each case:
- {5,5,5,5} = 9+4 =13 questions
- {5,5,10} = 9+3 =12 questions
- {10,10} = 9+2 =11 questions
- {5,15} = 9+2 =11 questions
Considering MT and ET together, each faculty member prepared the same number of questions.
The total number of questions should be multiple of 6, thus the total number of questions will be 24.
| MT | ET | Total Marks | |
|---|---|---|---|
| 5 marks | ≥ 4 | ≥ 4 | ≥ 20 marks |
| 10 marks | ≥ 3 | ≥ 3 | ≥ 30 marks |
| 15 marks | ≥ 2 | ≥ 2 | ≥ 30 marks |
For ET and MT, there are 2 cases :
- {5,5,5,5}{5,15}
- {5,5,5,5}{10,10}
According to the statement (i), Annie prepared the fifth question for both MT and ET.
For MT, this question carried 5 marks.
Thus {10,10} case is not possible.
=> MT -{5,5,5,5,5,10,10,10,15,15,15}
And, ET - {5,5,5,5,5,5,5,5,10,10,10,15,15}
| MT Q.No | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Marks | 5 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 15 | 15 | 15 |
| Setter |
| ET Q.No | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Marks | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 15 | 15 |
| Setter |
We can make these two tables for MT and ET respectively, containing marks and setters name in second and third row respectively.
And, directly fill the data from previous slide over here.
| MT Q.No | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Marks | 5 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 15 | 15 | 15 |
| Setter | A |
| ET Q.No | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Marks | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 15 | 15 |
| Setter | A |
Clue 1 says,
Annie prepared the fifth question for both MT and ET. For MT, this question carried 5 marks.
We can directly add this to the table.
| MT Q.No | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Marks | 5 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 15 | 15 | 15 |
| Setter | C | A | E |
| ET Q.No | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Marks | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 15 | 15 |
| Setter | C | A | E |
Clue 4 says,
Chetan prepared the third question in both MT and ET; and Esha prepared the eighth question in both.
We can directly add this to the table.
| MT Q.No | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Marks | 5 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 15 | 15 | 15 |
| Setter | F | C | A | E | D |
| ET Q.No | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Marks | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 15 | 15 |
| Setter | D | C | A | E | F |
Clue 5 says,
Fakir prepared the first question of MT and the last one in ET. Dave prepared the last question of MT and the first one in ET.
We can directly add this to the table.
| MT Q.No | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Marks | 5 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 15 | 15 | 15 |
| Setter | F | F | C | C | A | B | B | E | E | D | D |
| ET Q.No | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Marks | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 15 | 15 |
| Setter | D | C | A | E | F |
Clue 2 states that Annie prepared only one question for the Midterm (MT), while every other faculty member prepared more than one question.
According to Clue 3, all questions prepared by a faculty member appear consecutively in both MT and ET.
Based on this:
Fakir must have prepared Questions 1 and 2.
Chetan must have prepared Questions 3 and 4.
Annie prepared only one question, which is Question 5.
Now, for Dave to have prepared more than one question, and given that he prepared the last question of the MT, he must have also prepared Question 10. So, Dave is assigned to Questions 10 and 11.
For Beti to have more than one question and still maintain consecutiveness, the only possibility is that she prepared Questions 6 and 7.
Lastly, for Esha to meet the condition of preparing multiple consecutive questions, she must have prepared Questions 8 and 9.
| MT Q.No | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Marks | 5 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 15 | 15 | 15 |
| Setter | F | F | C | C | A | B | B | E | E | D | D |
| ET Q.No | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Marks | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 15 | 15 |
| Setter | D | D | C | C | A | A | A | E | E | B | B | F | F |
Now, Doing the same for ET , also, we know,
There are 24 questions in total so each faculty will make 4 questions in total.
We have each faculty (except A ) preparing 2 questions in MT.
=> In ET all faculties (except A) preparing 2 more questions to make the total 4.
and A will make 3 questions in ET as she has only prepared one question in MT.
- D - for consecutive multiple question D will prepare question 1 and 2.
- C - question 3 and 4
- For F - question 10 will be prepared by him, along with question 11
- A - her 3 questions will be - question 5, 6, 7.
Therefore, only place left for B's two questions are question number 10 and 11.
| MT Q.No | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Marks | 5 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 15 | 15 | 15 |
| Setter | F | F | C | C | A | B | B | E | E | D | D |
| ET Q.No | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Marks | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 15 | 15 |
| Setter | D | D | C | C | A | A | A | E | E | B | B | F | F |
As shown above ,5-mark questions were there in MT and ET combined were 8+5=13