Question

There are only three female students - Amala, Koli and Rini - and only three male students - Biman, Mathew and Shyamal - in a course. The course has two evaluation components, a project and a test. The aggregate score in the course is a weighted average of the two components, with the weights being positive and adding to 1 . 
The projects are done in groups of two, with each group consisting of a female and a male student. Both the group members obtain the same score in the project. 
The following additional facts are known about the scores in the project and the test.
1. The minimum, maximum and the average of both project and test scores were identical – 40, 80 and 60 , respectively. 
2. The test scores of the students were all multiples of 10 ; four of them were distinct and the remaining two were equal to the average test scores.
3. Amala's score in the project was double that of Koli in the same, but Koli scored 20 more than Amala in the test. Yet Amala had the highest aggregate score. 
4. Shyamal scored the second highest in the test. He scored two more than Koli, but two less than Amala in the aggregate. 
5. Biman scored the second lowest in the test and the lowest in the aggregate. 
6. Mathew scored more than Rini in the project, but less than her in the test.
What was Mathew's score in the test?(This Question was asked as TITA)

• A. 40 Marks
• B. 50 Marks
• C. 60 Marks
• D. 55 Marks

Answer 1

The Correct Option is A

Given :
In a course, there are only three female students named Amala, Koli, and Rini, and only three male students named Biman, Mathew, and Shyamal.
It is known that the total score in the course is calculated as a weighted average of two components, with both weights being positive and summing up to 1.
Let's assume that the project score component be x, while the test score is represented by (1-x).Projects are completed in pairs, with each pair consisting of one female and one male student, totaling three pairs.Both members of each pair receive the same score for the project. So, the scores achieved in the project are 40, 60, and 80, respectively.
Hence, it can be concluded that each female student will belong to a unique group, and no two male or female students will be assigned to the same group.
Regarding the test scores, there are six scores provided for six students, with four being unique and the remaining two being average scores, both of which are 60. Additionally, it is understood that the highest possible score is 80, while the lowest is 40.
Therefore, the unique scores are 80, 70, 50, and 40 (as all test scores are multiples of 10), while the remaining two scores are both 60.
Based on point 3, we deduce that Amala's project score was twice that of Koli's, while Koli scored 20 points higher than Amala in the test. Therefore, Amala's project score is 80, and Koli's is 40, resulting in Rini's project score being 60. Koli's test score, being 20 points higher than Amala's, could be either 80, 70, or 60.
So, the score obtained by them is as follows :
 

Students	Test scores	Project scores
Amala	40/50/60	80
Koli	60/70/80	40
Rini		60
Biman
Mathew
Shyamal

It is given that Amala attained the highest overall score, while Shyamal achieved the second highest on the test. His score surpassed Koli's by two points, yet fell short of Amala's aggregate by two points.
Therefore, Shyamal's test score is 70, which means Koli cannot score 70 in the test, leading to the inference that Amala cannot score 50 in the test.

Students	Test scores	Project scores
Amala	40/60	80
Koli	60/80	40
Rini		60
Biman
Mathew
Shyamal	70

As stated, Shyamal's aggregate score surpassed Koli's by two points but fell short of Amala's by two points. Consequently, Amala's aggregate score is four points higher than Koli's, and she holds the highest aggregate score.
Case (i) : The test of Amala is 40

Students	Test scores	Project scores	Aggregate score
Amala	40	80	40(1-x) + 80x
Koli	60	40	60(1-x) + 40x
Rini		60
Biman
Mathew
Shyamal	70

Hence, ⇒ 40(1 - x) + 80x = 60(1 - x) + 40x + 4
⇒ 60x = 24
⇒ x = 0.4
Therefore, Amala's aggregate score is calculated as :
= 40(1 - 0.4) + 80×0.4
⇒ 24 + 32 = 56
Shyamal's minimum aggregate score, calculated as 70(1 - 0.4) + 40×0.4, equals 58, which surpasses Amala's.
Therefore, Case 1 is not possible.
So, the below table is as follows :

Students	Test scores	Project scores	Aggregate score
Amala	60	80	60(1-x) + 80x
Koli	80	40	80(1-x) + 40x
Rini		60
Biman
Mathew
Shyamal	70

Hence, 60(1 - x) + 80x = 80(1 - x) + 40x + 4
⇒ 60 + 20x = 84 - 40x
⇒ 60x = 24
⇒ x = 0.24
Therefore, Amala's aggregate score, calculated as 60(1-0.4) + 80×0.4, amounts to 68, indicating that Shyamal's aggregate score is (68-2) = 66.
Thus, Shyamal's project score is calculated as $\frac{66-70\times(0.6)}{0.4} = 60.$
It is further understood that Biman achieved the second lowest score in the test, indicating his test score to be 50, and he attained the lowest aggregate score. Additionally, Mathew's project score exceeded Rini's but fell short of her test score. Consequently, Mathew's project score is 80 (as Rini scored 60 in the project), while Biman's project score is 40.
Likewise, Rini outperformed Mathew on the test, indicating Rini's score to be 60 and Mathew's to be 40.
Therefore, the final table will be as follows :

Students	Test scores (T)	Project scores (P)	Aggregrate score (T×0.6+P×0.4)	Project pair
Amala	60	80	68	Amala, Mathew
Koli	80	40	64	Koli, Biman
Rini	60	60	60	Rini, Shyamal
Biman	50	40	46	Biman, Koli
Mathew	40	80	56	Mathew, Amala
Shyamal	70	60	66	Shyamal, Rini

From the above table , we can see that Mathew has got a score of 40 in the test.
So, the correct answer is 40 Marks.