Mindyard | 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?

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?

Correct Answer :

Solution :

The correct answer is 40.

Let the female students be Amala (A), Koli (K), and Rini (R), and the male students be Biman (B), Mathew (M), and Shyamal (S). Let the weight of the project component be $w_{p}$ and the weight of the test component be $w_{t}$ , where $w_{p} + w_{t} = 1$ and both are positive.

Step 1: Finding Project and Test Scores

From Fact 1, the minimum, maximum, and average scores for both project and test were 40, 80, and 60 respectively.
Since there are 6 students, the sum of all project scores is:
$6 \times 60 = 360$
Since projects are done in groups of two (consisting of one female and one male student who get the same score), there are exactly three project groups. Let their scores be $P_{1}, P_{2}, and P_{3}$ .
Thus, $2 (P_{1} + P_{2} + P_{3}) = 360 \Rightarrow P_{1} + P_{2} + P_{3} = 180$ .
Given the minimum is 40 and the maximum is 80, we must have:
$40 + P_{2} + 80 = 180 \Rightarrow P_{2} = 60$ .
So, the project scores for the three pairs are exactly 40, 60, and 80.

For the test scores, Fact 2 states that all scores are multiples of 10, four are distinct, and the remaining two equal the average (60).
Since the minimum is 40 and the maximum is 80, the set of 6 test scores must be:
{40, 50, 60, 60, 70, 80} (summing to 360, with average 60).

Step 2: Assigning Project Scores and Finding Weights

From Fact 3, Amala's project score was double that of Koli. Since the project scores are 40, 60, and 80, we must have:
$P_{A} = 80$
$P_{K} = 40$
This leaves Rini's project score as:
$P_{R} = 60$

Fact 3 also states that Koli scored 20 more than Amala in the test:
$T_{K} = T_{A} + 20$
From Fact 4, Shyamal scored the second highest in the test, which corresponds to 70. Thus, $T_{S} = 70$ .
Since 70 is taken by Shyamal, the pair $(T_{A}, T_{K})$ with a difference of 20 can either be $(40, 60)$ or $(60, 80)$ .

According to Fact 4, Shyamal scored two more than Koli, but two less than Amala in the aggregate:
$A g g_{S} = A g g_{K} + 2$
$A g g_{S} = A g g_{A} - 2$
Subtracting the two equations gives:
$A g g_{A} - A g g_{K} = 4$
Substituting the aggregate formula:
$(80 w_{p} + T_{A} w_{t}) - (40 w_{p} + T_{K} w_{t}) = 4$
Using $T_{K} = T_{A} + 20$ and $w_{t} = 1 - w_{p}$ :
$40 w_{p} - 20 (1 - w_{p}) = 4 \Rightarrow 60 w_{p} = 24 \Rightarrow w_{p} = 0.4 and w_{t} = 0.6$

Step 3: Determining Test Scores

Using $A g g_{S} = A g g_{A} - 2$ :
$0.4 P_{S} + 0.6 (70) = 0.4 (80) + 0.6 T_{A} - 2$
$0.4 P_{S} + 42 = 30 + 0.6 T_{A} \Rightarrow 2 P_{S} + 60 = 3 T_{A}$

If we test our two cases for $T_{A}$ :
- If $T_{A} = 40$ , then $2 P_{S} = 120 - 60 = 60 \Rightarrow P_{S} = 30$ (not a valid project score).
- If $T_{A} = 60$ , then $2 P_{S} = 180 - 60 = 120 \Rightarrow P_{S} = 60$ (which is a valid project score).
Thus, we establish:
$T_{A} = 60, T_{K} = 80, and P_{S} = 60$ .

Step 4: Pairing Students and Final Calculations

Since Shyamal has a project score of 60, he must be paired with Rini (whose project score is 60).
From Fact 6, Mathew scored more than Rini in the project. Since Rini's project score is 60, Mathew's project score must be 80 (paired with Amala).
Consequently, Biman's project score must be 40 (paired with Koli).
So, the project scores are:
$P_{M} = 80$
$P_{B} = 40$

The remaining test scores to be assigned to Mathew, Biman, and Rini are {40, 50, 60} (since 60, 70, and 80 are already assigned to Amala, Shyamal, and Koli).
Fact 5 states that Biman scored the second lowest in the test. The second lowest test score overall is 50, so $T_{B} = 50$ .
This leaves 40 and 60 for Mathew and Rini.
Since Mathew scored less than Rini in the test (Fact 6), we must have:
Mathew's test score = 40
Rini's test score = 60