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 the weight of the test component?

The correct answer is 0.60.

Let us break down the information step-by-step to determine the weight of the test component.

Step 1: Determine the Project and Test Scores

There are 6 students: three female (Amala, Koli, Rini) and three male (Biman, Mathew, Shyamal).

The aggregate score is calculated as a weighted average:
$Aggregate\; =\; (1\; -\; w)\; \backslash times\; Project\; +\; w\; \backslash times\; Test$
where $w$ is the weight of the test component, and .

From the first fact, the project and test scores both have a minimum of 40, maximum of 80, and an average of 60.

Project Scores:
Since projects are done in groups of two (each containing one male and one female student) and partners get the same score, there are three distinct project scores for the three groups.
Let these group scores be $P\_1$, $P\_2$, and $P\_3$ with $P\_1\; \backslash le\; P\_2\; \backslash le\; P\_3$.
Since the minimum score is 40 and the maximum is 80:
$P\_1\; =\; 40$ and $P\_3\; =\; 80$.
Since the average project score is 60:
$(40\; +\; P\_2\; +\; 80)\; /\; 3\; =\; 60\; \backslash Rightarrow\; P\_2\; =\; 60$.
Thus, the three project group scores are 40, 60, and 80.

Test Scores:
From fact 2, test scores are multiples of 10. Four of them are distinct, and the remaining two are equal to the average test score (60).
Let the test scores be $T\_1,\; T\_2,\; T\_3,\; T\_4,\; T\_5,\; T\_6$, where two of them are 60.
The sum of all six test scores must be $6\; \backslash times\; 60\; =\; 360$.
Subtracting the two scores of 60 gives $360\; -\; 120\; =\; 240$ as the sum of the remaining four distinct test scores.
Since the minimum test score is 40 and the maximum is 80, the four distinct test scores must be selected from $\backslash \{40,\; 50,\; 70,\; 80\backslash \}$.
Summing these gives: $40\; +\; 50\; +\; 70\; +\; 80\; =\; 240$, which matches perfectly.
Thus, the set of test scores is {40, 50, 60, 60, 70, 80}.

Step 2: Relate Project and Test Scores of Amala and Koli

From fact 3:
Amala's project score ($P\_A$) is double that of Koli's ($P\_K$).
Given the project scores are 40, 60, and 80, the only possible pair is:
$P\_K\; =\; 40$ and $P\_A\; =\; 80$.
This also implies that Rini's project score is $P\_R\; =\; 60$.

Also from fact 3, Koli scored 20 more than Amala in the test:
$T\_K\; =\; T\_A\; +\; 20$.

Step 3: Analyze the Aggregate Scores

From fact 4:
Shyamal scored the second highest in the test. Looking at our test scores $\backslash \{40,\; 50,\; 60,\; 60,\; 70,\; 80\backslash \}$, the second highest is 70.
So, Shyamal's test score is $T\_S\; =\; 70$.

Shyamal's aggregate score ($A\_S$) is two more than Koli's aggregate score ($A\_K$) and two less than Amala's aggregate score ($A\_A$):
$A\_S\; =\; A\_K\; +\; 2$
$A\_A\; =\; A\_S\; +\; 2$
This gives the relation:
$A\_A\; -\; A\_K\; =\; 4$.

Step 4: Solve for the Weight of the Test Component ($w$)

Let's write down the aggregate score formulas for Amala and Koli:
$A\_A\; =\; (1\; -\; w)P\_A\; +\; w\; T\_A\; =\; 80(1\; -\; w)\; +\; w\; T\_A$
$A\_K\; =\; (1\; -\; w)P\_K\; +\; w\; T\_K\; =\; 40(1\; -\; w)\; +\; w(T\_A\; +\; 20)$

Now, calculate the difference $A\_A\; -\; A\_K$:
$A\_A\; -\; A\_K\; =\; [80(1\; -\; w)\; +\; w\; T\_A]\; -\; [40(1\; -\; w)\; +\; w\; T\_A\; +\; 20w]$
$A\_A\; -\; A\_K\; =\; 40(1\; -\; w)\; -\; 20w$
$A\_A\; -\; A\_K\; =\; 40\; -\; 40w\; -\; 20w$
$A\_A\; -\; A\_K\; =\; 40\; -\; 60w$

Since we established $A\_A\; -\; A\_K\; =\; 4$:
$40\; -\; 60w\; =\; 4$
$60w\; =\; 36$
$w\; =\; 36\; /\; 60\; =\; 0.60$.

Thus, the weight of the test component is 0.60.