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?

Question

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?

Accepted Answer

To determine the weight of the test component ($$ w_t $$), we analyze the problem step by step using the given information and equations.

### Key Information Recap:
1. The aggregate score is calculated as:
   $$
   	ext{Aggregate} = w_p \cdot (	ext{Project Score}) + w_t \cdot (	ext{Test Score}),
   $$
   where $$ w_p + w_t = 1 $$.

2. From the problem:
   - Amala's aggregate score is the highest.
   - Shyamal's aggregate score is **2 less than Amala's**.
   - Amala's project score = **80**, test score = **80**.
   - Shyamal's project score = **60**, test score = **62**.

---

### Step-by-Step Solution:

#### Step 1: Write Aggregate Scores
- Amala's aggregate score:
  $$
  	ext{Amala's Aggregate} = w_p \cdot 80 + w_t \cdot 80.
  $$

- Shyamal's aggregate score:
  $$
  	ext{Shyamal's Aggregate} = w_p \cdot 60 + w_t \cdot 62.
  $$

#### Step 2: Difference Between Aggregates
From the problem, the difference between Amala's and Shyamal's aggregates is **2**:
$$
	ext{Amala's Aggregate} - 	ext{Shyamal's Aggregate} = 2.
$$

Substitute the aggregate formulas:
$$
(w_p \cdot 80 + w_t \cdot 80) - (w_p \cdot 60 + w_t \cdot 62) = 2.
$$

#### Step 3: Simplify the Equation
Using $$ w_p = 1 - w_t $$ (since $$ w_p + w_t = 1 $$):
$$
((1 - w_t) \cdot 80 + w_t \cdot 80) - ((1 - w_t) \cdot 60 + w_t \cdot 62) = 2.
$$

Expand terms:
$$
(80 - 80w_t + 80w_t) - (60 - 60w_t + 62w_t) = 2.
$$

Simplify:
$$
80 - (60 + 2w_t) = 2.
$$

Solve for $$ w_t $$:
$$
20 - 2w_t = 2.
$$

$$
2w_t = 18.
$$

$$
w_t = 0.60.
$$

---

### Conclusion:
The weight of the test component ($$ w_t $$) is **0.60**, making option **C** correct.

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