Siya, Presh, and Nene decide to help each other study during the final exams. Siya's favorite subject is
Economics, Presh loves Statistics, and Nene knows his Information System (IT). Each studies his own
subject as well as helps the others learn their subjects. After the finals, they realize that Siya spent 40% of
his time studying his own subject Economics, 30% of his time helping Presh learn economics and 30% of the
time helping Nene learn economics. Presh spent 30% of his time studying his own subject statistics, 30% of
his time helping Siya learn Stats, and 40% of the time helping Nene learn Stats. Nene spent 20% of his time
studying his own subject information System (IT), 40% of his time helping Siya learn information system, and
40% of the time helping Presh learn information system.
1. Determine the study/learning matrix

2. If they originally agreed that each should work about 33 hours, how long did each work?

### 1. **Determine the study/learning matrix:**

This problem can be modeled using a matrix that shows how much time each person spends on studying their own subject and helping the others study. The rows of the matrix will represent Siya, Presh, and Nene, while the columns represent the subjects: Economics (Siya's subject), Statistics (Presh's subject), and Information Systems (Nene's subject). The elements of the matrix will be the percentages of time each person spends on each subject.

- **Siya**:
  - Spent 40% of his time on Economics (his own subject).
  - Spent 30% of his time helping Presh with Economics.
  - Spent 30% of his time helping Nene with Economics.

- **Presh**:
  - Spent 30% of his time on Statistics (his own subject).
  - Spent 30% of his time helping Siya with Statistics.
  - Spent 40% of his time helping Nene with Statistics.

- **Nene**:
  - Spent 20% of his time on Information Systems (his own subject).
  - Spent 40% of his time helping Siya with Information Systems.
  - Spent 40% of his time helping Presh with Information Systems.

Thus, the **study/learning matrix** \(A\) will be:

\[
A = 
\begin{pmatrix}
\text{Siya's time on Economics} & \text{Siya's time on Statistics} & \text{Siya's time on IT} \\
\text{Presh's time on Economics} & \text{Presh's time on Statistics} & \text{Presh's time on IT} \\
\text{Nene's time on Economics} & \text{Nene's time on Statistics} & \text{Nene's time on IT} \\
\end{pmatrix}
\]

Substituting the given values, we get:

\[
A = 
\begin{pmatrix}
0.40 & 0.30 & 0.30 \\
0.30 & 0.30 & 0.40 \\
0.40 & 0.40 & 0.20 \\
\end{pmatrix}
\]

This matrix represents how each person distributes their study time across the three subjects.

### 2. **How long did each work?**

If each person originally agreed to work for 33 hours, we can determine how many hours each person spent on each subject by multiplying their percentage of time spent on each subject by 33 hours.

Let's calculate the hours for each:

- **Siya**:
  - Time on Economics: \( 0.40 \times 33 = 13.2 \) hours.
  - Time on Statistics: \( 0.30 \times 33 = 9.9 \) hours.
  - Time on IT: \( 0.30 \times 33 = 9.9 \) hours.

  **Total time** spent by Siya: \( 13.2 + 9.9 + 9.9 = 33 \) hours.

- **Presh**:
  - Time on Economics: \( 0.30 \times 33 = 9.9 \) hours.
  - Time on Statistics: \( 0.30 \times 33 = 9.9 \) hours.
  - Time on IT: \( 0.40 \times 33 = 13.2 \) hours.

  **Total time** spent by Presh: \( 9.9 + 9.9 + 13.2 = 33 \) hours.

- **Nene**:
  - Time on Economics: \( 0.40 \times 33 = 13.2 \) hours.
  - Time on Statistics: \( 0.40 \times 33 = 13.2 \) hours.
  - Time on IT: \( 0.20 \times 33 = 6.6 \) hours.

  **Total time** spent by Nene: \( 13.2 + 13.2 + 6.6 = 33 \) hours.

Thus, all three spent the full 33 hours as agreed.

### Do you want more details or have any questions?

Here are 5 questions to consider:
1. How would the study matrix change if they spent different proportions of time on the subjects?
2. What is the interpretation of the diagonal elements of the study matrix?
3. How can the total time each person spent studying be represented using matrix multiplication?
4. What would happen if the agreed study time changed from 33 hours to another value?
5. How could this model be expanded to include additional subjects or people?

**Tip:** Matrices are a powerful tool in modeling interactions like these, where multiple agents influence each other's work or learning.

Siya, Presh, and Nene decide to help each other study during the final exams. Siya's favorite subject is Economics, Presh loves Statistics, and Nene knows his Information System (IT). Each studies his own subject as well as helps the others learn their subjects. After the finals, they realize that Siya spent 40% of his time studying his own subject Economics, 30% of his time helping Presh learn economics and 30% of the time helping Nene learn economics. Presh spent 30% of his time studying his own subject statistics, 30% of his time helping Siya learn Stats, and 40% of the time helping Nene learn Stats. Nene spent 20% of his time studying his own subject information System (IT), 40% of his time helping Siya learn information system, and 40% of the time helping Presh learn information system. 1. Determine the study/learning matrix 2. If they originally agreed that each should work about 33 hours, how long did each work?

Explore the study matrix and time allocation problem where Siya, Presh, and Nene share their study time across Economics, Statistics, and Information Systems. Learn how to build a study/learning matrix and calculate each person's total hours spent studying, based on a 33-hour agreement. Suitable for students in Grades 10-12.

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