BACKGROUND
Three teachers at Nyamatikiti High School bought two different kinds of fruits at the market.

When they got to school, they could not remember the individual prices per kilogram, but fortunately, two of the teachers could remember the total amount that they each paid. Taking advantage of the knowledge you acquired on simultaneous equations and matrices, you volunteer to calculate the price per kilogram for each fruit.

The CALA consists of two parts, PART A and PART B. You are required to respond to both parts.

PART A
The weights of the fruits and the total amounts paid are shown in the following table. (Copy the table)

\begin{tabular}{|l|l|l|l|}
\hline & Mrs Ndoro & Mrs Kori & Miss Ndlovu \\
\hline Bananas (kg) & 2 & 0.2 & 0.8 \\
Apples (kg) & 0.8 & 0 & 6 \\
\hline Total Paid (\$) & 40 & 20 & \\
\hline
\end{tabular}

Assuming that for each variety of fruit, the price per kilogram paid by each of the teachers is the same:
(a) Copy and complete the table.
(b) Define the variables.
(c) Form separate equations using the information from each column.

PART B
Solve for the unknown variables using the matrix method by:
(a) Pulling out the matrix.
(b) Finding the determinant of the drawn matrix.



Answer :

Let's break down the entire problem step by step, translating the information into a structured mathematical solution.


## PART A

### Given Data Table

Given the data table, it looks something like this:

[tex]\[ \begin{array}{|l|c|c|c|} \hline & \text{Mrs Ndoro} & \text{Mrs Kori} & \text{Miss Ndlovu} \\ \hline \text{Bananas (kg)} & 2 & 0.2 & 0.8 \\ \text{Apples (kg)} & 0.8 & 1 & 6 \\ \hline \text{Total Paid (\$)} & 40 & 20 & T \\ \hline \end{array} \][/tex]

### (a) Copy and Complete the Table

We are given that [tex]\( T \)[/tex] is the total amount paid by Miss Ndlovu, which we will calculate later.

### (b) Define the Variables

Let's define our variables:

- Let [tex]\( x \)[/tex] be the price per kilogram (kg) of bananas.
- Let [tex]\( y \)[/tex] be the price per kilogram (kg) of apples.

### (c) Form Separate Equations

Given the weights and total amounts paid by Mrs Ndoro, Mrs Kori, and Miss Ndlovu, we can form the following equations:

1. For Mrs Ndoro:
[tex]\[ 2x + 0.8y = 40 \][/tex]

2. For Mrs Kori:
[tex]\[ 0.2x + y = 20 \][/tex]

3. For Miss Ndlovu:
[tex]\[ 0.8x + 6y = T \][/tex]

## PART B

### (a) Pulling Out the Matrix

The system of equations can be represented in matrix form. The coefficient matrix (let's call it [tex]\( A \)[/tex]) and the constant matrix (let's call it [tex]\( B \)[/tex]) are extracted from the first two equations.

Matrix [tex]\( A \)[/tex]:
[tex]\[ A = \begin{pmatrix} 2 & 0.8 \\ 0.2 & 1 \\ 0.8 & 6 \end{pmatrix} \][/tex]

Matrix [tex]\( B \)[/tex]:
[tex]\[ B = \begin{pmatrix} 40 \\ 20 \\ T \end{pmatrix} \][/tex]

### Solving the System and Finding Determinant

To find the price per kilogram of bananas and apples ([tex]\( x \)[/tex] and [tex]\( y \)[/tex]), we take the coefficient matrix from the first two equations and solve using the matrix method:

Sub-matrix [tex]\( A_{subst} \)[/tex]:
[tex]\[ A_{subst} = \begin{pmatrix} 2 & 0.8 \\ 0.2 & 1 \end{pmatrix} \][/tex]

Sub-matrix [tex]\( B_{subst} \)[/tex]:
[tex]\[ B_{subst} = \begin{pmatrix} 40 \\ 20 \end{pmatrix} \][/tex]

Solving for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] using matrix methods:
[tex]\[ \begin{bmatrix} x \\ y \end{bmatrix} = A^{-1}_{subst} B_{subst} \][/tex]

The solutions are:
- Price per kg of bananas ([tex]\( x \)[/tex]) = 13.043478260869563
- Price per kg of apples ([tex]\( y \)[/tex]) = 17.39130434782609

So, [tex]\( x \approx 13.04 \)[/tex] and [tex]\( y \approx 17.39 \)[/tex].

### Finding Total Paid by Miss Ndlovu

Using the obtained values [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the third equation:
[tex]\[ 0.8 x + 6 y = T \][/tex]

Substituting:
[tex]\[ 0.8 \times 13.04 + 6 \times 17.39 = T \][/tex]

This yields the total amount paid by Miss Ndlovu:
[tex]\[ T \approx 114.78 \][/tex]

### (b) Finding the Determinant of the Drawn Matrix

To find the determinant of the sub-matrix [tex]\( A_{subst} \)[/tex]:
[tex]\[ \text{det}\left( A_{subst} \right) = \left| \begin{matrix} 2 & 0.8 \\ 0.2 & 1 \end{matrix} \right| \][/tex]
[tex]\[ \text{det}\left( A_{subst} \right) = (2 \times 1) - (0.8 \times 0.2) \][/tex]
[tex]\[ \text{det}\left( A_{subst} \right) = 2 - 0.16 = 1.84 \][/tex]

So, the determinant of the sub-matrix is [tex]\( 1.84 \)[/tex].

## Summary

- Price per kg of bananas: [tex]\( 13.04 \)[/tex]
- Price per kg of apples: [tex]\( 17.39 \)[/tex]
- Total amount paid by Miss Ndlovu: [tex]\( 114.78 \)[/tex]
- Determinant of sub-matrix: [tex]\( 1.84 \)[/tex]

This solves the problem using matrices as required.