An amount of 65,000 is invested in three investments at rates of 6%, 8% and 9% per

annum respectively. The total annual income is 4800. The income from the third

investment is 600 more than the income from the second investment. Using matrix

algebra, determine the amount ofeach investment.

Let's denote the amounts invested in the three investments as $x$, $y$, and $z$ respectively. The problem can be formulated using the following three equations:

1. **Total Investment**: 
   \[
   x + y + z = 65000
   \]

2. **Total Annual Income**: 
   \[
   0.06x + 0.08y + 0.09z = 4800
   \]

3. **Income Relationship**: 
   \[
   0.09z = 0.08y + 600
   \]

### Step 1: Set up the system of equations

From the third equation, we can rearrange it to express $z$ in terms of $y$:
\[
0.09z - 0.08y = 600
\]

Thus, the system of equations is:

\[
\begin{aligned}
x + y + z &= 65000 \quad \text{(Equation 1)}\\
0.06x + 0.08y + 0.09z &= 4800 \quad \text{(Equation 2)}\\
-0.08y + 0.09z &= 600 \quad \text{(Equation 3)}
\end{aligned}
\]

### Step 2: Convert to matrix form

The system can be written in matrix form as:
\[
\begin{pmatrix}
1 & 1 & 1 \\
0.06 & 0.08 & 0.09 \\
0 & -0.08 & 0.09 
\end{pmatrix}
\begin{pmatrix}
x \\
y \\
z 
\end{pmatrix}
=
\begin{pmatrix}
65000 \\
4800 \\
600
\end{pmatrix}
\]

### Step 3: Solve the system using matrix algebra

Let $ A $ be the coefficient matrix, $ \mathbf{x} $ the vector of unknowns, and $ \mathbf{b} $ the constant vector:
\[
A = \begin{pmatrix}
1 & 1 & 1 \\
0.06 & 0.08 & 0.09 \\
0 & -0.08 & 0.09 
\end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 65000 \\ 4800 \\ 600 \end{pmatrix}
\]

To solve for $ \mathbf{x} $, we can compute $ \mathbf{x} = A^{-1} \mathbf{b} $, where $ A^{-1} $ is the inverse of matrix $ A $.

### Step 4: Calculation

I will now calculate the solution using matrix algebra.The amounts invested in each investment are:

- **Investment 1** (at 6%): \$30,000
- **Investment 2** (at 8%): \$15,000
- **Investment 3** (at 9%): \$20,000

These amounts satisfy the conditions provided in the problem.

Would you like further details or have any questions about the solution? 

Here are 5 related questions to explore:

1. How do you calculate the total income from all investments using the individual rates?
2. What is the impact of changing the interest rates on the distribution of investments?
3. How does the income relationship between the second and third investments affect the solution?
4. How can the solution be verified using substitution into the original equations?
5. What if the total investment was different—how would that change the matrix?

**Tip:** When solving systems of equations, matrix algebra provides a powerful and systematic approach, especially for complex systems.

Learn how to solve an investment allocation problem using matrix algebra. Find out the amounts invested in three different investments at 6%, 8%, and 9% per annum given a total investment of $65,000 and an annual income of $4,800. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation