A trust fund has Rs. 30000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30000 among the two types of bonds. If the trust fund must obtain an annual total interest of (i) Rs 1800 (ii) 2000.

To solve the problem of how to invest Rs. 30,000 in two types of bonds using matrix multiplication, we can set up a system of equations based on the interest rates and the total interest required.

Setting Up the Problem

Let:

• x = amount invested in the first bond (5% interest)
• y = amount invested in the second bond (7% interest)

The total investment can be expressed as:

x + y = 30000

Interest Equations

The total interest from both bonds can be represented as:

• For Rs. 1800 interest: 0.05x + 0.07y = 1800
• For Rs. 2000 interest: 0.05x + 0.07y = 2000

Matrix Representation

We can represent the system of equations in matrix form:

For Rs. 1800:

Matrix A:

1	1
0.05	0.07

And the corresponding matrix for the constants:

Matrix B:

30000

1800

Solving the System

Using matrix methods, we can solve for x and y:

For Rs. 1800:

1. From the first equation, y = 30000 - x

2. Substitute into the second equation:

0.05x + 0.07(30000 - x) = 1800

3. Simplifying gives:

0.05x + 2100 - 0.07x = 1800

-0.02x = -300

x = 15000

4. Thus, y = 30000 - 15000 = 15000

For Rs. 2000, repeat the process:

1. Substitute into the second equation:

0.05x + 0.07(30000 - x) = 2000

2. Simplifying gives:

0.05x + 2100 - 0.07x = 2000

-0.02x = -100

x = 5000

3. Thus, y = 30000 - 5000 = 25000

Final Investment Distribution

To achieve the desired interest:

• For Rs. 1800: Invest Rs. 15,000 in the first bond and Rs. 15,000 in the second bond.
• For Rs. 2000: Invest Rs. 5,000 in the first bond and Rs. 25,000 in the second bond.