Question

A trust fund has Rs. 30,000 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. 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of
A) Rs. 1800 . B) Rs. 2000

Answer 1

It is given that $Rs.30,000$ must be invested into two types of bonds with $5\%$ and $7\%$ interest rates.

Let $Rs.x$ be invested in bonds of the first type.

Thus, $Rs.(30000-x)$ will be invested in the other type.

Hence, the amount invested in each type of the bonds can be represented in matrix form with each column corresponding to a different type of bond as :

$X=\left[\begin{array}{cc}x& 30000-x\end{array}\right]$

$A)$ Annual interest obtained is $Rs.1800.$

We know, $Interest=\frac{PTR}{100}$

Here, the time is one year and thus $T=1$

Hence, the interest obtained after one year can be expressed in matrix representation as -

$\left[\begin{array}{cc}x& 30000-x\end{array}\right]\left[\begin{array}{c}\frac{5}{100}\\ \frac{7}{100}\end{array}\right]=\left[1800\right]$

$\Rightarrow$ $[x\times \frac{5}{100}+(30000-x)\times \frac{7}{100}]=\left[1800\right]$

$\Rightarrow$ $\frac{5x}{100}+\frac{7(30000-x)}{100}=1800$

$\Rightarrow$ $5x+210000-7x=180000$

$\Rightarrow$ $-2x=-30000$

$\therefore$ $x=15000$

Amount invested in the first bond $=x=Rs.15000$

$\Rightarrow$ Amount invested second bond $=Rx(30000-x)=Rs.(30000-15000)=Rs.15000$

$\therefore$ The trust has to invest $Rs.15000$ each in both the bonds in order to obtain an annual interest of $Rs.1800$

$B)$ Annual interest obtained is $Rs.2000.$

Hence, the interest obtained after one year can be expressed in matrix representation as -

$\left[\begin{array}{cc}x& 30000-x\end{array}\right]\left[\begin{array}{c}\frac{5}{100}\\ \frac{7}{100}\end{array}\right]=\left[2000\right]$

$\Rightarrow$ $[x\times \frac{5}{100}+(30000-x)\times \frac{7}{100}]=\left[2000\right]$

$\Rightarrow$ $\frac{5x}{100}+\frac{7(30000-x)}{100}=2000$

$\Rightarrow$ $5x+210000-7x=200000$

$\Rightarrow$ $-2x=-10000$

$\therefore$ $x=5000$

Amount invested in the first bond $=x=Rs.5000$

$\Rightarrow$ Amount invested second bond $=Rx(30000-x)=Rs.(30000-5000)=Rs.25000$

$\therefore$ The trust has to invest $Rs.5000$ in the first bond and $Rs.25000$ in the second bond in order to obtain an annual interest of $Rs.2000$