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
$
  {\text{A}}{\text{. 1800}} \\
  {\text{B}}{\text{. 2000}} \\
 $

Answer 1

 Hint:-In this question first we have to have the amount invested in bonds of first type be Rs. $x$. Thus, Rs.(30000-$x$) will be invested in the second type of bond. Then , we represent it in the form of a matrix. And then solve the respective questions using matrix multiplication method.
Complete step-by-step solution -
Given, Rs 30,000 must be invested into two types of bonds with 5% and 7% interest rates.
Let Rs. $x$ invested in bonds of the first type.
Thus, Rs.(30,000-$x$) will be invested in the other type.
Hence, the amount invested in each type of bonds can be represented in matrix form with each column corresponding to a different type of bond as:
$X = [x{\text{ }}30000 - x]$

Rs. 1800
Annual interest obtained is Rs. 1800
We know
Interest=$\dfrac{{Principle \times Time \times Rate}}{{100}}$
Here, the time is one year
Thus $Time = 1$
Hence, the interest obtained after one year can be expressed in matrix representation as-
$ \Rightarrow \left[ {x{\text{ 30,000 - }}x} \right]\left[ \dfrac{5}{{100}} {\text{ }} \dfrac{7}{{100}} \right] = \left[ {1800} \right]$
Now, with the help of matrix multiplication we can write the above form as
$ \Rightarrow \left[ {x \times \dfrac{5}{{100}}{\text{ + (30000 - }}x) \times \dfrac{7}{{100}}} \right] = [1800]$
On opening the above matrix form into equation form, we get

 $ \Rightarrow \dfrac{{5x}}{{100}} + \dfrac{{7(30000 - x)}}{{100}} = 1800 \\
   \Rightarrow 5x + 210000 - 7x = 180000 \\
   \Rightarrow - 2x = - 30000 \\
   \Rightarrow x = 15000 \\ $

Amount in first bond=$x$=Rs.15000
$ \Rightarrow $Amount invested second bond=\[Rs.(30000 - x)\]
                                                             =$Rs.(30000 - 15000)$
                                                          =\[Rs.15000\]
Therefore, the trust has to invest Rs. 15000 each in both the bonds in order to obtain an annual interest of Rs.1800

Rs.2000
Annual interest obtained is Rs. 2000
Hence, the interest obtained after one year can be expressed in matrix representation as-
$ \Rightarrow \left[ {x{\text{ 30000 - }}x} \right] [ \dfrac{5}{{100}} \dfrac{7}{{100}} ] = \left[ {2000} \right] $

Now, with the help of matrix multiplication we can write the above form as
$ \Rightarrow \left[ {x \times \dfrac{5}{{100}} + ({\text{30000 - }}x) \times \dfrac{7}{{100}}} \right] = \left[ {2000} \right]$

On converting above matrix form into equation form
We get
$
   \Rightarrow \dfrac{{5x}}{{100}} + \dfrac{{7(30000 - x)}}{{100}} = 2000 \\
   \Rightarrow 5x + 210000 - 7x = 200000 \\
   \Rightarrow - 2x = - 10000 \\
   \Rightarrow x = 5000 \\
$
Therefore invested amount in the first bond = $x = Rs.5000$
$ \Rightarrow $Amount invested in the second bond=$Rs.(30000 - x)$
                                                                       =$Rs.(30000 - 5000)$
                                                                       =$Rs.25000$
Therefore, 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

Note:- Whenever you get this type of question the key concept to solve is to let one of things be $x$ and form a matrix . And remember one thing more than matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication the number of columns in the first matrix must be equal to the number of rows in the second matrix.