Question

A trust fund had ₹ 50000 that is to be invested into two types of bonds. First bond pays 5% interest per year and the second bond pays 6% interest per year. Using matrix multiplication determines how to divide ₹ 50000 among the two types of bonds so as to obtain an annual total interest of ₹ 2780. Find the difference of the two amounts in ₹.

Answer 1

Hint: Convert word problem into mathematical equation. Don’t focus on the complete question at the same time. Form equations by reading the question in parts.

Complete Step by step solution:
Let $x$ be invested in the first trust fund and $y$ be invested in the second trust fund then,
$x + y = 5000$
$y = 5000 - x.$
Now, matrix A represents the amount invested in the two trust funds.
Then $A = \left[ {\begin{array}{*{20}{c}}
  x&y
\end{array}} \right]$
$A = \left[ {\begin{array}{*{20}{c}}
  x&{50000 - x}
\end{array}} \right]$
Now, the interest for the first type of bond is $5\% $ i.e. $\dfrac{5}{{100}}$
And the interest for the second type of bond is $6\% $i.e. $\dfrac{6}{{100}}.$
Let Matrix B represents the interest on both the bonds, then $B = \left[
  \dfrac{5}{{100}}
  \dfrac{6}{{100}}
  \right]$
Then $AB = \left[ {\begin{array}{*{20}{c}}
  x&{50000 - x}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  {\dfrac{5}{{100}}} \\
  {\dfrac{6}{{100}}}
\end{array}} \right]$
$ \Rightarrow AB = \left[ {\dfrac{{5x}}{{100}} + \left( {5000 - x} \right) \times \dfrac{6}{{100}}} \right]$
$ = \left[ {\dfrac{{5x}}{{100}} + \dfrac{{300000}}{{100}} - \dfrac{{6x}}{{100}}} \right]$
$AB = \left[ {3000 - \dfrac{x}{{100}}} \right]$
This represents the interest gained by the two bonds.
It is given that the interest obtained is ₹ 2780.
$\therefore AB = \left[ {3000 - \dfrac{x}{{100}}} \right] = [2780]$
If two matrices are equal, then the element corresponding elements of the matrices will also be equal.
$3000 - \dfrac{x}{{100}} = 2780$
$3000 - 2780 = \dfrac{x}{{100}}$
$220 = \dfrac{x}{{100}}$
Rearranging it, we get
$x = 220 \times 100$
$ \Rightarrow x = $₹$22000$
$y = 50000 - 22000$
$ \Rightarrow y = $₹ $28000$
Thus the trust fund needs to invest ₹ $22000$in the first bond and ₹ $28000$ in the second bond.
The difference, between the amount invested is $28000 - 22000$
$ = $₹ $6000$

Note: This question can also be solved by writing A as column matrix and B as a row matrix. But we cannot solve it by writing both A and B in column or row matrix as in that case, the matrix multiplication will not exist.