question_answer

                      Mr. X has invested a part of his investment in 10% bond A and a part in 15% bond B. His interest income during first year is Rs. 4000. If he invests 20% more in 10% bond A and 10% more in 15% bond B, his income during second year increases by Rs. 500. Find the initial amount of investment in respective bonds, using matrix method. Is investment necessary a middle class?

Answer:

Let Rs. x and Rs. y be the initial investment in 10% bond A and 15% bond B, respectively. Then, according to the given condition, we have             \[(0.10)x+(0.15)y=4000\] and \[(0.10)\left( x+\frac{20}{100}x \right)+(0.15)\left( y+\frac{10}{100}y \right)=4500\] \[\Rightarrow \]   \[2x+3y=80000\]                          ? (i)                         [Multiplying both sides by 20] and \[(0.10)\,(120x)+(0.15)\,(110y)=450000\] and       \[8x+11y=300000\]                   ? (ii)                         [divide both sides by 1.5] Now, these equations can be written in matrix form as                     AX = B                         ? (iii) Where, \[A=\left[ \begin{matrix}    2 & 3  \\    8 & 11  \\ \end{matrix} \right],\] \[X=\left[ \begin{matrix}    x  \\    y  \\ \end{matrix} \right]\] and \[B=\left[ \begin{matrix}    80000  \\    300000  \\ \end{matrix} \right]\] Here, \[|A|\,\,=\left[ \begin{matrix}    2 & 3  \\    8 & 11  \\ \end{matrix} \right]=22-24=-\,2\ne 0\] \[\therefore \] \[{{A}^{-1}}\] exists. Now, \[{{A}^{-1}}=\frac{1}{|A|}\,\left[ \begin{matrix}    11 & -\,3  \\    -\,8 & 2  \\ \end{matrix} \right]\] \[\left[ \because \,i\text{f}\,A=\left[ \begin{matrix}    a & b  \\    c & d  \\ \end{matrix} \right]\,\,\text{then}\,{{A}^{-1}}=\frac{1}{|A|}\,\left[ \begin{matrix}    d & -b  \\    -c & a  \\ \end{matrix} \right] \right]\] \[=\frac{-1}{2}\,\left[ \begin{matrix}    11 & -\,3  \\    -\,8 & 2  \\ \end{matrix} \right]\] Now from Eq. (iii) we have \[X={{A}^{-1}}\,\,B=\frac{-1}{2}\left[ \begin{matrix}    11 & -\,3  \\    -\,8 & 2  \\ \end{matrix} \right]\,\,\left[ \begin{matrix}    80000  \\    300000  \\ \end{matrix} \right]\] \[=\frac{-1}{2}\left[ \begin{matrix}    880000-900000  \\    -\,640000+600000  \\ \end{matrix} \right]\] \[=\frac{-1}{2}\left[ \begin{matrix}    -\,20000  \\    -\,40000  \\ \end{matrix} \right]=\left[ \begin{matrix}    10000  \\    20000  \\ \end{matrix} \right]\] \[\Rightarrow \]   \[\left[ \begin{matrix}    x  \\    y  \\ \end{matrix} \right]=\left[ \begin{matrix}    10000  \\    20000  \\ \end{matrix} \right]\] \[\Rightarrow \]   \[x=10000\] and \[y=20000\]| Hence, Mr. X invested Rs. 10000 in bond A and Rs. 20000 in bond B.                         Value: Yes, for a person investment is very necessary because investing is very much important to secure his future life. To build wealth, a person has to invest money.