A trust invested some money in two type of bonds. The first bond pays 10% interest and second bond pays 12% interest. The trust received Rs 2,800 as interest. However, if trust had interchanged money in bonds, they would have got Rs 100 less as interest. Using matrix method, find the amount invested by the trust. Interest received on this amount will be given to Helpage India as donation. Which value is reflected in this question?
Solution
Let Rs x be invested in the first bond and Rs y be invested in the second bond.
Let A be the investment matrix and B be the interest per rupee matrix. Then,
`A=[[x, y]] and B=[[10/100],[12/100]]`
Total annual interest `=AB=[x y][[10/100],[12/100]]=(10x)/100+(12y)/100`
`(10x)/100+(12y)/100=2800`
`10x+12y=280000 ...................(1)`
If the rates of interest had been interchanged, then the total interest earned is Rs 100 less than the previous interest.
`(12x)/100+(12y)/100=`
`270012x+10y=270000 ...................(2)`
The system of equations (1) and (2) can be expressed as
`PX = Q, where P=[[10,12],[12,10]], x=[[x],[y]], Q=[[280000],[270000]]`
`P=|[10,12],[12,10]|=100-144=-44 ne 0`
Thus, P is invertible.
`therefore X=p^-1 Q`
`X=(adjP)/|P| Q`
`[[x],[y]]=1/(-44)|[10,-12],[-12,10]|^T[[280000],[270000]]`
`[[x],[y]]=1/(-44)|[10,-12],[-12,10]|[[280000],[270000]]`
`[[x],[y]]=[[(280000-3240000)/-44],[(-3360000+270000)/-44]]=[[10000],[15000]]`
x=10000 and y=15000
Therefore, Rs 10,000 be invested in the first bond and Rs 15,000 be invested in the second bond.
Thus, the total amount invested by the trust is Rs 10,000 + Rs 15,000 = Rs 25,000.
The interest received will be given to Helpage India as donation reflects the helping and caring nature of the trust.
