SOLUTION: A bank wishes to invest a $100,000 trust fund in three sources: bonds paying an 8% annual return, certificates of deposit paying 7%, and mortgages paying 10%. The bank wishes to

Question 145884: A bank wishes to invest a $100,000 trust fund in three sources: bonds paying an 8% annual return, certificates of deposit paying 7%, and mortgages paying 10%. The bank wishes to realize an $7800 annual return from the investment. A condition of the trust is that three times as much money must be invested in CDs as in mortgages. How much should the bank invest in each category?
Answer by nerdybill(7384) (Show Source): You can put this solution on YOUR website!
A bank wishes to invest a $100,000 trust fund in three sources: bonds paying an 8% annual return, certificates of deposit paying 7%, and mortgages paying 10%. The bank wishes to realize an $7800 annual return from the investment. A condition of the trust is that three times as much money must be invested in CDs as in mortgages. How much should the bank invest in each category?
.
This is best solved using matrices.
.
Let b = amount invested in bonds
and c = amount invested in cds
and m = amount invested in mortages
.
Since we have 3 unknowns, we'll need to find 3 equations:
Eq 1 is from the fact that "A bank wishes to invest a $100,000":
b + c + m = 100000
.
Eq 2 is from "bonds paying an 8% annual return, certificates of deposit paying 7%, and mortgages paying 10%. The bank wishes to realize an $7800 annual return from the investment.":
.08b + .07c + .10m = 7800
.
Eq 3 is from "three times as much money must be invested in CDs as in mortgages.":
c = 3m
we can rewrite the above as:
0b + 1c - 3m = 0
.
Our equations are then:
b + c + m = 100000
.08b + .07c + .10m = 7800
0b + 1c - 3m = 0
.
Our coefficient matrix is:
1 1 1
.08 .07 .10
0 1 -3
.
The determinant of the coefficient matrix is:
-(.10 - .08) - 3(.07 - .08)
-(.02) - 3(-.01)
-.02 + .03
.01
.
For the bond matrix, we replace the coefficients of the first column and replace it with the numbers on the right of the equal sign. Thus:
100000 1 1
7800 .07 .10
0 1 -3
.
The determinant of the "bond matrix" is:
-(10000 - 7800) - 3(7000 - 7800)
-(2200) - 3(-800)
-2200 + 2400
200
.
b = "det of bond matrix"/"det of coefficients"
b = 200/.01
b = $20000
.
To find the other two amounts, it would be very similar -- that is:
c = "det of cd matrix"/"det of coefficients"
m = "det of mortgage matrix"/"det of coefficients"
.
Can you complete it from here?

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