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?
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This is best solved using matrices.
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Let b = amount invested in bonds
and c = amount invested in cds
and m = amount invested in mortages
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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
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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
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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
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Our equations are then:
   b +    c +    m = 100000
.08b + .07c + .10m = 7800
  0b +   1c -   3m = 0
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Our coefficient matrix is:
  1   1   1
.08 .07 .10
  0   1  -3
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The determinant of the coefficient matrix is:
-(.10 - .08) - 3(.07 - .08)
-(.02) - 3(-.01)
-.02 + .03
.01
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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
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The determinant of the "bond matrix" is:
-(10000 - 7800) - 3(7000 - 7800)
-(2200) - 3(-800)
-2200 + 2400
200
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b = "det of bond matrix"/"det of coefficients"
b = 200/.01
b = $20000
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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"
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Can you complete it from here?