an investment firm recommends that a client invest in bonds rated aaa, a, and b.  the average yield on aaa bonds is 4 %, on a bonds 6 %, and on b bonds 11 %.  the client wants to invest twice as much in aaa bonds as in b bonds.  how much should be invested in each type of bond under the following conditions? a. the total investment is $26,000 , and the investor wants an annual return of $1,620 on the three investments. b. the values in part a are changed to $38,000 and $2,360 , respectively a. the client should invest $ in aaa bonds, $ a bonds, and $ b bonds.

Part A.

1) Variables:

X: amount invested in AAA bonds

Y: amount invested in A bonds

Z: amount invested in B bonds.

2) Return:

4%* X + 6% * Y + 11% * Z = 1620

3) Investment:

 X + Y + Z = 26,000

4) Ratio between AAA and B bonds.

X = 2*Z

5) Solution of the system of equations:

Replace X with 2Z in the two first equations:

2Z * 4% + Y * 6% + Z * 11% = 1620

=> 0.08Z + 0.06Y + 0.11Z = 1620

=> 0.19Z + 0.06Y = 1620

2Z + Y + Z = 26,000

=> 3Z + Y = 26,000 => Y = 26,000 - 3Z

Replace Y with 26,000 - 3Z

0.19Z + 0.06(26,000 - 3Z) = 1620

0.19Z + 1560 - 0.18Z = 1620

0.01Z = 1620 - 1560

0.01Z = 60

Z = 60 / 0.01 = 6000

=> X = 2*6000 = 12,000

=> Y = 26,000 - 12,000 - 6,000 = 8,000

12,000 in bonds AAA, 8,000 in bonds A, and 6000 in bonds B.

Part B.

2) Return:

0.04X + 0.06 Y + 0.11Z = 2360

3) Investment:

 X + Y + Z = 38,000

4) Ratio between AAA and B bonds.

X = 2Z

5) Solution of the system of equations:

Replace X with 2Z in the two first equations:

=> 0.08Z + 0.06Y + 0.11Z = 2360

=> 0.19Z + 0.06Y = 2360

2Z + Y + Z = 38,000

=> 3Z + Y = 38,000 => Y = 38,000 - 3Z

Replace Y with 38,000 - 3Z

0.19Z + 0.06(38,000 - 3Z) = 2360

0.19Z + 2280 - 0.18Z = 2360

0.01Z = 2360 - 2280

0.01Z = 80

Z = 80 / 0.01 = 8000

=> X = 2*8000 = 16,000

=> Y = 38,000 - 16,000 - 8000 = 14,000

16,000 in AAA bonds, 14,000 in A bonds, and 8,000 in B  bonds.

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