Question 1120364: An investor has $39,000 to invest in three types of bonds: short-term, intermediate-term, and long-term. How much should she invest in each type to satisfy the given conditions?
Short-term bonds pay 4% annually, intermediate-term bonds pay 5%, and long-term bonds pay 6%. The investor wishes to realize a total annual income of 5.00%, with equal amounts invested in short- and intermediate-term bonds.
solve
short-term bonds = $
intermediate-term bonds = $
long-term bonds= $
Answer by greenestamps(13203)   (Show Source):
You can put this solution on YOUR website!
Let x be the amount invested in each of the short- and intermediate-term bonds. Then the amount invested in the long-term bonds is (39000-2x). To realize an average of 5% income on the investments, the equation is
.04(x)+.05(x)+.06(39000-2x) = .05(39000)
We could solve that equation using formal algebra; however, a little logical analysis will give us the answer almost immediately.
(1) The percent return on the intermediate-term bonds is the same as the desired average percent return.
(2) That means the investments in the long-term bonds at 6% and the short-term bonds at 4% must average out to a 5% return.
(3) But 5% is the average of 4% and 6%; that means the amounts invested in short- and long-term bonds must be equal.
(4) But the amounts invested in short- and intermediate-term bonds are the same; therefore the amounts invested in all three types of bonds must be the same.
So 1/3 of the total $39000, $13000, needs to be invested in each type of bond.
If we continue with the formal algebra, we get that result:
4x + 5x + 6(39000-2x) = 5(39000)
9x + 234000-12x = 195000
39000 = 3x
x = 13000
The amount to be invested in each of short- and intermediate-term bonds is x = $13000; the amount to be invested in long-term bonds is $39000-2x = $39000-$26000 = $13000.
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