Question 1009023: Mark has $100,000 to invest. His financial consultant advises him to diversify his investment in three types of bonds: short-term, intermediate-term, and long-term. The short-term bonds pay 5%, the intermediate-term bonds pay 6%, and the long-term bonds pay 7% simple interest per year. Mark wishes to realize a total annual income of 5.95%, with equal amounts invested in short- and intermediate-term bonds. How much should he invest in each type of bond?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! x = short range
y = intermediate range
z = long range.
x + y + z = 100,000
.05x + .06y + .07z = .0595 * 100,000 = 5950
x = y
replace y with x to get:
2x + z = 100,000
.11x + .07z = 5950
note that .11x is equivalent to .05*x + .06*x.
multiply both sides of the second equation by 100 and multiply both sides of the first equation by 7 to get:
14x + 7z = 700,000
11x + 7z = 595,000
subtract second equation from the first to get:
3x = 105,000
solve for x to get:
x = 35,000
in the first original equation of 2x + z = 100,000, replace x with 35,000 to get 2 * 35,000 + z = 100,000.
this becomes 70,000 + z = 100,000.
solve for z to get z = 30,000
in the second original equation of .11x + .07z = 5950, replace x with 35,000 and replace z with 30,000 to get .11 * 35,000 + .07 * 30,000 = 5950.
simplify to get 5950 = 5950.
this confirms the solution is correct.
.11 * x is equivalent to .05 * x + .06 * x
since y = x, this is equivalent to .05 * x + .06 * y
both x and y are equal to 35,000, so this is equivalent to .05 * 35,000 + .06 * 35,000.
you have:
.05 * 35,000 dollars in short term bonds + .06 * 35,000 in intermediate term bonds + .07 * 30,000 dollars in long term bonds.
the total money invested is 100,000.
the total interest earned is 5,950
5,950 / 100,000 = .0595
.0595 * 100 = 5.95%.
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