Question 1179825: A private investment club has $500,000 earmarked for investment in stocks. To arrive at an acceptable overall level of risk, the stocks that management is considering have been classified into three categories: high-risk, medium-risk, and low-risk. Management estimates that high-risk stocks will have a return rate of 15%; medium-risk stocks, 10%; and low-risk stocks, 6%. The members have decided that the investment in low-risk stocks should be equal to the sum of the investments in the stocks of the other two categories. Determine how much the club should invest in each type of stock if the investment goal is to have a return of $50,000 on the total investment. (Assume that all the money available for investment is invested. Let x, y, and z denote the amount, in dollars, invested in high-, medium-, and low-risk stocks, respectively.)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! x + y + z = 500,000
x = high risk investment
y = medium risk investment
z = low risk investment
high risk gets 15%
medium risk gets 10%
low risk gets 6%
low risk investment equals sum of the other two investments.
z = x + y
goal is to get a return of 50,000 on the total investment.
.15x + .10y + .06z = 50,000
your two equations that need to be solved simultaneously are:
x + y + z = 500,000
.15x + .10y + .06z = 50,000
since z = x + y, then replace z with x + y in both equations to get:
x + y + x + y = 500,000
.15x + .10y + .06 * (x + y) = 50,000
combine like terms in the first equation and simplify the second equation to get:
2x + 2y = 500,000
.15x + .10y + .06x + .06y = 50,000
leave the first equation as is and combine like terms in the second equation to get:
2x + 2y = 500,000
.21x + .16y = 50,000
multiply both sides of the first equation by .08 and leave the second equation as is to get:
.16x + .16y = 40,000
.21x + .16y = 50,000
subtract the first equation from the second to get:
.05x = 10,000
solve for x to get:
x = 10,000 / .05 = 200,000
replace x with 200,000 in the first equation to get:
.16x + .16y = 40,000 becomes:
.16 * 200,000 + .16 * y = 40,000
simplify to get:
32000 + .16 * y = 40,000
subtract 32000 from both sides to get:
.16 * y = 8000
solve for y to get:
y = 8000 / .16 = 50,000
you have:
x = 200,000
y = 50,000
go back to your original equations and replace x and y with their respective values to get:
x + y + z = 500,000 becomes 200,000 + 50,000 + z = 500,000 which becomes 250,000 + z = 500,000.
solve for z to get:
z = 250,000
.15x + .10y + .06z = 50,000 becomes .15 * 200,000 + .10 * 50,000 + .06 * z = 50,000 which becomes 30,000 + 5,000 + .06 * z = 50,000 which becomes 35000 + .06 * z = 50,000.
subtract 35,000 from both sides to get:
.06 * z = 50,000 - 35,000 = 15,000.
solve for z to get:
z = 15,000 / .06 = 250,000
you now have;
x = 200,000
y = 50,000
z = 250,000
.15 * x = 30,000
.10 * y = 5,000
.06 * z = 15,000
z = x + y so that part is good.
your solution is:
amount invested in high risk is 200,000.
amount invested in medium risk is 50,000
amount invested in low risk is 250,000
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