Question 1201131: Formulate a system of equations for the situation below and solve.
A private investment club has $400,000 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 rate of return of 15%/year; medium-risk stocks, 10%/year; and low-risk stocks, 6%/year. 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 $40,000/year on the total investment. (Assume that all the money available for investment is invested.)
high-risk stocks $
medium-risk stocks $
low-risk stocks $
Found 4 solutions by mananth, MathTherapy, ikleyn, greenestamps:
Answer by mananth(16949)   (Show Source):
You can put this solution on YOUR website! high-risk stocks $ amount invested be z ----15%
medium-risk stocks $ amount invested be y-----10%
low-risk stocks $ amount invested be x-------6%
Total investment $400,000
Condition
x = y+z...................(1)
x+y+z =400000..............(2)
6%x+10%y+15%z =40,000..........(3)
x+y+z =400000 (x=y+z)
x+x = 400000
x=200000
Therefore y+z =200 000....................(4)
6%x+10%y+15%z =40,000 multiply by 100
6x+10y+15z =4000000
6(y+z)+10y+15z = 4000000
16y+21z = 4000000................(5)
Solv (4) &(5)
5z =800000
z= 160000
y= 200000-160000 =40000
high-risk stocks $ amount invested 160,000
medium-risk stocks $ amount invested 40,000
low-risk stocks $ amount invested 200,000
Answer by MathTherapy(10682)  (Show Source):
You can put this solution on YOUR website!
Formulate a system of equations for the situation below and solve.
A private investment club has $400,000 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 rate of return of 15%/year; medium-risk stocks, 10%/year; and low-risk stocks, 6%/year. 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 $40,000/year on the total investment. (Assume that all the money available for investment is invested.)
high-risk stocks $
medium-risk stocks $
low-risk stocks $
Let high, medium, and low-risk amounts be H, M, and L, respectively. We then get the following equations:
H + M + L = 400,000 -- eq (i)
L = H + M____H + M - L = 0 -------- eq (ii)
.15H + .1M + .06L = 40,000 --- eq (iii)
2L = 400,000 ---- Subtracting eq (ii) from eq (i)
Amount invested in the low-risk stocks, or 
H + M = 200,000 ---- Substituting 200,000 for L in eq (ii)
M = 200,000 - H ----- eq (iv)
.15H + .1(200,000 - H) + .06(200,000) = 40,000 ----- Substituting 200,000 - H for M, and 200,000 for L, in eq (iii)
.15H + 20,000 - .1H + 12,000 = 40,000
.15H - .1H = 40,000 - 32,000
.05H = 8,000
Amount invested in the high-risk stocks, or 
You should now be able to determine the amount invested in the medium-risk by substituting 160,000 for H in eq (iv)
Answer by ikleyn(53541)  (Show Source):
You can put this solution on YOUR website! .
Formulate a system of equations for the situation below and solve.
A private investment club has $400,000 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 rate of return of 15%/year; medium-risk stocks, 10%/year; and low-risk stocks, 6%/year. 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 $40,000/year on the total investment. (Assume that all the money available for investment is invested.)
high-risk stocks $
medium-risk stocks $
low-risk stocks $
~~~~~~~~~~~~~~~~~~~~~~~~~~
Probably, you will be very amazed if I tell you that this problem can be easily solved
using only one unknown and one equation instead of using three unknowns and three equations.
But I will show it right now, in the next 3 minutes.
As the problem says, the total $400,000 is split in two equal parts: low-risk stock and the sum
the investments in the stocks of the other two categories.
So, $200,000 go to the low risk stock at 6%, and another $200,000 go to the other two categories
at 10% and 15%.
Let 'x' be the amount invested at 15%.
Then the amount invested at 10% is (200000-x) dollars.
Write the total interest equation
0.15x + 0.1*(200000-x) + 0.06*200000 = 40000.
Simplify and find x
0.15x + 20000 - 0.1x + 12000 = 40000
0.15x - 0.1x = 40000 - 20000 - 12000
0.05x = 8000
x = 8000/0.05 = 160000
Thus $160,000 invested at 15% (high-risk); $200,000 - $160,000 = $40,000 invested at 10% (medium-risk);
and $200,000 invested at 6% (low-risk).
At this point, the problem is solved completely using only one unknown and one equation.
It's immeasurably simpler, isn't it?
So, there are two ways to solve this problem: one way is to pretend that you are a stupid person
and follow blindly the problem's stupid instruction. Or learn the advanced way and reduce calculations.
Answer by greenestamps(13287)  (Show Source):
You can put this solution on YOUR website!
The problem requires that half of the $400,000 be invested in the low-risk stocks that yield a return of 6%. $200,000 invested at 6% yields .06($200000) = $12,000.
The total return required is $40,000, so the $200,000 invested in the stocks that have returns of 10% and 15% must yield a return of $40,000-$12,000 = $28,000.
A yield of $28,000 on an investment of $200,000 requires an average interest rate of $28,000/$200,000 = 0.14 = 14%.
Simple mental arithmetic shows that 14% is four-fifths of the way from 10% to 15%; that means 4/5 of the remaining $200,000, or $160,000, must be invested at the higher rate.
ANSWER:
$160,000 at 15%
$40,000 at 10%
$200,000 at 6%
CHECK: .15($160,000) + .10($40,000) + .06($200,000) = $24,000 + $4000 + $12,000 = $40,000
|
|
|
