The club should invest $1450 in high-risk stocks, $950 in medium-risk stocks, and $600 in low-risk stocks to achieve the investment goal of a return of $363 per year.
How to determine how much to invest
Denote the amount invested in high-risk stocks as H, in medium-risk stocks as M, and in low-risk stocks as L.
Given:
Total investment amount: $3000
Rate of return for high-risk stocks: 18%
Rate of return for medium-risk stocks: 9%
Rate of return for low-risk stocks: 6%
Investment goal: Return of $363/year
Based on the given information, set up the following equations:
Equation 1: H + M + L = $3000 (Total investment amount)
Equation 2: 0.18H + 0.09M + 0.06L = $363 (Return on investment goal)
Additionally, we are given that the investment in low-risk stocks should be a quarter of the sum of the investments in the other two categories. This can be represented by the equation:
Equation 3: L = 0.25(H + M)
Now solve the system of equations to find the values of H, M, and L.
Substitute Equation 3 into Equation 1
H + M + 0.25(H + M) = $3000
1.25H + 1.25M = $3000
Divide both sides of the equation by 1.25, we have:
H + M = $2400
Next, substitute this value into Equation 2:
0.18H + 0.09M + 0.06L = $363
0.18H + 0.09M + 0.06(0.25(H + M)) = $363
0.18H + 0.09M + 0.015H + 0.015M = $363
0.195H + 0.105M = $363
Now we have a system of two linear equations:
H + M = $2400
0.195H + 0.105M = $363
Solving this system of equations will give us the values of H and M.
Multiply the first equation by -0.105 and the second equation by 1.25:
-0.105H - 0.105M = -$252
0.244H + 0.13125M = $453.75
Add the two equations together
0.139H = $201.75
Divide both sides by 0.139
H = $1450
Substitute this value back into the first equation, we can solve for M:
$1450 + M = $2400
M = $950
Now, find L by substituting the values of H and M into Equation 3:
L = 0.25(H + M)
L = 0.25($1450 + $950)
L = $600
Therefore, the club should invest $1450 in high-risk stocks, $950 in medium-risk stocks, and $600 in low-risk stocks to achieve the investment goal of a return of $363 per year.