Let's assume the amounts of the three parts of the investment are x, y, and z.
According to the given information, the investment was split into three parts, so we know that x + y + z = 72,000 (equation 1).
The first part of the investment earned 8% interest, which can be calculated by multiplying the amount of the first part (x) by 0.08. Therefore, the interest from the first investment is 0.08x.
The second part of the investment earned 6% interest, which can be calculated by multiplying the amount of the second part (y) by 0.06. Therefore, the interest from the second investment is 0.06y.
The third part of the investment earned 9% interest, which can be calculated by multiplying the amount of the third part (z) by 0.09. Therefore, the interest from the third investment is 0.09z.
The total interest from the investments is given as $5640, so we have the equation: 0.08x + 0.06y + 0.09z = 5640 (equation 2).
Additionally, we are given that the interest from the first investment (0.08x) is 3 times the interest from the second investment (0.06y). So we can express this relationship as: 0.08x = 3 * 0.06y.
Now we can solve the equations to find the values of x, y, and z:
From equation 1, we have: x + y + z = 72,000
Simplifying equation 2 using the relationship between x and y:
0.08x = 3 * 0.06y
0.08x = 0.18y
Now we can substitute 0.18y for 0.08x in equation 2:
0.18y + 0.06y + 0.09z = 5640
0.24y + 0.09z = 5640
We can multiply this equation by 100 to eliminate decimals:
24y + 9z = 564,000
Since we have two equations and two unknowns (x and y), we can solve the system of equations:
x + y + z = 72,000
24y + 9z = 564,000
From the first equation, we can rewrite it as: x = 72,000 - y - z.
Substituting this into the second equation, we have:
24y + 9z = 564,000
24(72,000 - y - z) + 9z = 564,000
1,728,000 - 24y - 24z + 9z = 564,000
-24y - 15z = -1,164,000
24y + 15z = 1,164,000 (multiplied by -1)
Now we can solve the system of equations:
24y + 15z = 1,164,000
24y + 9z = 564,000
By subtracting the second equation from the first equation, we eliminate y:
(24y + 15z) - (24y + 9z) = 1,164,000 - 564,000
15z - 9z = 600,000
6z = 600,000
z = 600,000 / 6
z = 100,000
Substituting the value of z back into the equation 24y + 9z = 564,000:
24y + 9(100,000) = 564,000
24y + 900,000 = 564,000
24y = 564,000 - 900,000
24y = -336,000
y = -336,000 / 24
y = -14,000
Now we can substitute the values of x = 72,000 - y - z:
x = 72,000 - (-14,000) - 100,000
x = 72,000 + 14,000 - 100,000
x = -14,000
The three parts of the investment are x = -14,000, y = -14,000, and z = 100,000.
Please note that it is unusual to have negative amounts for investments, so please double-check the problem statement or calculations to ensure accuracy.