An investment of $180,000 was made by a business club. The investment was split into three parts and lasted for one year. The first part of the investment earned 8% interest, the second 6%, and the third 9%. Total interest from the investments was $8520. The interest from the first investment was 6 times the interest from the second. Find the amounts of the three parts of the investment.
If the amounts invested are x,y,z then they have told us that
x+y+z = 180000
.08x + .06y + .09z = 8520
.08x = 6 * .06y
I suspect a typo, since even if the whole amount were invested at 6%, the interest would be 10,800. In other words, there's no way to solve these equations and get positive amounts.
To solve this problem, we can break it down into steps:
Step 1: Define the variables
Let's define the three parts of the investment as x, y, and z.
Step 2: Set up the equations
We are given the following information:
- The total investment amount is $180,000, so we have the equation: x + y + z = 180,000.
- The total interest earned is $8,520, so we have the equation: 0.08x + 0.06y + 0.09z = 8,520.
- The interest from the first investment (0.08x) is 6 times the interest from the second investment (0.06y), so we have the equation: 0.08x = 6 * 0.06y.
Step 3: Solve the equations
Let's rearrange the third equation and solve for x in terms of y:
0.08x = 6 * 0.06y
0.08x = 0.36y
x = 0.36y / 0.08
x = 4.5y
Now substitute this value of x into the first equation:
4.5y + y + z = 180,000
5.5y + z = 180,000
Substitute the values of x and z into the second equation:
0.08x + 0.06y + 0.09z = 8,520
0.08(4.5y) + 0.06y + 0.09z = 8,520
0.36y + 0.06y + 0.09z = 8,520
0.42y + 0.09z = 8,520
Step 4: Apply substitution method to solve the system of equations
Now we have two equations:
Equation 1: 5.5y + z = 180,000
Equation 2: 0.42y + 0.09z = 8,520
We can solve this system of equations to find the values of y and z.
Multiply Equation 1 by -0.09 to match the coefficients of z:
-0.495y - 0.09z = -16,200
Add Equation 2 and the modified Equation 1:
-0.495y - 0.09z + 0.42y + 0.09z = -16,200 + 8,520
-0.075y = -7,680
Solve for y:
y = -7,680 / (-0.075)
y = 102,400
Substitute the value of y back into Equation 1 to solve for z:
5.5(102,400) + z = 180,000
563,200 + z = 180,000
z = 180,000 - 563,200
z = -383,200
Step 5: Interpret the solution
Since the investment amounts cannot be negative, the value of z is not valid. This means there is no solution to the problem.