An investment of $93,000 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 earned 6% and the third 9%. total interest from the investments was $7110. The interest from the first investment was 3 times the interest from the second. Find the amounts of the three parts of the investment??
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To solve this problem, let's break it down into steps:
Step 1: Understand the problem.
We are given that the total investment is $93,000, split into three parts, each earning different interest rates. The total interest earned from the investments is $7,110. The interest earned from the first investment is three times the interest earned from the second investment. We need to find the amounts of the three parts of the investment.
Step 2: Define the variables.
Let's use variables to represent the unknown amounts of the three investments. We can call the first investment amount "x," the second investment amount "y," and the third investment amount "z."
Step 3: Set up the equations.
From the information given, we can set up the following equations:
Equation 1: x + y + z = $93,000 (the sum of the three investment amounts is equal to the total investment)
Equation 2: 0.08x + 0.06y + 0.09z = $7,110 (the total interest earned from the investments is $7,110)
Equation 3: 0.08x = 3(0.06y) (the interest earned from the first investment is three times the interest earned from the second investment)
Step 4: Solve the equations.
We have three equations and three unknowns, so we can solve them simultaneously using substitution or elimination. Let's solve them using substitution:
From Equation 3, we can simplify it to: 0.08x = 0.18y
Now we can substitute this into Equation 2:
0.08x + 0.06y + 0.09z = $7,110
Substituting 0.18y for 0.08x:
0.18y + 0.06y + 0.09z = $7,110
0.24y + 0.09z = $7,110
Next, let's use Equation 1 to express one variable in terms of the other two:
x = $93,000 - (y + z)
Substituting this into Equation 2:
0.08($93,000 - (y + z)) + 0.06y + 0.09z = $7,110
7,440 - 0.08y - 0.08z + 0.06y + 0.09z = $7,110
Now let's simplify this equation:
0.02y + 0.01z = $330
Multiplying both sides by 100 to clear decimals:
2y + z = $33,000
Now we have two equations:
0.24y + 0.09z = $7,110
2y + z = $33,000
We can solve these two equations using substitution or elimination to find the values of y and z.
Step 5: Solve the equations for y and z.
Using substitution, let's solve Equation 2 for y:
y = ($33,000 - z) / 2
Substituting this into Equation 1:
0.24($33,000 - z) / 2 + 0.09z = $7,110
7,920 - 0.12z + 0.09z = $7,110
Combining like terms:
-0.03z = -$810
Dividing both sides by -0.03:
z = $27,000
Now we can substitute this value of z back into Equation 2 to solve for y:
2y + $27,000 = $33,000
2y = $6,000
y = $3,000
Finally, we can substitute the values of y and z into Equation 1 to solve for x:
x + $3,000 + $27,000 = $93,000
x = $63,000
So, the amounts of the three parts of the investment are:
The first part: $63,000
The second part: $3,000
The third part: $27,000