an investment of 72,000 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 $5640. the interest from the first investment was 3 times the interest from the second. find the amounts of three parts of the investment
$X-in 1st inv.
$Y-in 2nd inv.
$Z-in 3rd inv.
0.08X + 0.06Y + 0.09Z = $5640.
Eq1: 0.08X + 0.06Y + 0.09Z = 5640.
Eq2: X + Y + Z = 72000.
Multiply Eq1 by 100 and Eq2 by -9:
8X + 6Y + 9Z = 564000,
-9X - 9Y - 9Z = - 648000,
Add the Eqs:
-X - 3Y = -84,000,
0.08X = 3*0.06Y,
X = 2.25Y
Substitute 2.25Y for X:
-2.25Y - 3Y = -84000,
-5.25Y = -84000,
Y = $16000.
X = 2.25Y = 2.25 * 16000 = $36,000.
X + Y + Z = $72000,
36000 + 16000 + Z = 72000,
Z = $20,000.
Well, well, well, it seems like the business club decided to play some investment bingo! Let's solve this riddle, shall we?
Let's call the amount invested at 8% "x", the amount invested at 6% "y", and the amount invested at 9% "z". Now let's get to the heart of the matter:
According to the problem, the interest from the first investment was three times the interest from the second investment. So we can set up an equation: 0.08x = 3(0.06y).
Now let's focus on the sum of the interest earned by all three investments. We know it totaled $5,640. So we can set up another equation: 0.08x + 0.06y + 0.09z = 5,640.
Lastly, we know that the total investment made by the business club was $72,000. Therefore, we have one more equation: x + y + z = 72,000.
Looks like we've got a system of three equations. Time to bring out the big guns and solve it!
Using the first equation, we have: 0.08x = 3(0.06y).
Simplifying that, we get: 0.08x = 0.18y.
Now, let's rearrange the equation to be in terms of x: x = (0.18y) / 0.08.
Substituting this value of x into the third equation, we have: (0.18y) / 0.08 + y + z = 72,000.
Now, let's tackle the second equation: 0.08x + 0.06y + 0.09z = 5,640.
We can substitute the expression for x from the first equation into the second one, resulting in: (0.18y) / 0.08 * 0.08 + 0.06y + 0.09z = 5,640.
Simplifying, we get: 0.18y + 0.06y + 0.09z = 5,640.
Combining the y terms, we have: 0.24y + 0.09z = 5,640.
Now, we can substitute the value of x from the first equation into this equation as well, leading us to: (0.18y) / 0.08 * 0.24 + 0.09z = 5,640.
Simplifying again, we get: 0.54y + 0.09z = 5,640.
Great job! Now we have a system of two equations with two variables: 0.18y + 0.09z = 5,640 and 0.54y + 0.09z = 5,640.
Let's solve this system! Subtracting the first equation from the second, we get: (0.54y + 0.09z) - (0.18y + 0.09z) = 5,640 - 5,640.
Simplifying, we have: 0.36y = 0.
Now, here comes the punchline! If 0.36y = 0, it means y must equal 0 as well. But hey, hang on a second! That means the second investment, represented by y, is a big fat goose egg! So, the second part of the investment was $0.
Moving on, let's solve for the remaining variables. Substituting y = 0 into the third equation, we have: z = 72,000 - x - 0.
Lastly, substituting y = 0 into the first equation, we have: 0.08x = 3(0).
Therefore, x = 0.
So, as it turns out, the first part (x) and second part (y) of the investment were both $0.
And last, but not least, the third part (z) of the investment is calculated as: z = 72,000.
So, the amounts of the three parts of the investment are $0, $0, and $72,000. I hope this financial comedy routine didn't leave you feeling too deflated!
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.
To find the amounts of the three parts of the investment, let's assign variables to represent the unknown amounts. Let's say the first part of the investment is x dollars, the second part is y dollars, and the third part is z dollars.
We know that the total investment is $72,000, so we can create the equation:
x + y + z = 72,000 (Equation 1)
We are also given information about the interest earned by each part of the investment. The first part of the investment earned 8% interest, the second part earned 6% interest, and the third part earned 9% interest.
The total interest earned from all parts of the investment is $5,640. We can write this as another equation:
0.08x + 0.06y + 0.09z = 5,640 (Equation 2)
Finally, we are told that the interest earned from the first part of the investment is three times the interest earned from the second part. We can represent this as another equation:
0.08x = 3(0.06y) (Equation 3)
Now, we have a system of three equations (Equations 1, 2, and 3) that we can solve simultaneously to find the values of x, y, and z.
Using Equation 3, we can simplify it as follows:
0.08x = 0.18y
8x = 18y (distributing the 3)
Now, we can rewrite Equation 2 in terms of x and y:
0.08x + 0.06y + 0.09z = 5,640
8x + 6y + 9z = 564,000 (multiplying both sides by 100)
We can now substitute the simplified Equation 3 into Equation 1 to eliminate x:
8x = 18y
18y + 6y + 9z = 564,000
24y + 9z = 564,000 (simplifying)
Now, we have the following two equations:
x + y + z = 72,000
24y + 9z = 564,000
We can solve this system using various methods, such as substitution or elimination, to find the values of y and z. Once we have those values, we can substitute them back into Equation 1 to find x and solve for all three parts of the investment.