An investment of $53,000 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 $4050. The interest from the first investment was 4 times the interest from the second. Find the amounts of the three
.08 a + .06 b + .09 c = 4050
a + b + c = 53,000
.08 a = 4 * .06b
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a = .24 b / .08 = 3 b
so
4 b + c = 53,000
c = 53,000 - 4 b
so
.08 a + .06 b + .09 c = 4050
is now
.08 (3b) + .06 b + .09 (53,000 -4b) = 4050
solve for b and go back for a and c
Let's assume the amount invested in the first part is x.
Then, the amount invested in the second part is y.
And the amount invested in the third part is z.
We are given the following information:
1. The total amount invested is $53,000:
x + y + z = 53000
2. The total interest earned is $4050:
0.08x + 0.06y + 0.09z = 4050
3. The interest earned from the first investment is 4 times the interest earned from the second:
0.08x = 4 * 0.06y
0.08x = 0.24y
Now, we have a system of equations:
Equation 1: x + y + z = 53000
Equation 2: 0.08x + 0.06y + 0.09z = 4050
Equation 3: 0.08x - 0.24y = 0
To solve this system of equations, we can use substitution or elimination method. Let's use substitution.
From Equation 3, we can rewrite it as:
0.24y = 0.08x
Divide both sides by 0.08:
3y = x
Now we can substitute this value of x in Equation 1 and solve for z:
x + y + z = 53000
3y + y + z = 53000
4y + z = 53000
z = 53000 - 4y
Substitute the value of z in Equation 2:
0.08x + 0.06y + 0.09z = 4050
0.08(3y) + 0.06y + 0.09(53000 - 4y) = 4050
0.24y + 0.06y + 4770 - 0.36y = 4050
-0.06y + 4770 - 0.36y = 4050
-0.42y = -720
y = (-720) / (-0.42)
y = 1714.29
Now, substitute the value of y in the equation z = 53000 - 4y:
z = 53000 - 4(1714.29)
z = 53000 - 6857.16
z = 46142.84
Finally, we can substitute the values of y and z into Equation 1 to solve for x:
x + y + z = 53000
x + 1714.29 + 46142.84 = 53000
x = 53000 - 1714.29 - 46142.84
x = 513.87
Therefore, the amounts invested in the three parts are:
x = $513.87
y = $1714.29
z = $46142.84
To solve this problem, let's break it down step by step:
Step 1: Set up the equations
Let's denote the amounts of the three investments as follows:
- Amount invested in the first part: X
- Amount invested in the second part: Y
- Amount invested in the third part: Z
Step 2: Translate the given information into equations
We know that the total investment was $53,000:
X + Y + Z = 53,000 (Equation 1)
We also know that the total interest earned from the investments was $4,050:
0.08X + 0.06Y + 0.09Z = 4,050 (Equation 2)
Finally, we are given that the interest earned from the first investment was 4 times the interest earned from the second investment:
0.08X = 4 * 0.06Y (Equation 3)
Step 3: Solve the system of equations
We will solve this system of equations using the method of substitution:
From Equation 3, we can express X in terms of Y:
X = (4 * 0.06Y) / 0.08
X = 3 * 0.06Y / 0.08
X = 3 * 0.75Y
X = 2.25Y
Now we substitute this value of X into Equation 1:
2.25Y + Y + Z = 53,000
3.25Y + Z = 53,000 (Equation 4)
Substituting X and Z into Equation 2, we get:
0.08(2.25Y) + 0.06Y + 0.09Z = 4,050
0.18Y + 0.06Y + 0.09Z = 4,050
0.24Y + 0.09Z = 4,050 (Equation 5)
Now we have a system of two equations with two variables (Equations 4 and 5). We can solve this system by substitution or elimination.
Step 4: Solve the system of equations using substitution or elimination
Let's solve it using the substitution method. From Equation 4, we express Z in terms of Y:
Z = 53,000 - 3.25Y
Substituting this value of Z into Equation 5, we get:
0.24Y + 0.09(53,000 - 3.25Y) = 4,050
0.24Y + 0.09 * 53,000 - 0.09 * 3.25Y = 4,050
0.24Y + 4,770 - 0.2925Y = 4,050
0.24Y - 0.2925Y = 4,050 - 4,770
-0.0525Y = -720
Dividing both sides by -0.0525, we get:
Y = (-720) / (-0.0525)
Y ≈ 13,714.29
Substituting this value of Y into Equation 4, we get:
3.25Y + Z = 53,000
3.25 * 13,714.29 + Z = 53,000
44,535.71 + Z = 53,000
Z = 53,000 - 44,535.71
Z ≈ 8,464.29
Finally, to find X, we substitute the values of Y and Z into Equation 1:
X + Y + Z = 53,000
X + 13,714.29 + 8,464.29 = 53,000
X = 53,000 - 13,714.29 - 8,464.29
X ≈ 30,821.43
Therefore, the amounts of the three investments are approximately as follows:
- Amount invested in the first part: $30,821.43
- Amount invested in the second part: $13,714.29
- Amount invested in the third part: $8,464.29