To solve this problem, we'll use a system of equations.
Let's define:
x = amount invested at 9%
y = amount invested at 10%
According to the information given, the total amount invested is $15,000, so:
x + y = 15,000 ---> Equation 1
The total interest earned is $1,432. Since the interest for each investment is calculated independently, we can write:
0.09x + 0.10y = 1,432 ---> Equation 2
Now, let's solve the system of equations.
From Equation 1, we can express y as:
y = 15,000 - x
Substituting this value of y into Equation 2, we have:
0.09x + 0.10(15,000 - x) = 1,432
0.09x + 1,500 - 0.10x = 1,432
-0.01x = 1,432 - 1,500
-0.01x = -68
Now, let's solve for x by dividing both sides of the equation by -0.01:
x = -68 / (-0.01)
x = 6,800
Substituting the value of x back into Equation 1 to find y:
6,800 + y = 15,000
y = 15,000 - 6,800
y = 8,200
Therefore, $6,800 was invested at a rate of 9%, and $8,200 was invested at a rate of 10%.
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Now let's solve the second problem.
Let's define:
a = amount invested at 12%
b = amount invested at 11%
According to the information given, the total amount invested is $1,150, so:
a + b = 1,150 ---> Equation 3
The total interest earned is $133.75. Using similar logic as before, we have:
0.12a + 0.11b = 133.75 ---> Equation 4
To solve this system of equations, we can use the substitution method.
From Equation 3, we can express a as:
a = 1,150 - b
Substituting this value of a into Equation 4, we have:
0.12(1,150 - b) + 0.11b = 133.75
138 - 0.12b + 0.11b = 133.75
0.11b - 0.12b = 133.75 - 138
-0.01b = -4.25
Now, let's solve for b by dividing both sides of the equation by -0.01:
b = -4.25 / (-0.01)
b = 425
Substituting the value of b back into Equation 3 to find a:
a + 425 = 1,150
a = 1,150 - 425
a = 725
Therefore, $725 was invested at a rate of 12%, and $425 was invested at a rate of 11%.