Sally invested a total of $4500, some at 9% per year and the rest at 6% per year. The return from the 9% investment exceeds that from the 6% investment by $180. How much did she invest at each rate?
I can't tell you how to set it up but I can tell you that she invested $3000 at 9% and $1500 at 6%.
3000 x .09 = 270
1500 x .06 = 90
270 - 90 = 180
let the amount invested at 9% be x
then the amount invested at 6% is 4500-x
.09x + .06(4500-x) = 180
solve for x
Let's solve this step-by-step.
Step 1: Let's assume that Sally invested x dollars at 9% per year.
Step 2: The remaining amount she invested at 6% per year would be 4500 - x dollars.
Step 3: The return from the 9% investment is given by 0.09x (0.09 is the decimal representation of 9%).
Step 4: The return from the 6% investment is given by 0.06(4500 - x) (0.06 is the decimal representation of 6%).
Step 5: The problem states that the return from the 9% investment exceeds the return from the 6% investment by $180, so we can set up the following equation:
0.09x = 0.06(4500 - x) + 180
Step 6: Let's solve for x.
0.09x = 0.06(4500 - x) + 180
0.09x = 270 + 0.06x + 180
0.09x - 0.06x = 450
0.03x = 450
x = 450 / 0.03
x ≈ 15000
Step 7: Therefore, Sally invested approximately $15000 at 9% per year.
Step 8: Now, let's find out how much she invested at 6% per year.
4500 - x = 4500 - 15000
4500 - 15000 = -10500
Step 9: Sally did not invest a negative amount, so there must be an error in the calculation.
This means that the equation in Step 5 is not correct.
Step 10: Let's correct the equation:
0.09x = 0.06(4500 - x) + 180
0.09x = 270 - 0.06x + 180
0.09x + 0.06x = 450
0.15x = 450
x = 450 / 0.15
x ≈ 3000
Step 11: Therefore, Sally invested approximately $3000 at 9% per year.
Step 12: Now, let's find out how much she invested at 6% per year.
4500 - x = 4500 - 3000
4500 - 3000 = 1500
Step 13: Sally invested approximately $1500 at 6% per year.
In conclusion, Sally invested approximately $3000 at 9% per year and $1500 at 6% per year.
To solve this problem, we can set up a system of equations.
Let's assume Sally invested x dollars at 9% and (4500-x) dollars at 6%.
The return from the 9% investment can be calculated as:
0.09x (return = investment * rate)
The return from the 6% investment can be calculated as:
0.06(4500-x)
According to the given information, the return from the 9% investment exceeds that from the 6% investment by $180. Therefore, we can write the equation:
0.09x = 0.06(4500-x) + 180
Now we can solve this equation to find the value of x.
0.09x = 0.06(4500-x) + 180
0.09x = 270 - 0.06x + 180
0.09x + 0.06x = 450
0.15x = 450
x = 450/0.15
x = 3000
Therefore, Sally invested $3000 at 9% and the remaining $4500 - $3000 = $1500 at 6%.