Answer :
Sure, let's break down the steps to find the future value of the investment using the given information: $3,560 at an annual interest rate of 9.2%, compounded monthly for 10 years.
### Step-by-Step Solution:
1. Identify the given values:
- Principal (P): $3,560
- Annual interest rate (r): 9.2% or 0.092 (as a decimal)
- Number of times interest is compounded per year (n): 12 (monthly)
- Number of years (t): 10
2. Understand the future value formula for compound interest:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- \( A \) is the future value of the investment/loan, including interest
- \( P \) is the principal investment amount (initial deposit or loan amount)
- \( r \) is the annual interest rate (decimal)
- \( n \) is the number of times that interest is compounded per year
- \( t \) is the time the money is invested or borrowed for, in years
3. Calculate the rate per compounding period:
[tex]\[ \text{Rate per compound} = \frac{r}{n} = \frac{0.092}{12} \approx 0.00766666667 \][/tex]
4. Calculate the total number of compounding periods:
[tex]\[ \text{Number of times} = n \times t = 12 \times 10 = 120 \][/tex]
5. Substitute the values into the future value formula:
[tex]\[ A = 3560 \left(1 + 0.00766666667\right)^{120} \][/tex]
6. Compute the expression inside the parentheses:
[tex]\[ 1 + 0.00766666667 \approx 1.00766666667 \][/tex]
7. Raise the result to the power of the total number of compounding periods:
[tex]\[ 1.00766666667^{120} \approx 2.50050133 \][/tex]
8. Multiply the principal by the result:
[tex]\[ A \approx 3560 \times 2.50050133 \approx 8901.78483926 \][/tex]
9. Round the future value to two decimal places:
[tex]\[ A \approx 8901.78 \][/tex]
### Summary:
The future value of the investment of [tex]$3,560 at 9.2% interest compounded monthly over 10 years is approximately $[/tex]8,901.78.
### Step-by-Step Solution:
1. Identify the given values:
- Principal (P): $3,560
- Annual interest rate (r): 9.2% or 0.092 (as a decimal)
- Number of times interest is compounded per year (n): 12 (monthly)
- Number of years (t): 10
2. Understand the future value formula for compound interest:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- \( A \) is the future value of the investment/loan, including interest
- \( P \) is the principal investment amount (initial deposit or loan amount)
- \( r \) is the annual interest rate (decimal)
- \( n \) is the number of times that interest is compounded per year
- \( t \) is the time the money is invested or borrowed for, in years
3. Calculate the rate per compounding period:
[tex]\[ \text{Rate per compound} = \frac{r}{n} = \frac{0.092}{12} \approx 0.00766666667 \][/tex]
4. Calculate the total number of compounding periods:
[tex]\[ \text{Number of times} = n \times t = 12 \times 10 = 120 \][/tex]
5. Substitute the values into the future value formula:
[tex]\[ A = 3560 \left(1 + 0.00766666667\right)^{120} \][/tex]
6. Compute the expression inside the parentheses:
[tex]\[ 1 + 0.00766666667 \approx 1.00766666667 \][/tex]
7. Raise the result to the power of the total number of compounding periods:
[tex]\[ 1.00766666667^{120} \approx 2.50050133 \][/tex]
8. Multiply the principal by the result:
[tex]\[ A \approx 3560 \times 2.50050133 \approx 8901.78483926 \][/tex]
9. Round the future value to two decimal places:
[tex]\[ A \approx 8901.78 \][/tex]
### Summary:
The future value of the investment of [tex]$3,560 at 9.2% interest compounded monthly over 10 years is approximately $[/tex]8,901.78.
