Let's begin by breaking down the problem and solving each part systematically.
1. Account A: Simple Interest Calculation
Account A pays 4% annual simple interest. Simple interest is calculated using the formula:
[tex]\[ \text{Simple Interest} = \text{Principal} \times \text{Interest Rate} \times \text{Time (in years)} \][/tex]
Principal = [tex]$1500
Simple annual interest rate = 4% = 0.04
Time = 20 years
Using the formula for simple interest, we calculate the total interest earned over 20 years:
Total simple interest = $[/tex]1500 \times 0.04 \times 20
Total simple interest = [tex]$1500 \times 0.8
Total simple interest = $[/tex]1200
Now, we add the simple interest earned to the original principal to get the total value of Account A after 20 years:
Total value of Account A = Principal + Total simple interest
Total value of Account A = [tex]$1500 + $[/tex]1200
Total value of Account A = [tex]$2700
So, at the end of 20 years, Account A will be worth $[/tex]2700.
2. Account B: Compound Interest Calculation
Account B pays 4% interest compounded annually. Compound interest is calculated using the formula:
[tex]\[ \text{Compound Interest} = \text{Principal} \times \left(1 + \text{Interest Rate}\right)^{\text{Number of times interest is compounded}} \][/tex]
Since interest is compounded annually, the number of times interest is compounded each year is 1.
Principal = [tex]$1500
Annual compound interest rate = 4% = 0.04
Time = 20 years
Using the formula for compound interest, we calculate the total amount in Account B after 20 years:
Total value of Account B = $[/tex]1500 \times (1 + 0.04)^{20}
Total value of Account B = [tex]$1500 \times (1 + 0.04)^{20}
Total value of Account B = $[/tex]1500 \times (1.04)^{20}
Now we need to calculate [tex]\( (1.04)^{20} \)[/tex]:
[tex]\( (1.04)^{20} \approx 2.191123 \)[/tex] (rounded to six decimal places for precision)
Using this value in our formula:
Total value of Account B = [tex]$1500 \times 2.191123
Total value of Account B = $[/tex]3286.6845 (approximately)
Rounding to the nearest cent, we have:
Total value of Account B ≈ [tex]$3286.68
So, at the end of 20 years, Account B will be worth approximately $[/tex]3286.68.
To present both answers using dollars and cents:
At the end of 20 years:
- The total value of Account A will be [tex]$2700.00
- The total value of Account B will be $[/tex]3286.68
