Compare the future amounts (A) you would have if the money were invested at simple interest and if it were invested with annual compounding.
$3,000 at 11% for 20 years
(a) Calculate the future amount if the money were invested at simple interest.
$
(b) Calculate the future amount if the money were invested with annual compounding. (Round your answer to the nearest cent.)
$
(a) To calculate the future amount if the money were invested at simple interest, we can use the formula:
A = P(1 + rt)
Where:
A = Future amount
P = Principal amount ($3,000)
r = Interest rate (11% = 0.11)
t = Time period (20 years)
Plugging in the values, we have:
A = $3,000 * (1 + 0.11 * 20)
A = $3,000 * (1 + 2.2)
A = $3,000 * 3.2
A = $9,600
Therefore, the future amount if the money were invested at simple interest would be $9,600.
(b) To calculate the future amount if the money were invested with annual compounding, we can use the formula:
A = P(1 + r/n)^(n*t)
Where:
A = Future amount
P = Principal amount ($3,000)
r = Interest rate (11% = 0.11)
n = Number of times interest is compounded per year (annually = 1)
t = Time period (20 years)
Plugging in the values, we have:
A = $3,000 * (1 + 0.11/1)^(1 * 20)
A = $3,000 * (1 + 0.11)^(20)
A = $3,000 * (1.11)^(20)
Using a calculator, we get:
A ≈ $3,000 * 3.818804969
A ≈ $11,456.41
Therefore, the future amount if the money were invested with annual compounding would be approximately $11,456.41.
To calculate the future amounts under different investment scenarios, we need to use the formulas for simple interest and compound interest.
(a) Simple Interest:
The formula to calculate simple interest is:
A = P(1 + rt), where A is the future amount, P is the principal (initial amount), r is the interest rate, and t is the time period.
In this case, the principal (P) is $3,000, the interest rate (r) is 11%, and the time period (t) is 20 years.
Using the formula, we can calculate the future amount with simple interest:
A = 3000(1 + 0.11*20)
A = 3000(1 + 2.2)
A ≈ $9,600
Therefore, the future amount if the money were invested at simple interest would be approximately $9,600.
(b) Compound Interest:
The formula to calculate compound interest is:
A = P(1 + r/n)^(nt), where A is the future amount, P is the principal (initial amount), r is the interest rate, n is the number of times interest is compounded per year, and t is the time period.
For annual compounding, the interest is compounded once a year, so n in this case would be 1.
Using the formula, we can calculate the future amount with annual compounding:
A = 3000(1 + 0.11/1)^(1*20)
A = 3000(1 + 0.11)^20
A ≈ $20,855.22
Therefore, the future amount if the money were invested with annual compounding would be approximately $20,855.22.
Please note that the amounts may vary slightly depending on the rounding and compounding method used.