Here is the question on compound interest. It is not graded, but is bugging me because I can't figure it out. After the third year, $2662 was in Samantha's account. If the account continues to earn 10% interest, how much money will be in the account (1)after the tenth year and (2) after the twentieth year? Round answers to the nearest cent. I need to know the steps to the problem, because I have to have the steps down to do the next part. Help
Here is the question on compound interest. It is not graded, but is bugging me because I can't figure it out. After the third year, $2662 was in Samantha's account. If the account continues to earn 10% interest, how much money will be in the account (1)after the tenth year and (2) after the twentieth year? Round answers to the nearest cent. I need to know the steps to the problem, because I have to have the steps down to do the next part. Help
S = P(1 + i)^n where S = the accumulated sum, P = the principal, or amount invested, i = the annual interest rate divided by 100n and n = the number of interest bearing periods.
If the interest is compounded annually,
S = 2662(1 + .10)^10 = $6904.54 after 10 years.
S = 2662(1 + .1) ^20 = $17,908.60 after 20 years.
If the interest is compounded monthly,
S = 2662(1 + .1/12)^120 = $7206.14 after 10 years
S = 2662(1.08333)^240 = $19,507.33 after 20 years.
To solve this problem, you need to use the compound interest formula, which is:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the account
P = the principal amount (initial amount)
r = the annual interest rate (as a decimal)
n = the number of times interest is compounded per year
t = the number of years
In this case, we know that after the third year, $2662 was in Samantha's account. So, we can set up an equation to find the initial principal amount (P).
Given:
A = $2662
r = 10% or 0.10 (as a decimal)
n = interest is not specified to be compounded more frequently than once a year, so we can assume n = 1
t = 3 years
Now we can solve for P:
$2662 = P(1 + 0.10/1)^(1*3)
$2662 = P(1 + 0.10)^3
To solve this equation, we need to find the value of (1 + 0.10)^3, which simplifies to 1.1^3 = 1.331.
Now, divide both sides of the equation by 1.331:
$2662 / 1.331 = P
P ≈ $2000 (rounded to the nearest dollar)
So, the initial principal amount is approximately $2000.
To find the amount of money in the account after the tenth year, we can substitute the values into the formula:
A = $2000(1 + 0.10/1)^(1*10)
A = $2000(1.1)^10
Calculate (1.1)^10 ≈ 2.594:
A = $2000 * 2.594
A ≈ $5188.00 (rounded to the nearest cent)
Therefore, after the tenth year, approximately $5188.00 will be in Samantha's account.
To find the amount of money in the account after the twentieth year, use the same formula:
A = $2000(1.1)^20
Calculate (1.1)^20 ≈ 6.727:
A = $2000 * 6.727
A ≈ $13454.00 (rounded to the nearest cent)
Therefore, after the twentieth year, approximately $13454.00 will be in Samantha's account.