at the end of every 3 months, Rita deposits $100 into an account that pays 5% compounded quarterly. After 5 yrs, she puts the accumulated amount into a certificate of deposit paying 8.5% compounded semiannually for 1 yr. when this certificate matures, how much will Teresa have accumulated?
amount after 5 yrs = 100 (1.0125^20 - 1)/.0125
then multiply that result by (1.0425)^2
To calculate the accumulated amount, we need to break down the problem into two parts:
1. Calculating the accumulated amount after 5 years of quarterly deposits.
2. Calculating the accumulated amount after 1 year of compound interest on the initial amount.
Let's start with the first part:
1. Calculating the accumulated amount after 5 years of quarterly deposits:
The formula to calculate compound interest is A = P(1 + r/n)^(nt), where:
A = the accumulated amount
P = the principal amount
r = the annual interest rate
n = the number of compounding periods per year
t = the number of years
Given:
P = $100 (the quarterly deposit)
r = 5% or 0.05 (annual interest rate)
n = 4 (quarterly compounding)
t = 5 years
We'll calculate the accumulated amount using the formula above:
A = 100(1 + 0.05/4)^(4*5)
A = 100(1.0125)^(20)
A ≈ $611.61
So, after 5 years of quarterly deposits, Rita will accumulate approximately $611.61.
Now, let's move on to the second part:
2. Calculating the accumulated amount after 1 year of compound interest on the initial amount:
Given:
P = $611.61 (the accumulated amount after 5 years)
r = 8.5% or 0.085 (annual interest rate)
n = 2 (semiannual compounding)
t = 1 year
Using the compound interest formula:
A = 611.61(1 + 0.085/2)^(2*1)
A ≈ $673.13
Therefore, after the certificate matures, Teresa will have accumulated approximately $673.13.
To calculate the accumulated amount, we need to break down the problem into two parts:
1. The initial deposit compounded quarterly for 5 years.
2. The accumulated amount from the first part, compounded semiannually for another year.
Let's calculate each part step by step:
1. Compounding the initial deposit quarterly for 5 years:
The formula to calculate the future value with compound interest is:
FV = P(1 + r/n)^(nt)
Where:
FV = Future Value (accumulated amount)
P = Principal amount (initial deposit)
r = Annual interest rate (5%)
n = Number of compounding periods in a year (quarterly, so 4)
t = Time in years (5 years)
Plugging in the given values:
FV1 = $100(1 + 0.05/4)^(4*5)
Calculating this, we get:
FV1 = $100(1.0125)^(20)
FV1 ≈ $128.35
After 5 years, Rita will accumulate approximately $128.35.
2. Compounding the accumulated amount semiannually for 1 year:
Now, we need to take the accumulated amount from the first part and calculate the future value with the new interest rate for 1 year.
The formula remains the same, except:
r = Annual interest rate (8.5%)
n = Number of compounding periods in a year (semiannually, so 2)
t = Time in years (1 year)
Plugging in the given values:
FV2 = $128.35(1 + 0.085/2)^(2*1)
Calculating this, we get:
FV2 = $128.35(1.0425)^(2)
FV2 ≈ $142.05
When the certificate matures, Teresa will have accumulated approximately $142.05.
Therefore, after the certificate matures, Teresa will have accumulated approximately $142.05.