In order to have money for their daughter's college education, a young couple started a savings plan into which they made intermittent deposits. They started the account with a deposit of 128.44 dollars (in year zero) and then added 421.97 dollars in year 3, and 723.46 dollars in years 8 and 9. What is the amount they have in the account in year 16 if they earned interest at 4.59% per year and never made any withdrawals? (Accuracy is set at the first decimal.)
To calculate the amount in the account in year 16, we need to calculate the compound interest for each deposit separately and then sum them up.
Step 1: Calculate the compound interest for the initial deposit of $128.44 in year 0.
Compound Interest = Principal Amount * (1 + Interest Rate)^Number of years
Compound Interest = $128.44 * (1 + 0.0459)^16
Compound Interest = $128.44 * 1.85965881
Compound Interest = $239.03 (rounded to the nearest cent)
Step 2: Calculate the compound interest for the deposit of $421.97 in year 3.
Compound Interest = Principal Amount * (1 + Interest Rate)^Number of years
Compound Interest = $421.97 * (1 + 0.0459)^13
Compound Interest = $421.97 * 1.75987894
Compound Interest = $742.32 (rounded to the nearest cent)
Step 3: Calculate the compound interest for the deposits of $723.46 in years 8 and 9.
Compound Interest for year 8 = Principal Amount * (1 + Interest Rate)^Number of years
Compound Interest for year 8 = $723.46 * (1 + 0.0459)^8
Compound Interest for year 8 = $723.46 * 1.42093865
Compound Interest for year 8 = $1,029.47 (rounded to the nearest cent)
Compound Interest for year 9 = Principal Amount * (1 + Interest Rate)^Number of years
Compound Interest for year 9 = $723.46 * (1 + 0.0459)^7
Compound Interest for year 9 = $723.46 * 1.37646722
Compound Interest for year 9 = $994.38 (rounded to the nearest cent)
Step 4: Sum up the total compound interest for all the deposits.
Total Compound Interest = $239.03 + $742.32 + $1,029.47 + $994.38
Total Compound Interest = $3,005.20 (rounded to the nearest cent)
Step 5: Calculate the total amount in the account in year 16.
Total Amount in the Account = Sum of Deposits + Total Compound Interest
Total Amount in the Account = $128.44 + $421.97 + $723.46 + $723.46 + $3,005.20
Total Amount in the Account = $4,002.53 (rounded to the nearest cent)
Therefore, the amount in the account in year 16 is $4,002.53.
To calculate the amount in the account in year 16, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/amount in the account
P = the principal amount (initial deposit)
r = annual interest rate (expressed as a decimal)
n = number of times interest is compounded per year
t = number of years
Let's break down the information given in the problem:
1. Initial deposit: $128.44 (year 0)
2. Deposit after 3 years: $421.97 (year 3)
3. Deposit in year 8 and 9: $723.46
First, we need to calculate the number of years between the initial deposit (year 0) and year 16. Since we have deposits in year 3, 8, and 9, we can consider these points in determining the time span.
Time span:
Years 0-3 = 3 years
Years 3-8 = 5 years
Years 8-9 = 1 year
Years 9-16 = 7 years
Now, let's calculate the values at each milestone:
1. Calculate the amount after 3 years:
P1 = $128.44
r = 4.59% = 0.0459 (interest rate as a decimal)
n = 1 (interest compounded annually)
t1 = 3 years
A1 = P1(1 + r/n)^(nt1)
A1 = $128.44(1 + 0.0459/1)^(1*3)
A1 = $128.44(1 + 0.0459)^3
A1 = $128.44(1.0459)^3
A1 = $143.760291972
2. Calculate the amount after 8 years:
P2 = P1 + $421.97
r = 4.59%
n = 1
t2 = 5 years
A2 = P2(1 + r/n)^(nt2)
A2 = ($128.44 + $421.97)(1 + 0.0459/1)^(1*5)
A2 = $550.41(1 + 0.0459)^5
A2 = $550.41(1.0459)^5
A2 = $644.365404649
3. Calculate the amount after 9 years:
P3 = P2 + $723.46
r = 4.59%
n = 1
t3 = 1 year
A3 = P3(1 + r/n)^(nt3)
A3 = ($550.41 + $723.46)(1 + 0.0459/1)^(1*1)
A3 = $1273.87(1 + 0.0459)^1
A3 = $1273.87(1.0459)^1
A3 = $1331.6693533
4. Calculate the amount after 16 years:
P4 = A3
r = 4.59%
n = 1
t4 = 7 years
A4 = P4(1 + r/n)^(nt4)
A4 = $1331.6693533(1 + 0.0459/1)^(1*7)
A4 = $1331.6693533(1.0459)^7
A4 = $1331.6693533(1.0459^7)
A4 ≈ $1932.7
Therefore, the amount they have in the account in year 16 is approximately $1932.7.