john deposited 3000 into an account with 5% annual interest rate compounde quarterly at the beginning of 2011. the bank guarantees that 5% annual interest rate for the next 10 years if john deposits 3000 every two years. assume that john does deposit 3000 every two years, what will the total amount of this account at the end of 2021?
To calculate the total amount in John's account at the end of 2021, we need to consider the compound interest he will earn over the years.
First, let's calculate the interest earned on the initial deposit of $3000 in 2011 compounded quarterly at a 5% annual interest rate.
The formula to calculate compound interest is:
A = P (1 + r/n)^(nt)
Where:
A = the future amount (total amount in the account)
P = the principal amount (initial deposit)
r = annual interest rate (in decimal form)
n = number of times the interest is compounded per year
t = number of years
In this case, we have:
P = $3000
r = 5% (or 0.05 in decimal form)
n = 4 (quarterly compounding)
t = 10 years
Using the formula, we can calculate the future value of the initial deposit after 10 years:
A = $3000 (1 + 0.05/4)^(4*10)
A = $3000 (1.0125)^(40)
A ≈ $4431.08 (rounded to two decimal places)
So, after 10 years, the initial deposit of $3000 will grow to approximately $4431.08.
Now, let's consider the additional deposits of $3000 made every two years, starting from 2013 till 2021.
Between 2013 and 2021, there are five two-year periods (2013-2014, 2015-2016, 2017-2018, 2019-2020, 2021), during which John makes an additional deposit of $3000.
For each two-year period, we can calculate the future value of the deposit using the same compound interest formula:
A = P (1 + r/n)^(nt)
Where:
P = $3000
r = 5% (or 0.05 in decimal form)
n = 1 (annual compounding, as John will deposit only once in two years)
t = 2 years
Calculating the future value for each two-year period:
A1 = $3000 (1 + 0.05)^(1*2) ≈ $3315
A2 = $3000 (1 + 0.05)^(1*2) ≈ $3315
A3 = $3000 (1 + 0.05)^(1*2) ≈ $3315
A4 = $3000 (1 + 0.05)^(1*2) ≈ $3315
A5 = $3000 (1 + 0.05)^(1*1) ≈ $3150
Now, to calculate the total amount at the end of 2021, we add up the initial deposit and the future values of all the additional deposits made:
Total amount = $4431.08 + $3315 + $3315 + $3315 + $3315 + $3150
Total amount ≈ $20,831.08 (rounded to two decimal places)
Therefore, the total amount in John's account at the end of 2021, considering the additional deposits and compound interest, will be approximately $20,831.08.
To calculate the total amount in John's account at the end of 2021, we need to consider two scenarios:
1. From 2011 to 2015: John deposited $3000 at the beginning of 2011. The interest is compounded quarterly at a 5% annual interest rate. Since John deposits $3000 every two years, there will be three additional deposits in 2013, 2015, and 2017.
Let's calculate the amount at the end of 2015:
Principal (P) = $3000
Annual Interest Rate (r) = 5% = 0.05
Compounding Frequency (n) = 4 (quarterly compounding)
Time Period (t) = 5 years (2011 to 2015)
Using the compound interest formula:
Total Amount = P * (1 + (r / n))^(n * t)
Total Amount = $3000 * (1 + (0.05 / 4))^(4 * 5)
Now, calculate the future value of this account at the end of 2021 by using the same formula but with different values:
Principal (P) = Total Amount = $3000 * (1 + (0.05 / 4))^(4 * 5)
Annual Interest Rate (r) = 5% = 0.05
Compounding Frequency (n) = 4 (quarterly compounding)
Time Period (t) = 6 years (2015 to 2021)
Total Amount at the end of 2021 = Principal (P) * (1 + (r / n))^(n * t)
2. From 2017 to 2021: John deposited $3000 at the beginning of 2017. Since he does not deposit any more money after this, there will be no additional deposits. The interest for these years is compounded quarterly at a 5% annual interest rate.
Let's calculate the amount at the end of 2021:
Principal (P) = $3000
Annual Interest Rate (r) = 5% = 0.05
Compounding Frequency (n) = 4 (quarterly compounding)
Time Period (t) = 4 years (2017 to 2021)
Total Amount at the end of 2021 = Principal (P) * (1 + (r / n))^(n * t)
Now, add the total amounts from both scenarios to get the final total amount in John's account at the end of 2021.