Question

how much will be in the account after 4 years, if $5, 000 is invested into any account bearing 9% p.a compounded monthly and after this first investment a further $300 is paid into that account at the end of every month

Answer 1

The formula to calculate the future value of an investment with compound interest is:

FV = PV * (1 + r/n)^(nt)

Where:
FV = Future Value
PV = Present Value ($5,000 in this case)
r = annual interest rate (9%)
n = number of times interest is compounded per year (monthly compounding = 12)
t = number of years (4 years)

First, we calculate the future value of the initial $5,000 investment:

FV_initial = $5,000 * (1 + 0.09/12)^(12*4)
FV_initial = $5,000 * (1 + 0.0075)^48
FV_initial = $5,000 * (1.0075)^48
FV_initial = $5,000 * 1.473901
FV_initial = $7,369.50

Next, we calculate the future value of the $300 monthly payments that are deposited into the account at the end of each month. This requires calculating the future value of a series of monthly deposits. The formula for calculating the future value of a series of deposits is:

FV_series = PMT * ((1 + r/n)^(nt) - 1) / (r/n)

Where:
PMT = monthly deposit amount ($300)
r = annual interest rate (9%)
n = number of times interest is compounded per year (monthly = 12)
t = number of years (4 years)

FV_series = $300 * ((1 + 0.09/12)^(12*4) - 1) / (0.09/12)
FV_series = $300 * ((1 + 0.0075)^48 - 1) / 0.0075
FV_series = $300 * (1.473901 - 1) / 0.0075
FV_series = $300 * 0.473901 / 0.0075
FV_series = $300 * 63.1868
FV_series = $18,955.04

Therefore, the total amount in the account after 4 years would be:

Total = FV_initial + FV_series
Total = $7,369.50 + $18,955.04
Total = $26,324.54

So, the total amount in the account after 4 years would be $26,324.54.