Question

R14 000 is deposited into a bank account and three years later R8 000 is withdrawn from the account. The interest paid on money in the account is 5,5% p.a. compounded annually for the first two years and then the interest rate is decreased to 4% p.a. compounded monthly. Calculate the balance in the account at the end of six years.​

Answer 1

well, first off, let's see how much was made in the first 3 years.

for the first 2 years

`~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$14000\\ r=rate\to 5.5\%\to (5.5)/(100)\dotfill &0.055\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\dotfill &1\\ t=years\dotfill &2 \end{cases} \\\\\\ A=14000\left(1+(0.055)/(1)\right)^(1\cdot 2)\implies A=15582.35`

now for the following year, at 4% compounded monthly

`~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$15582.35\\ r=rate\to 4\%\to (4)/(100)\dotfill &0.04\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\dotfill &12\\ t=years\dotfill &1 \end{cases} \\\\\\ A=15582.35\left(1+(0.04)/(12)\right)^(12\cdot 1)\implies A\approx 16217.20`

now, we're 3 years later and 8000 bucks get pulled out of it, so the remaining balance is simply 16217.20 - 8000 ≈ 8217.20.

now, let's get the amount of that new balance of 8217.20 still at 4% rate compounded monthly for the last 3 years.

`~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$8217.20\\ r=rate\to 4\%\to (4)/(100)\dotfill &0.04\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\dotfill &12\\ t=years\dotfill &3 \end{cases} \\\\\\ A=8217.20\left(1+(0.04)/(12)\right)^(12\cdot 3)\implies \boxed{A\approx 9263.02}`