Question

Raymond invested $2,000 in an account that earns an 8% annual interest rate. Find a balance of his account after 4 years compounded the following ways round your answers to the nearest cent.

Answer 1

The rule of the compounded interest is

$A=P(1+\frac{r}{n})^{nt}$

A is the new value

P is the initial value

r is the rate in decimal

n is the period

t is the time

He invested $2000

P = 2000

The account earns an 8% annual interest

r = 8% = 8/100 = 0.08

The balance is for 4 years

t = 4

For no 1 it is annual, then

n = 1

Substitute these values in the rule above

$\begin{gathered} A=2000(1+\frac{0.08}{1})^{(1\times4)} \\ A=2000(1.08)^4 \\ A=2720.97792 \end{gathered}$

Round it to the nearest cent means 2 d.p

$A=2720.98$

The balance is $2720.98

2. Semi-annual means

n = 2

So we will substitute n by 2

$\begin{gathered} A=2000(1+\frac{0.08}{2})^{(2\times4)} \\ A=2000(1.04)^8 \\ A=2737.138101 \end{gathered}$

Round it to the nearest cent

$A=2737.14$

The balance is $2737.14 in semi-annual

3. Quarterly means

n = 4

So we will substitute n by 4

$\begin{gathered} A=2000(1+\frac{0.08}{4})^{(4\times4)} \\ A=2000(1.02)^{16} \\ A=2745.57141 \end{gathered}$

Round it to the nearest cents

$A=2745.57$

The balance is $2745.57 quarterly

4. Monthly means

n = 12

Substitute n by 12

$\begin{gathered} A=2000(1+\frac{0.08}{12})^{(12\times4)} \\ A=2000(1.006666667)^{48} \\ A=2751.332201 \end{gathered}$

Round it to the nearest cent

$A=2751.33$

The balance is $2751.33 monthly