Question

How much time will be needed for $32,000 to grow to $33,467.27 if deposited at 3% compounded quarterly?

Answer 1

We need the time t in the compound interest formula:

`A=P(1+(r)/(n))^(nt)`

Then, if we move P to the left hand side, we get

`(A)/(P)=(1+(r)/(n))^(nt)`

If we apply logarithm in both sides, we obtain

`\log (A)/(P)=nt\cdot\log (1+(r)/(n))`

therefore, the time is given by

`t=(\log (A)/(P))/(n\cdot\log (1+(r)/(n)))`

In our case A= $33467.27 (Amount), P=$32000 (Principal), n=0.03 (interest rate) and n=4 (quarterly interest).

By substituting these values into the last formula, we have

`t=(\log (33467.27)/(32000))/(4\cdot\log (1+(0.03)/(4)))`

which gives

`t=(\log 1.04585)/(4\cdot\log 1.0075)`

the, the time is

`\begin{gathered} t=(0.04482)/(0.02988) \\ t=1.5 \end{gathered}`

that is, the time will be equal to 1.5 years.