Question

How much time will be needed for $39,000 to grow to $43,278.39 if deposited at 7% compounded quarterly?Do not round until the final answer. Then round to the nearest tenth as needed.

Answer 1

The compound interest formula is given by

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

where A is the final amount including the principal P, r is the rate, n is the number or times interes and t is the time.

In our case, A= $43,278.39, P=$39,000, n=4 (for quarterly) and t is the unknown time.

We must find t in our formula. First, if we move P to the left hand side, we ger

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

If we apply logarithm in both sides, we obtain

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

which gives,

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

then, t is equal to

$t=\frac{\log \frac{A}{P}}{n\log (1+\frac{r}{n})}$

Therefore, by means of this formula, we can find t. If we substitute the given values into this formula ,we get

$t=\frac{\log \frac{43278.39}{39000}}{4\cdot\log (1+\frac{0.07}{4})}$

then, we have

$t=\frac{\log 1.1097}{4\cdot\log 1.0175}$

which is equal to

$\begin{gathered} t=\frac{0.0173}{0.06939} \\ t=0.249 \end{gathered}$

that is, by rounding up, the times is 0.3 years.