Question

Suppose that $3000 is placed in a savings account at an annual rate of 10.2%, compounded monthly. Assuming that no withdrawals are made, how long will it take for the account to grow to $3738? Do not round any Intermediate computations, and round your answer to the nearest hundredth.

Answer 1

Answer:

$2.17\text{years}$

Explanations:

The formula for calculating compound amount is expressed according to the formula;

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

where;

P is the principal (amount saved)

A is the compounded amount

t is the time (in years)

r is the rate (in decimal)

n is the compounding time

Given the following parameters

A = $3738

P = $3000

r = 10.2% = 0.102

n = 12 (compounded monthly)

Substitute the given parameters into the formula to get the required time.

$3738=3000(1+\frac{0.102}{12})^{12t}$

Make "t" the subject of the formula as shown;

$\begin{gathered} \frac{3738}{3000}=(1+0.0085)^{12t} \\ 1.246=(1.0085)^{12t} \\ \end{gathered}$

Take the natural logarithm of both sides

$\begin{gathered} \log 1.246=12t\log (1.0085) \\ 12t=\frac{\log 1.246}{log1.0085} \\ 12t=\frac{0.095518}{0.003676} \\ 12t=25.9913 \\ t=\frac{25.9913}{12} \\ t=2.1659 \\ t\approx2.17\text{years} \end{gathered}$

This shows that it will take 2.17 years for the account to grow to $3738