Question

You decide to put $2,000 in a savings account to save for a $3,000 downpayment on a new car. If the account has an interest rate of 4% per year and is compounded monthly
how long does it take until you have $3.000 without depositing any additional funds?
121.862 years
1
12.1862 years
|
10.155 years
1.0155 years

Answer 1

Answer:

1.0155 years is the correct answer

Answer 2

Answer:

$t \approx 10.15351... \ (years)$

Step-by-step explanation:

Interest compound is defined as

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

Where,

$P$ is the principal.

$r$ is the rate in decimal.

$n$ is the number of compounded periods within a year.

$t$ is the time in years.

In this case, we have

$P=2000\\A=3000\\r=4\%=0.04\\ n=12\\t=?$

Replacing all values, we have

$3000=2000(1+\frac{0.04}{12})^{12t}$

Now, we solve for $t$

$\frac{3000}{2000}=(1+ 0.003)^{12t}$

$1.5=(1.003)^{12t}$

$ln(1.5)=ln((1.003)^{12t} )\\ln(1.5)=12t \times ln(1.003)\\12t=\frac{ln(1.5)}{ln(1.003)} \\12t=\frac{0.4}{0.003}\\ 12t=133.33\\t=\frac{133.33}{12}\\ t \approx 11.11$

Our result is different because at each step we approximated results.

If you use a calculater, you would find a more exact result would be

$t \approx 10.15351...$

Therefore, the right answer is the third choice.