Answer:
Option c - 3%
Step-by-step explanation:
Given : Scott currently has an account balance of $2,147.39. He opened the account five years ago with a deposit of $1,852.10.
To find : If the interest compounds monthly, what is the interest rate on the account?
Solution :
The compound interest formula is
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]
Where, A is the amount A=$2,147.39
P is the principal P=$1,852.10
t is the time t= 5 years
n is the number of time compounded n=12
r is the interest rate
Substitute all the values in the formula,
[tex]2147.39=1852.10(1+\frac{r}{12})^{12\times 5}[/tex]
[tex]\frac{2147.39}{1852.10}=(1+\frac{r}{12})^{60}[/tex]
[tex]1.159=(1+\frac{r}{12})^{60}[/tex]
Taking ln both side,
[tex]\ln (1.159)=60\ln (1+\frac{r}{12})[/tex]
[tex]0.1479=60\ln (1+\frac{r}{12})[/tex]
[tex]\frac{0.1479}{60}=\ln (1+\frac{r}{12})[/tex]
[tex]0.00245=\ln (1+\frac{r}{12})[/tex]
Taking exponential both side,
[tex]e^{0.00245}=1+\frac{r}{12}[/tex]
[tex]1.00245=1+\frac{r}{12}[/tex]
[tex]1.00245-1=\frac{r}{12}[/tex]
[tex]0.00245\times 12=r[/tex]
[tex]0.0294=r[/tex]
Into percentage, [tex]r=0.0294\times 100=2.94\%[/tex]
Approximately, The interest rate on the account is 3%.
Therefore, Option c is correct.
