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
![A=P(1+\frac{r}{n})^{nt}](https://tex.z-dn.net/?f=A%3DP%281%2B%5Cfrac%7Br%7D%7Bn%7D%29%5E%7Bnt%7D)
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,
![2147.39=1852.10(1+\frac{r}{12})^{12\times 5}](https://tex.z-dn.net/?f=2147.39%3D1852.10%281%2B%5Cfrac%7Br%7D%7B12%7D%29%5E%7B12%5Ctimes%205%7D)
![\frac{2147.39}{1852.10}=(1+\frac{r}{12})^{60}](https://tex.z-dn.net/?f=%5Cfrac%7B2147.39%7D%7B1852.10%7D%3D%281%2B%5Cfrac%7Br%7D%7B12%7D%29%5E%7B60%7D)
![1.159=(1+\frac{r}{12})^{60}](https://tex.z-dn.net/?f=1.159%3D%281%2B%5Cfrac%7Br%7D%7B12%7D%29%5E%7B60%7D)
Taking ln both side,
![\ln (1.159)=60\ln (1+\frac{r}{12})](https://tex.z-dn.net/?f=%5Cln%20%281.159%29%3D60%5Cln%20%281%2B%5Cfrac%7Br%7D%7B12%7D%29)
![0.1479=60\ln (1+\frac{r}{12})](https://tex.z-dn.net/?f=0.1479%3D60%5Cln%20%281%2B%5Cfrac%7Br%7D%7B12%7D%29)
![\frac{0.1479}{60}=\ln (1+\frac{r}{12})](https://tex.z-dn.net/?f=%5Cfrac%7B0.1479%7D%7B60%7D%3D%5Cln%20%281%2B%5Cfrac%7Br%7D%7B12%7D%29)
![0.00245=\ln (1+\frac{r}{12})](https://tex.z-dn.net/?f=0.00245%3D%5Cln%20%281%2B%5Cfrac%7Br%7D%7B12%7D%29)
Taking exponential both side,
![e^{0.00245}=1+\frac{r}{12}](https://tex.z-dn.net/?f=e%5E%7B0.00245%7D%3D1%2B%5Cfrac%7Br%7D%7B12%7D)
![1.00245=1+\frac{r}{12}](https://tex.z-dn.net/?f=1.00245%3D1%2B%5Cfrac%7Br%7D%7B12%7D)
![1.00245-1=\frac{r}{12}](https://tex.z-dn.net/?f=1.00245-1%3D%5Cfrac%7Br%7D%7B12%7D)
![0.00245\times 12=r](https://tex.z-dn.net/?f=0.00245%5Ctimes%2012%3Dr)
![0.0294=r](https://tex.z-dn.net/?f=0.0294%3Dr)
Into percentage, ![r=0.0294\times 100=2.94\%](https://tex.z-dn.net/?f=r%3D0.0294%5Ctimes%20100%3D2.94%5C%25)
Approximately, The interest rate on the account is 3%.
Therefore, Option c is correct.