Answer:
9.42 years (= 113 months)
Step-by-step explanation:
Use the compound rate interest formula:
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]
where:
- A = amount
- P = principal
- r = interest rate (in decimal format)
- n = number of times interest is compounded per unit t
- t = time
Given:
- A = $7850
- P = $5000
- r = 4.8% = 0.048
- n = 12
- t = years
[tex]\implies 7850=5000(1+\frac{0.048}{12})^{12t}[/tex]
[tex]\implies 7850=5000(1.004)^{12t}[/tex]
[tex]\implies \dfrac{7850}{5000}=(1.004)^{12t}[/tex]
[tex]\implies 1.57=(1.004)^{12t}[/tex]
Take natural logs:
[tex]\implies \ln1.57=\ln(1.004)^{12t}[/tex]
[tex]\implies \ln1.57=12t\ln(1.004)[/tex]
[tex]\implies t=\dfrac{\ln 1.57}{12 \ln1.004}[/tex]
[tex]\implies t=9.42\textsf{ years (nearest hundredth)}[/tex]
[tex]\implies t=113 \textsf{ months}[/tex]
