Answer
Monthly compounding: $22.41$ years
Continuous compounding: $22.38$ years
Work Step by Step
The amount A after t years due to a principal P
invested at an annual interest rate r, expressed as a decimal,
compounded n times per year is $A=P\displaystyle \cdot(1+\frac{r}{n})^{nt}$
If the compounding is continuous, then $A=Pe^{rt}$
---
$a.$
Compounding $n=12$ times per year$, r=0.025, P=100,A=175$
$175=100\displaystyle \cdot(1+\frac{0.025}{12})^{12t}\qquad.../\div 100$
$1.75=(1+\displaystyle \frac{0.025}{12})^{12t}\qquad.../\ln(...)$
$\displaystyle \ln 1.75=12t\ln(1+\frac{0.025}{12})\qquad.../\times\frac{1}{12\ln(1+\frac{0.025}{12})}$
$t=\displaystyle \frac{\ln 1.75}{12\ln(1+\frac{0.025}{12})}\approx 22.41$ years
$b.$
Continuous compounding: $A=Pe^{rt}$
$175=100e^{0.025t}\qquad.../\div 100$
$1.75=e^{0.025t}\qquad.../\ln(...)$
$\ln 1.75=0.025t$
$t=\displaystyle \frac{\ln 1.75}{0.025}\approx 22.38$ years