[Math] Find initial value and growth in exponential growth model

Question

I was given two models each with specific time($t$) and end amount($P$). $100$ after $3$ hours and $400$ after $5$ hours. Using the exponential growth model I get these $100=Pe^{3r}$ and $400=Pe^{5r}$. I heard you are suppose to divide these two so after that I got this: 4$=Pe^{5/3}$. Then I did the natural log of both and got this $r=0.831774$. After words I have to find the the end amount but I have no Initial Amount($P$). Only the rate and time. Advice?

Answer 1

$$100=Pe^{3r} \\400=Pe^{5r} \\ \implies 4 = e^{5r-3r}=e^{2r} \\ \implies r=\ln2\approx0.693$$ Substitute $r$ in any equation to get the $P$.