The population of a bacterial culture grows at a rate proportional to the number of bacteria present at any time. After 3 hours it is observed that there are 400 bacteria present. After 10 hours there are 2000 present. What was the initial number of bacteria?

My attempt:

It is implied by context that a model for the rate of growth is

which implies that .

But the fact that , and we see that

and

Solving for in the first equation gives

. Substituting into the second...

and we see that

Now, again using the fact that we can conclude that

So that

and

Have I done this right. If so, was there a faster way?