Confusing text about Radioactive decay

Question

I have a question about relative growth rate and relative decay rate. In my textbook I see this:

and

k makes sense in the first image. If the population is 50,000 and $\frac{dp}{dt}$ is 5000 then k is 10%. If the population is 100000 and $\frac{dp}{dt}$ is 10000, then k is still 10%. This makes the relative growth rate constant.

But the second image text is a bit confusing. On one hand it says "the relative decay rate is positive" represented by this:

$$\frac{-1}{m} \cdot \frac{dm}{dt}$$

if $\frac{dm}{dt}$ is negative, then k is positive. But then moments later it says "k is a negative constant". What is going on?

Answer 1

I see what your problem is. $\frac{dm}{dt}$ is the relative growth rate, and that's negative. The decay rate is the negative of the growth rate. Forget about the words and trust the math! $\frac{dm}{dt}$ is the derivative of mass w.r.t time, and we know the mass decreases.

Answer 2

There is not contradiction since for k negative

$$\frac{-1}{m} \cdot \frac{dm}{dt}=-k \iff \frac{dm}{dt}=km$$