I am currently studying the textbook *Ordinary Differential Equations* by Morris Tenenbaum and Harry Pollard. The section **LESSON 1. How Differential Equations Originate.** has the following exercise:

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> 1. The radium in a piece of lead decomposes at a rate which is proportional to the amount present. If 10 percent of the radium decomposes in 200 years, what percent of the original amount of radium will be present in a piece of lead after 1000 years?
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> 
> 

The answer is said to be 59.05 percent.

Let $x$ be the amount of radium. Since the amount of radium in a piece of lead *decomposes* at a rate which is *proportional* to the amount present, we have the differential equation $\dfrac{dx}{dt} = -kx$, where $k > 0$ is a constant of proportionality.

Using the usual algebraic manipulations and then integration, we get that $x = Ae^{-kt}$, where $A$ is a constant.

So we now need to find the values of the constants, $A$ and $k$. Since we are told that 10 percent of the radium decomposes in 200 years, it seems to me that we have $0.9x = Ae^{-200k}$; but this still doesn't allow us to find $A$ and $k$. So what am I misunderstanding here? And what is the correct way to solve this?

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EDIT
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When $t = 0$, we have that $x = A$.

If I'm not mistaken, By plugging in $t = 0$ and concluding that $x = A$, this means that $x = A$ for all $t$, since $A$ is constant.

So we have that $$0.9x = xe^{-200k} \Rightarrow \dfrac{\ln(0.9)}{-200} = k = 0.00053$$

If we say that $C$ is the percent of the original amount of radium after $t$ years, then we have that $$Cx = xe^{-1000(0.00053)} \Rightarrow C = 0.5886,$$ which is 58.86% – not 59.05, as the textbook states.