Step 1:

Let $x$ be the quantity of radium at time $t$ (in years)

At $t=0$,the quantity was $A_0$

Step 2:

The rate of change of $x,\large\frac{dx}{dt}\propto x$

$\therefore \large\frac{dx}{dt}$$=kx$

Where $k$ is the constant of proportionality .

$\large\frac{dx}{x}$$=kdt$

Step 3:

Integrating on both sides we get,

$\int\large\frac{dx}{x}$$=\int kdt+\log c$

$\log x=kt+\log c$

$\log \large\frac{x}{c}$$=kt$

$\Rightarrow x=ce^{kt}$

Step 4:

When $t=0,x=A_0$

$\therefore A_0=c$

When $t=50,x=0.95A_0$

$\therefore 0.95A_0=A_0e^{50k}$

$\Rightarrow e^{50k}=0.95$

Step 5:

When $t=100$

$x=A_0e^{100k}$

$\;\;\;=A_0e^{50k}.e^{50k}$

$\;\;\;=A_0(0.95)(0.95)$

$\;\;\;=0.9025A_0$

$\therefore$ the amount remaining after 100 years =$0.9025A_0$