During the decay of a radioactive substance, the rate at which mass is lost is proportional to the mass present at the instant. Use $m$ for the mass of the substance in grams and $t$ for the time in seconds. Initially there is 24 g of the substance and the mass is decreasing at a rate of 1.2$g$$s^{-1}$. Write a differential equation for the rate of change of mass.
My workings go as follows:
$\frac{dm}{dt}$ $∝$ $m$
$m$ = 24 when $\frac{dm}{dt}$ = -1.2
therefore, $\frac{dm}{dt}$ = $km$
However, as $\frac{dm}{dt}$ is negative, and $m$ is positive, $k$ must be negative for the initial condition to be true.
Therefore:
$\frac{dm}{dt}$ = -$km$
-1.2 = -24$k$
$k$ = $\frac{1}{20}$
Therefore:
$\frac{dm}{dt}$ = - $\frac{m}{20}$
or $\frac{dm}{dt}$ = - 0.05$m$
Is my method correct? Because the textbook claimed the answer was $\frac{dm}{dt}$ = -0.005$m$
Your method looks essentially correct. Maybe the book has a typographical error.