Q.Radium decomposes at a rate proportional to the amount present at any time. If $p$ percent of the amount disappears in one year, what percent of the amount of radium will be left after 2 years?

Let $x=x_0e^{-kt}$ ($t$ in years). If $p$ percent disappears in 1 year, $(100-p)\%$ remains, so $e^{-k}=1-\dfrac{p}{100}=\dfrac{100-p}{100}$. After 2 years: $x=x_0e^{-2k}=x_0\left(\dfrac{100-p}{100}\right)^2$. So the percentage of …

Model the situation as dx/dt proportional to x (or to (θ-θ0) for cooling), solve the resulting first-order linear/separable equation to get x = c·e^{kt} (or the Newton's-law form), then use the gi …

Mixing up which measured instant corresponds to t (measuring time from the WRONG reference point); forgetting that a ratio like x(t2)/x(t1)=e^{k(t2-t1)} lets k cancel out algebraically, and instead trying …