I4.2 The ratio of two numbers is $5: a$. If 12 is added to each of them, the ratio becomes $3: 4$. If $b$ is the difference of the original numbers and $b>0$, find the value of $b$.
Let the two numbers be $5 k, 8 k$.
$$
\begin{array}{l}
\frac{5 k+12}{8 k+12}=\frac{3}{4} \\
\Rightarrow 20 k+48=24 k+36 \\
\Rightarrow 4 k=12 \\
\Rightarrow k=3 \\
5 k=15,8 k=24 \\
b=24-15=9
\end{array}
$$