jmc

Since $\pi >3.14$, we know that $\pi -3.14>0$, and so $|\pi -3.14|=\pi -3.14$. Also, since $\pi <22/7=3.\overline{142857}$, we have $|\pi -\frac{22}{7}|=\frac{22}{7}-\pi$. The exact value of the sum is $$\begin{array}{rl}|\pi -3.14|+\left|\pi -\frac{22}{7}\right|& =\pi -3.14+\frac{22}{7}-\pi \\ & =\frac{22}{7}-3.14\\ & =\frac{22}{7}-\frac{314}{100}\\ & =\frac{2200}{700}-\frac{7(314)}{700}\\ & =\frac{2200-2198}{700}\\ & =\frac{2}{700}\\ & =\boxed{\frac{1}{350}}.\end{array}$$