If $\pi=3.1415926...$, what is the exact value of $|\pi-3.14|+|\pi-\frac{22}{7}|$? Express your answer as a common fraction.
To find the exact value of this expression, I need to compare the given approximations of $\pi$ to the actual value. I know that $3.14$ is a decimal approximation that rounds $\pi$ to two decimal places, so it is slightly less than $\pi$. Therefore, $|\pi-3.14|=\pi-3.14$, since the absolute value function removes the negative sign. I also know that $\frac{22}{7}$ is a rational approximation that overestimates $\pi$, since $\frac{22}{7}$ is about $3.142857$. Therefore, $|\pi-\frac{22}{7}|=\frac{22}{7}-\pi$, since the absolute value function flips the sign of a negative number. Now I can simplify the expression by combining the two terms. I get $|\pi-3.14|+|\pi-\frac{22}{7}|=\pi-3.14+\frac{22}{7}-\pi$. The terms with $\pi$ cancel out, leaving me with $\frac{22}{7}-3.14$. To convert $3.14$ to a fraction with denominator $7$, I can multiply it by $\frac{7}{7}$. This gives me $\frac{22}{7}-\frac{21.98}{7}$. Now I can subtract the fractions by subtracting the numerators and keeping the same denominator. I get $\frac{22}{7}-\frac{21.98}{7}=\frac{22-21.98}{7}=\frac{0.02}{7}$. This is the exact value of the expression, but I can simplify it further by dividing both the numerator and the denominator by $0.01$. This gives me $\frac{0.02}{7}\div\frac{0.01}{0.01}=\frac{2}{700}=\frac{1}{350}$.

