To add the fractions \(-\frac{3}{16} + \frac{5}{34}\), we first need to find a common denominator.
The denominators are 16 and 34. The least common multiple (LCM) of 16 and 34 can be found as follows:
- The prime factorization of 16 is \(2^4\).
- The prime factorization of 34 is \(2^1 \times 17^1\).
- The LCM takes the highest power of each prime factor: \(2^4 \times 17^1 = 16 \times 17 = 272\).
Now we can convert each fraction to have a denominator of 272.
For \(-\frac{3}{16}\):
\[ -\frac{3}{16} \times \frac{17}{17} = -\frac{3 \times 17}{16 \times 17} = -\frac{51}{272} \]
For \(\frac{5}{34}\):
\[ \frac{5}{34} \times \frac{8}{8} = \frac{5 \times 8}{34 \times 8} = \frac{40}{272} \]
Now we can add the two fractions:
\[ -\frac{51}{272} + \frac{40}{272} = \frac{-51 + 40}{272} = \frac{-11}{272} \]
To write this as a mixed number, we note that \(-\frac{11}{272}\) is already in its simplest form and cannot be converted to a mixed number since it's a proper fraction.
Thus, the final answer is:
\[ -\frac{11}{272} \]