To add the mixed numbers \(-3 \frac{1}{6}\) and \(5 \frac{3}{4}\), we can follow these steps:
- Convert the mixed numbers into improper fractions.
- Find a common denominator.
- Add the fractions.
- Convert back to a mixed number, if necessary.
Step 1: Convert to improper fractions
\(-3 \frac{1}{6}\): \[ -3 \frac{1}{6} = -\left(3 \times 6 + 1\right)/6 = -\frac{19}{6} \]
\(5 \frac{3}{4}\): \[ 5 \frac{3}{4} = 5 \times 4 + 3/4 = \frac{20 + 3}{4} = \frac{23}{4} \]
Step 2: Find a common denominator
The denominators are \(6\) and \(4\). The least common multiple (LCM) of \(6\) and \(4\) is \(12\).
Step 3: Rewrite each fraction with the common denominator
Convert \(-\frac{19}{6}\) to have a denominator of \(12\): \[ -\frac{19}{6} = -\frac{19 \times 2}{6 \times 2} = -\frac{38}{12} \]
Convert \(\frac{23}{4}\) to have a denominator of \(12\): \[ \frac{23}{4} = \frac{23 \times 3}{4 \times 3} = \frac{69}{12} \]
Step 4: Add the two fractions
Now we can add: \[ -\frac{38}{12} + \frac{69}{12} = \frac{69 - 38}{12} = \frac{31}{12} \]
Step 5: Convert to a mixed number
Since \(\frac{31}{12}\) is an improper fraction, we convert it to a mixed number: \[ 31 \div 12 = 2 \quad \text{(remainder 7)} \] So, \(\frac{31}{12} = 2 \frac{7}{12}\).
Final Answer
Thus, the final answer is: \[ 2 \frac{7}{12} \]