To add the mixed numbers \(-3 \frac{1}{6}\) and \(5 \frac{3}{4}\), we'll first convert them into improper fractions.
Step 1: Convert the mixed numbers to improper fractions.
For \(-3 \frac{1}{6}\): \[ -3 \frac{1}{6} = -\left(3 \times 6 + 1\right)/6 = -\left(18 + 1\right)/6 = -19/6 \]
For \(5 \frac{3}{4}\): \[ 5 \frac{3}{4} = (5 \times 4 + 3)/4 = (20 + 3)/4 = 23/4 \]
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: Convert each fraction to have a denominator of 12.
For \(-\frac{19}{6}\): \[ -\frac{19}{6} = -\frac{19 \times 2}{6 \times 2} = -\frac{38}{12} \]
For \(\frac{23}{4}\): \[ \frac{23}{4} = \frac{23 \times 3}{4 \times 3} = \frac{69}{12} \]
Step 4: Add the fractions.
Now we can add: \[ -\frac{38}{12} + \frac{69}{12} = \frac{69 - 38}{12} = \frac{31}{12} \]
Step 5: Convert the improper fraction back to a mixed number.
To convert \(\frac{31}{12}\) to a mixed number:
- Divide 31 by 12, which gives 2 with a remainder of 7.
So, \[ \frac{31}{12} = 2 \frac{7}{12} \]
Thus, the sum \(-3 \frac{1}{6} + 5 \frac{3}{4}\) is: \[ \boxed{2 \frac{7}{12}} \]