To add \( -3 \frac{1}{6} + 5 \frac{3}{4} \), we first need to convert the mixed numbers to improper fractions.
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For \( -3 \frac{1}{6} \): \[ -3 \frac{1}{6} = -\left(3 \times 6 + 1\right)/6 = -\left(18 + 1\right)/6 = -19/6 \]
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For \( 5 \frac{3}{4} \): \[ 5 \frac{3}{4} = 5 \times 4 + 3 = 20 + 3 = 23/4 \]
Next, we add these two improper fractions: \[ -\frac{19}{6} + \frac{23}{4} \]
To add the fractions, we need a common denominator. The least common multiple of 6 and 4 is 12.
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Convert \( -\frac{19}{6} \) to a denominator of 12: \[ -\frac{19}{6} = -\frac{19 \times 2}{6 \times 2} = -\frac{38}{12} \]
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Convert \( \frac{23}{4} \) to a denominator of 12: \[ \frac{23}{4} = \frac{23 \times 3}{4 \times 3} = \frac{69}{12} \]
Now we can add the fractions: \[ -\frac{38}{12} + \frac{69}{12} = \frac{69 - 38}{12} = \frac{31}{12} \]
Finally, convert \(\frac{31}{12}\) back to a mixed number:
- Divide 31 by 12, which results in 2 with a remainder of 7.
Thus: \[ \frac{31}{12} = 2 \frac{7}{12} \]
The answer, written as a reduced mixed number, is: \[ \boxed{2 \frac{7}{12}} \]