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