To add the mixed numbers \(-3 \frac{1}{6}\) and \(5 \frac{3}{4}\), 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 \(5 \frac{3}{4}\): \[ 5 \frac{3}{4} = \left(5 \times 4 + 3\right)/4 = \frac{23}{4} \]
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Find a common denominator.
- The denominators are \(6\) and \(4\). The least common multiple (LCM) of \(6\) and \(4\) is \(12\).
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Convert both fractions to have the common denominator of \(12\).
- For \(-\frac{19}{6}\): \[ -\frac{19}{6} = -\frac{19 \times 2}{6 \times 2} = -\frac{38}{12} \]
- For \(\frac{23}{4}\): \[ \frac{23}{4} = \frac{23 \times 3}{4 \times 3} = \frac{69}{12} \]
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Add the fractions. \[ -\frac{38}{12} + \frac{69}{12} = \frac{-38 + 69}{12} = \frac{31}{12} \]
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Convert the improper fraction back to a mixed number. \(\frac{31}{12}\) is equal to \(2 \frac{7}{12}\) because:
- \(31 \div 12 = 2\) with a remainder of \(7\).
Thus, the final result is: \[ \boxed{2 \frac{7}{12}} \]