To add the mixed numbers \(-1 \frac{3}{10}\) and \(4 \frac{2}{15}\), we'll first convert each mixed number into an improper fraction.
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Convert \(-1 \frac{3}{10}\) to an improper fraction: \[ -1 \frac{3}{10} = -\left(1 \times 10 + 3\right)/10 = -\frac{10 + 3}{10} = -\frac{13}{10} \]
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Convert \(4 \frac{2}{15}\) to an improper fraction: \[ 4 \frac{2}{15} = 4 \times 15 + 2/15 = \frac{60 + 2}{15} = \frac{62}{15} \]
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Now add the two improper fractions: To add \(-\frac{13}{10}\) and \(\frac{62}{15}\), we need a common denominator. The least common multiple of \(10\) and \(15\) is \(30\).
Convert \(-\frac{13}{10}\) to a denominator of \(30\): \[ -\frac{13}{10} = -\frac{13 \times 3}{10 \times 3} = -\frac{39}{30} \]
Convert \(\frac{62}{15}\) to a denominator of \(30\): \[ \frac{62}{15} = \frac{62 \times 2}{15 \times 2} = \frac{124}{30} \]
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Now we have the fractions: \[ -\frac{39}{30} + \frac{124}{30} = \frac{-39 + 124}{30} = \frac{85}{30} \]
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Simplify \(\frac{85}{30}\): The greatest common divisor of \(85\) and \(30\) is \(5\). \[ \frac{85 \div 5}{30 \div 5} = \frac{17}{6} \]
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Convert \(\frac{17}{6}\) back to a mixed number: \[ \frac{17}{6} = 2 \frac{5}{6} \]
So, the final result of \(-1 \frac{3}{10} + 4 \frac{2}{15}\) is: \[ 2 \frac{5}{6} \]