To add \( -1 \frac{3}{10} + (-4 \frac{2}{15}) \), we first convert the mixed numbers to improper fractions.
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Convert \( -1 \frac{3}{10} \): \[ -1 \frac{3}{10} = -\left(\frac{10 \times 1 + 3}{10}\right) = -\left(\frac{10 + 3}{10}\right) = -\left(\frac{13}{10}\right) \]
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Convert \( -4 \frac{2}{15} \): \[ -4 \frac{2}{15} = -\left(\frac{15 \times 4 + 2}{15}\right) = -\left(\frac{60 + 2}{15}\right) = -\left(\frac{62}{15}\right) \]
Now, we need to add \( -\frac{13}{10} + -\frac{62}{15} \). We first find a common denominator for the fractions. The least common multiple (LCM) of 10 and 15 is 30.
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Convert \( -\frac{13}{10} \) to a fraction with a denominator of 30: \[ -\frac{13}{10} = -\frac{13 \times 3}{10 \times 3} = -\frac{39}{30} \]
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Convert \( -\frac{62}{15} \) to a fraction with a denominator of 30: \[ -\frac{62}{15} = -\frac{62 \times 2}{15 \times 2} = -\frac{124}{30} \]
Now we can add the two fractions: \[ -\frac{39}{30} + -\frac{124}{30} = -\frac{39 + 124}{30} = -\frac{163}{30} \]
The result of adding \( -1 \frac{3}{10} + (-4 \frac{2}{15}) \) is: \[ -\frac{163}{30} \]
To express this as a mixed number, we divide 163 by 30: \[ 163 \div 30 = 5 \quad \text{(remainder } 13\text{)} \] Thus, \[ -\frac{163}{30} = -5 \frac{13}{30} \]
Therefore, the final answer is: \[ \boxed{-5 \frac{13}{30}} \]