To add the numbers \(-1 \frac{3}{10}\) and \(-4 \frac{2}{15}\), we'll first convert them to improper fractions.
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Convert \(-1 \frac{3}{10}\): \[ -1 \frac{3}{10} = -\left(1 \times 10 + 3\right) / 10 = -\frac{10 + 3}{10} = -\frac{13}{10} \]
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Convert \(-4 \frac{2}{15}\): \[ -4 \frac{2}{15} = -\left(4 \times 15 + 2\right) / 15 = -\frac{60 + 2}{15} = -\frac{62}{15} \]
Now we need to add these two improper fractions: \(-\frac{13}{10} + -\frac{62}{15}\).
To add the fractions, we need a common denominator. 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: \[ -\frac{39}{30} + -\frac{124}{30} = -\frac{39 + 124}{30} = -\frac{163}{30} \]
Now we can convert \(-\frac{163}{30}\) to a mixed number:
- \(163 \div 30 = 5\) with a remainder of \(13\).
So, \[ -\frac{163}{30} = -5 \frac{13}{30} \]
Finally, the result of \(-1 \frac{3}{10} + (-4 \frac{2}{15})\) is: \[ \boxed{-5 \frac{13}{30}} \]