To add the two numbers \(-1 \frac{3}{10}\) and \(-4 \frac{2}{15}\), we can first convert them into 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 \(-4 \frac{2}{15}\): \[ -4 \frac{2}{15} = -\left(4 \times 15 + 2\right)/15 = -\frac{60 + 2}{15} = -\frac{62}{15} \]
Next, we need a common denominator to add these two fractions. The denominators are 10 and 15, and the least common multiple of 10 and 15 is 30.
Now we can convert both fractions to have a denominator of 30:
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For \(-\frac{13}{10}\): \[ -\frac{13 \times 3}{10 \times 3} = -\frac{39}{30} \]
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For \(-\frac{62}{15}\): \[ -\frac{62 \times 2}{15 \times 2} = -\frac{124}{30} \]
Now we can add the two fractions: \[ -\frac{39}{30} + -\frac{124}{30} = -\frac{39 + 124}{30} = -\frac{163}{30} \]
So the result of the addition is: \[ -\frac{163}{30} \]
If you'd like to convert this back to a mixed number: \[ -\frac{163}{30} = -5 \frac{13}{30} \]
Thus, the final answer is: \[ -5 \frac{13}{30} \]