To find the result of \( -24 \frac{4}{5} - 6 \frac{7}{10} \) using the additive inverse, we start by rewriting the expression in terms of positive fractions.
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First, convert both mixed numbers to improper fractions.
For \( -24 \frac{4}{5} \): \[ -24 \frac{4}{5} = -\left(24 \times 5 + 4\right) / 5 = -\left(120 + 4\right) / 5 = -\frac{124}{5} \]
For \( -6 \frac{7}{10} \): \[ -6 \frac{7}{10} = -\left(6 \times 10 + 7\right) / 10 = -\left(60 + 7\right) / 10 = -\frac{67}{10} \]
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Now, we add the two negative fractions together: \[ -\frac{124}{5} - \frac{67}{10} \]
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To combine these fractions, we need a common denominator. The least common multiple of 5 and 10 is 10.
Convert \( -\frac{124}{5} \) to have a denominator of 10: \[ -\frac{124}{5} = -\frac{124 \times 2}{5 \times 2} = -\frac{248}{10} \]
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Now we can add: \[ -\frac{248}{10} - \frac{67}{10} = -\frac{248 + 67}{10} = -\frac{315}{10} \]
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Simplify \( -\frac{315}{10} \): \[ -\frac{315 \div 5}{10 \div 5} = -\frac{63}{2} \]
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To express \( -\frac{63}{2} \) as a mixed number: \[ -\frac{63}{2} = -31 \frac{1}{2} \]
Thus, the final result of \( -24 \frac{4}{5} - 6 \frac{7}{10} \) is: \[ \boxed{-31 \frac{1}{2}} \]