To solve the expression \(18 \frac{1}{5} + 1 \frac{3}{5} - 2 \frac{1}{5} - 10\), we will first convert all mixed numbers into improper fractions, then perform the arithmetic.
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Convert the mixed numbers to improper fractions:
\[ 18 \frac{1}{5} = \frac{18 \times 5 + 1}{5} = \frac{90 + 1}{5} = \frac{91}{5} \]
\[ 1 \frac{3}{5} = \frac{1 \times 5 + 3}{5} = \frac{5 + 3}{5} = \frac{8}{5} \]
\[ 2 \frac{1}{5} = \frac{2 \times 5 + 1}{5} = \frac{10 + 1}{5} = \frac{11}{5} \]
The integer \(10\) can be expressed as \(\frac{50}{5}\).
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Substitute these values into the expression:
\[ \frac{91}{5} + \frac{8}{5} - \frac{11}{5} - \frac{50}{5} \]
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Combine the fractions:
\[ = \frac{91 + 8 - 11 - 50}{5} = \frac{91 + 8 - 11 - 50}{5} = \frac{38}{5} \]
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Convert \(\frac{38}{5}\) back to a mixed number:
\[ 38 \div 5 = 7 \quad \text{(remainder } 3\text{)} \]
Thus,
\[ \frac{38}{5} = 7 \frac{3}{5} \]
Now, let's compare this result to the options provided:
- \(12 \frac{2}{5}\)
- \(17 \frac{2}{5}\)
- \(-10 \frac{1}{5}\)
- \(7 \frac{3}{5}\)
The final result is:
\[ \boxed{7 \frac{3}{5}} \]