To solve the expression \(12 \frac{8}{15} - 7 + 7 \frac{1}{5} + 15\), we will first convert the mixed numbers into improper fractions and then perform the arithmetic.
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Convert \(12 \frac{8}{15}\) into an improper fraction: \[ 12 \frac{8}{15} = \frac{12 \times 15 + 8}{15} = \frac{180 + 8}{15} = \frac{188}{15} \]
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Convert \(7 \frac{1}{5}\) into an improper fraction: \[ 7 \frac{1}{5} = \frac{7 \times 5 + 1}{5} = \frac{35 + 1}{5} = \frac{36}{5} \]
Now, rewrite the original expression using these improper fractions: \[ \frac{188}{15} - 7 + \frac{36}{5} + 15 \]
- Next, convert 7 and 15 to fractions with a common denominator of 15: \[ 7 = \frac{7 \times 15}{15} = \frac{105}{15} \] \[ 15 = \frac{15 \times 15}{15} = \frac{225}{15} \]
Now, our expression can be rewritten as: \[ \frac{188}{15} - \frac{105}{15} + \frac{36}{5} + \frac{225}{15} \]
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Combine the fractions with the common denominator of 15: \[ \frac{188 - 105 + 225}{15} + \frac{36}{5} \] First, simplify: \[ 188 - 105 = 83 \] So, \[ \frac{83 + 225}{15} = \frac{308}{15} \]
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Next, convert \(\frac{36}{5}\) to have a denominator of 15: \[ \frac{36}{5} = \frac{36 \times 3}{5 \times 3} = \frac{108}{15} \]
Now combine them: \[ \frac{308}{15} + \frac{108}{15} = \frac{308 + 108}{15} = \frac{416}{15} \]
- Finally, we simplify \(\frac{416}{15}\) if needed. Since 416 and 15 do not share any common factors besides 1, this fraction is already in simplest form.
Thus, the final answer is: \[ \boxed{\frac{416}{15}} \]