To simplify the expression \(12 \frac{8}{15} - 7 + \frac{7}{15} + 15\), we can follow these steps:
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Convert \(12 \frac{8}{15}\) to 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 \(7\) and \(15\) to fractions with a denominator of \(15\): \[ 7 = \frac{7 \times 15}{15} = \frac{105}{15} \] \[ 15 = \frac{15 \times 15}{15} = \frac{225}{15} \]
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Rewrite the expression using the fractions: \[ \frac{188}{15} - \frac{105}{15} + \frac{7}{15} + \frac{225}{15} \]
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Combine the fractions: \[ \frac{188 - 105 + 7 + 225}{15} = \frac{188 - 105 + 7 + 225}{15} = \frac{188 + 225 - 105 + 7}{15} = \frac{188 + 225 - 105 + 7}{15} \] Calculate this step by step:
- First, \(188 - 105 = 83\)
- Then, \(83 + 7 = 90\)
- Lastly, \(90 + 225 = 315\)
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So, we have: \[ \frac{315}{15} \]
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Simplifying \( \frac{315}{15} \): \[ 315 \div 15 = 21 \]
Thus, the simplified answer is: \[ \boxed{21} \]