To simplify the expression \( 12 \frac{8}{15} - 7 + \frac{7}{15} + 15 \), let's break it down step-by-step.
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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 \( 7 \) to a fraction with a denominator of 15: \[ 7 = \frac{7 \times 15}{15} = \frac{105}{15} \]
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Convert \( 15 \) to a fraction with a denominator of 15: \[ 15 = \frac{15 \times 15}{15} = \frac{225}{15} \]
Now we can rewrite the original expression: \[ \frac{188}{15} - \frac{105}{15} + \frac{7}{15} + \frac{225}{15} \]
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Combine all fractions: \[ \frac{188 - 105 + 7 + 225}{15} \]
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Perform the arithmetic in the numerator: \[ 188 - 105 = 83 \] \[ 83 + 7 = 90 \] \[ 90 + 225 = 315 \]
So we have: \[ \frac{315}{15} \]
- Simplify \( \frac{315}{15} \): \[ \frac{315 \div 15}{15 \div 15} = \frac{21}{1} = 21 \]
Thus, the simplified answer is: \[ \boxed{21} \]
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