To add the mixed numbers \(4 \frac{3}{5}\) and \(7 \frac{8}{15}\), we first convert them to improper fractions.
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Convert \(4 \frac{3}{5}\) to an improper fraction: \[ 4 \frac{3}{5} = \frac{4 \times 5 + 3}{5} = \frac{20 + 3}{5} = \frac{23}{5} \]
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Convert \(7 \frac{8}{15}\) to an improper fraction: \[ 7 \frac{8}{15} = \frac{7 \times 15 + 8}{15} = \frac{105 + 8}{15} = \frac{113}{15} \]
Next, we need a common denominator to add the fractions. The least common multiple of 5 and 15 is 15.
- Convert \(\frac{23}{5}\) to have a denominator of 15: \[ \frac{23}{5} = \frac{23 \times 3}{5 \times 3} = \frac{69}{15} \]
Now, we can add the two fractions: \[ \frac{69}{15} + \frac{113}{15} = \frac{69 + 113}{15} = \frac{182}{15} \]
Finally, convert \(\frac{182}{15}\) back to a mixed number:
- Divide 182 by 15: \(182 \div 15 = 12\) remainder \(2\).
- Therefore, \(182 = 15 \times 12 + 2\).
So, \[ \frac{182}{15} = 12 \frac{2}{15} \]
Thus, the sum is: \[ \boxed{12 \frac{2}{15}} \]