To add the mixed numbers \( 3 \frac{5}{7} \) and \( 7 \frac{1}{2} \), we can follow these steps:
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Convert the mixed numbers to improper fractions:
- For \( 3 \frac{5}{7} \): \[ 3 \frac{5}{7} = \frac{3 \times 7 + 5}{7} = \frac{21 + 5}{7} = \frac{26}{7} \]
- For \( 7 \frac{1}{2} \): \[ 7 \frac{1}{2} = \frac{7 \times 2 + 1}{2} = \frac{14 + 1}{2} = \frac{15}{2} \]
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Find a common denominator: The least common multiple of 7 and 2 is 14.
- Convert \( \frac{26}{7} \) to a fraction with a denominator of 14: \[ \frac{26}{7} = \frac{26 \times 2}{7 \times 2} = \frac{52}{14} \]
- Convert \( \frac{15}{2} \) to a fraction with a denominator of 14: \[ \frac{15}{2} = \frac{15 \times 7}{2 \times 7} = \frac{105}{14} \]
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Add the fractions: Now that we have both fractions with a common denominator: \[ \frac{52}{14} + \frac{105}{14} = \frac{52 + 105}{14} = \frac{157}{14} \]
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Convert the improper fraction back to a mixed number: Divide 157 by 14:
- \( 157 \div 14 = 11 \) (whole number part)
- Remainder: \( 157 - (11 \times 14) = 157 - 154 = 3 \)
So, \( \frac{157}{14} = 11 \frac{3}{14} \).
Thus, the sum of \( 3 \frac{5}{7} + 7 \frac{1}{2} \) is: \[ \boxed{11 \frac{3}{14}} \]