To divide the mixed number \( 3 \frac{5}{6} \) by the fraction \( \frac{7}{9} \), we first convert the mixed number into an improper fraction.
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Convert \( 3 \frac{5}{6} \) into an improper fraction:
- Multiply the whole number 3 by the denominator 6: \( 3 \times 6 = 18 \).
- Add the numerator 5: \( 18 + 5 = 23 \).
- Therefore, \( 3 \frac{5}{6} = \frac{23}{6} \).
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Now, divide \( \frac{23}{6} \) by \( \frac{7}{9} \) by multiplying \( \frac{23}{6} \) by the reciprocal of \( \frac{7}{9} \): \[ \frac{23}{6} \div \frac{7}{9} = \frac{23}{6} \times \frac{9}{7} \]
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Now, multiply the fractions: \[ \frac{23 \times 9}{6 \times 7} = \frac{207}{42} \]
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Simplify \( \frac{207}{42} \):
- Find the greatest common divisor (GCD) of 207 and 42. The GCD is 3.
- Divide the numerator and denominator by 3: \[ \frac{207 \div 3}{42 \div 3} = \frac{69}{14} \]
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Convert \( \frac{69}{14} \) into a mixed number:
- Divide 69 by 14, which is 4 with a remainder of 13.
- Thus, \( \frac{69}{14} = 4 \frac{13}{14} \).
So, the final answer is: \[ \boxed{\frac{69}{14}} \quad \text{or} \quad \boxed{4 \frac{13}{14}} \]
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