To divide the mixed number \(2 \frac{2}{3}\) by the fraction \(\frac{7}{9}\), follow these steps:
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Convert the mixed number to an improper fraction: \[ 2 \frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3} \]
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Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{8}{3} \div \frac{7}{9} = \frac{8}{3} \times \frac{9}{7} \]
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Multiply the fractions: \[ \frac{8 \times 9}{3 \times 7} = \frac{72}{21} \]
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Simplify the fraction:
- Find the greatest common divisor (GCD) of 72 and 21, which is 3.
- Divide both the numerator and the denominator by 3: \[ \frac{72 \div 3}{21 \div 3} = \frac{24}{7} \]
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Convert back to a mixed number (if desired): \(\frac{24}{7} = 3 \frac{3}{7}\) (since \(24 \div 7 = 3\) remainder \(3\)).
So, the final answer is: \[ \frac{24}{7} \text{ or } 3 \frac{3}{7} \]