2 2/3 • 6 7/7 : 9 9/9

To solve the expression \( 2 \frac{2}{3} \cdot 6 \frac{7}{7} : 9 \frac{9}{9} \), we first need to convert the mixed numbers into improper fractions.

1. Convert \( 2 \frac{2}{3} \) into an improper fraction: \[ 2 \frac{2}{3} = \frac{2 \cdot 3 + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3} \]

2. Convert \( 6 \frac{7}{7} \) into an improper fraction: \[ 6 \frac{7}{7} = \frac{6 \cdot 7 + 7}{7} = \frac{42 + 7}{7} = \frac{49}{7} = 7 \]

3. Convert \( 9 \frac{9}{9} \) into an improper fraction: \[ 9 \frac{9}{9} = \frac{9 \cdot 9 + 9}{9} = \frac{81 + 9}{9} = \frac{90}{9} = 10 \]

Now the expression becomes: \[ \frac{8}{3} \cdot 7 : 10 \]

Next, we interpret the colon as division: \[ \frac{8}{3} \cdot 7 \div 10 \]

We can rewrite this as: \[ \frac{8}{3} \cdot 7 \cdot \frac{1}{10} \]

Now let's calculate \( \frac{8}{3} \cdot 7 \): \[ \frac{8 \cdot 7}{3} = \frac{56}{3} \]

Now substitute back: \[ \frac{56}{3} \cdot \frac{1}{10} = \frac{56}{30} \]

Next, we can simplify \( \frac{56}{30} \) by finding the greatest common divisor (GCD), which is 2: \[ \frac{56 \div 2}{30 \div 2} = \frac{28}{15} \]

Thus, the final answer is: \[ \frac{28}{15} \]

If you prefer it as a mixed number: \[ \frac{28}{15} = 1 \frac{13}{15} \]

So, the expression \( 2 \frac{2}{3} \cdot 6 \frac{7}{7} : 9 \frac{9}{9} \) simplifies to: \[ \frac{28}{15} \text{ or } 1 \frac{13}{15} \]