(7)) \( \left(\frac{2}{3}-\frac{1}{8} \cdot \frac{2}{3}\right):\left(\frac{2}{9}+\frac{13}{14}: \frac{13}{7}\right)=x:\left(7-\frac{43}{7}\right. \)

To solve the equation \[ \left(\frac{2}{3}-\frac{1}{8} \cdot \frac{2}{3}\right):\left(\frac{2}{9}+\frac{13}{14}: \frac{13}{7}\right)=x:\left(7-\frac{43}{7}\right), \] we will break it down step by step. ### Step 1: Simplify the left side **Calculate \( \frac{1}{8} \cdot \frac{2}{3} \)**: \[ \frac{1}{8} \cdot \frac{2}{3} = \frac{2}{24} = \frac{1}{12}. \] **Now calculate \( \frac{2}{3} - \frac{1}{12} \)**: To subtract these fractions, we need a common denominator. The least common multiple of 3 and 12 is 12. \[ \frac{2}{3} = \frac{8}{12}, \] so \[ \frac{2}{3} - \frac{1}{12} = \frac{8}{12} - \frac{1}{12} = \frac{7}{12}. \] ### Step 2: Simplify the right side of the left side **Calculate \( \frac{13}{14} : \frac{13}{7} \)**: Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{13}{14} : \frac{13}{7} = \frac{13}{14} \cdot \frac{7}{13} = \frac{7}{14} = \frac{1}{2}. \] **Now calculate \( \frac{2}{9} + \frac{1}{2} \)**: To add these fractions, we need a common denominator. The least common multiple of 9 and 2 is 18. \[ \frac{2}{9} = \frac{4}{18}, \quad \frac{1}{2} = \frac{9}{18}, \] so \[ \frac{2}{9} + \frac{1}{2} = \frac{4}{18} + \frac{9}{18} = \frac{13}{18}. \] ### Step 3: Combine the results Now we have: \[ \frac{7}{12} : \frac{13}{18}. \] Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{7}{12} : \frac{13}{18} = \frac{7}{12} \cdot \frac{18}{13} = \frac{7 \cdot 18}{12 \cdot 13} = \frac{126}{156}. \] Now simplify \( \frac{126}{156} \): The greatest common divisor of 126 and 156 is 6. \[ \frac{126 \div 6}{156 \div 6} = \frac{21}{26}. \] ### Step 4: Simplify the right side **Calculate \( 7 - \frac{43}{7} \)**: Convert 7 to a fraction with a denominator of 7: \[ 7 = \frac{49}{7}, \] so \[ 7 - \frac{43}{7} = \frac{49}{7} - \frac{43}{7} = \frac{6}{7}. \] ### Step 5: Set up the proportion Now we have: \[ \frac{21}{26} = \frac{x}{\frac{6}{7}}. \] Cross-multiply to solve for \( x \): \[ 21 \cdot \frac{6}{7} = 26x. \] Calculating \( 21 \cdot \frac{6}{7} \): \[ 21 \cdot \frac{6}{7} = \frac{126}{7} = 18. \] So we have: \[ 18 = 26x. \] ### Step 6: Solve for \( x \) \[ x = \frac{18}{26} = \frac{9}{13}. \] Thus, the final answer is: \[ \boxed{\frac{9}{13}}. \]