il \( \quad \begin{array}{rlr} & \equiv \underline{216} \cdot \frac{1}{4}: \frac{3}{8}=\frac{5}{9}: x & \frac{9}{8}: \frac{3}{4}= \\ & \equiv \underline{217} \cdot \frac{3}{4}: \frac{5}{12}=x: \frac{4}{9} & x: \frac{2}{3}=\end{array} \)

To solve the equations given in the problem, we will break them down step by step. ### First Equation The first equation is: \[ \frac{1}{4} : \frac{3}{8} = \frac{5}{9} : x \] We can rewrite the ratio as a fraction: \[ \frac{\frac{1}{4}}{\frac{3}{8}} = \frac{\frac{5}{9}}{x} \] Calculating the left side: \[ \frac{\frac{1}{4}}{\frac{3}{8}} = \frac{1}{4} \cdot \frac{8}{3} = \frac{8}{12} = \frac{2}{3} \] Now we have: \[ \frac{2}{3} = \frac{\frac{5}{9}}{x} \] Cross-multiplying gives: \[ 2x = \frac{5}{3} \implies x = \frac{5}{6} \] ### Second Equation The second equation is: \[ \frac{3}{4} : \frac{5}{12} = x : \frac{4}{9} \] Again, we rewrite the ratio as a fraction: \[ \frac{\frac{3}{4}}{\frac{5}{12}} = \frac{x}{\frac{4}{9}} \] Calculating the left side: \[ \frac{\frac{3}{4}}{\frac{5}{12}} = \frac{3}{4} \cdot \frac{12}{5} = \frac{36}{20} = \frac{9}{5} \] Now we have: \[ \frac{9}{5} = \frac{x}{\frac{4}{9}} \] Cross-multiplying gives: \[ 9 \cdot \frac{4}{9} = 5x \implies 4 = 5x \implies x = \frac{4}{5} \] ### Third Equation The third equation is: \[ x : \frac{2}{3} = \frac{5}{9} : \frac{4}{9} \] We can rewrite this as: \[ \frac{x}{\frac{2}{3}} = \frac{\frac{5}{9}}{\frac{4}{9}} \] Calculating the right side: \[ \frac{\frac{5}{9}}{\frac{4}{9}} = \frac{5}{4} \] Now we have: \[ \frac{x}{\frac{2}{3}} = \frac{5}{4} \] Cross-multiplying gives: \[ 4x = \frac{5}{3} \implies x = \frac{5}{12} \] ### Summary of Results 1. From the first equation, \( x = \frac{5}{6} \). 2. From the second equation, \( x = \frac{4}{5} \). 3. From the third equation, \( x = \frac{5}{12} \). Thus, the values of \( x \) from each equation are: - First equation: \( x = \frac{5}{6} \) - Second equation: \( x = \frac{4}{5} \) - Third equation: \( x = \frac{5}{12} \)