3 2/3:a=4 8/9:1 5/7

To solve the proportion \( \frac{3 \frac{2}{3}}{a} = \frac{4 \frac{8}{9}}{1 \frac{5}{7}} \), we need to convert the mixed numbers into improper fractions first.

1. Convert \( 3 \frac{2}{3} \): \[ 3 \frac{2}{3} = \frac{3 \times 3 + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3} \]

2. Convert \( 4 \frac{8}{9} \): \[ 4 \frac{8}{9} = \frac{4 \times 9 + 8}{9} = \frac{36 + 8}{9} = \frac{44}{9} \]

3. Convert \( 1 \frac{5}{7} \): \[ 1 \frac{5}{7} = \frac{1 \times 7 + 5}{7} = \frac{7 + 5}{7} = \frac{12}{7} \]

Now, substituting these values into the proportion gives:

\[ \frac{\frac{11}{3}}{a} = \frac{\frac{44}{9}}{\frac{12}{7}} \]

Next, simplify the right side: \[ \frac{\frac{44}{9}}{\frac{12}{7}} = \frac{44}{9} \times \frac{7}{12} = \frac{44 \cdot 7}{9 \cdot 12} = \frac{308}{108} \] Now, simplify \( \frac{308}{108} \):

• The GCD of 308 and 108 is 4. \[ \frac{308 \div 4}{108 \div 4} = \frac{77}{27} \]

Now we have: \[ \frac{\frac{11}{3}}{a} = \frac{77}{27} \]

To solve for \( a \), cross-multiply: \[ 11 \cdot 27 = 77 \cdot 3a \] \[ 297 = 231a \]

Now, solve for \( a \): \[ a = \frac{297}{231} \]

Simplifying \( \frac{297}{231} \) involves finding the GCD of 297 and 231, which is 11: \[ a = \frac{297 \div 11}{231 \div 11} = \frac{27}{21} = \frac{9}{7} \]

Thus, the solution for \( a \) is: \[ \boxed{\frac{9}{7}} \]