Perform the operation and simplify the answer fully.

To solve the operation ( frac{5}{9} div frac{10}{7} ), you can follow these steps:

1. Change the division to multiplication: When dividing by a fraction, you multiply by its reciprocal. The reciprocal of ( frac{10}{7} ) is ( frac{7}{10} ).

[

frac{5}{9} div frac{10}{7} = frac{5}{9} times frac{7}{10}

]

2. Multiply the fractions: Multiply the numerators together and the denominators together.

[

frac{5 times 7}{9 times 10} = frac{35}{90}

]

3. Simplify the fraction: To simplify ( frac{35}{90} ), find the greatest common divisor (GCD) of 35 and 90, which is 5. Divide both the numerator and denominator by 5.
[

frac{35 div 5}{90 div 5} = frac{7}{18}

]

So the final answer is ( frac{7}{18} ).