"Rational numbers" are fractional numbers of the form #frac(x)(y)# where both the numerator and denominator are integers, i.e. #frac(x)(y);# #x, y in ZZ#.

We know that some rational number with a denominator of #9# is divided by #- frac(2)(3)#.

Let's consider this rational to be #frac(a)(9)#:

#" " " " " " " " " " " " " " " " " " " frac(a)(9) div - frac(2)(3)#

#" " " " " " " " " " " " " " " " " " " frac(a)(9) times - frac(3)(2)#

#" " " " " " " " " " " " " " " " " " " " - frac(3 a)(18)#

Now, this result is multiplied by #frac(4)(5)#, and then #- frac(5)(6)# is added to it:

#" " " " " " " " " " " " " " (- frac(3 a)(18) times frac(4)(5)) + (- frac(5)(6))#

#" " " " " " " " " " " " " " " " " " - frac(12 a)(90) - frac(5)(6)#

#" " " " " " " " " " " " " " " " " - (frac(12 a)(90) + frac(5)(6))#

#" " " " " " " " " " " " " " - (frac(6 times 12 a + 90 times 5)(90 times 6))#

#" " " " " " " " " " " " " " " " - (frac(72 a + 450)(540))#

Lastly, we know that the final value is #frac(1)(10)#:

#" " " " " " " " " " " " " " - (frac(72 a + 450)(540)) = frac(1)(10)#

#" " " " " " " " " " " " " " frac(72 a + 450)(540) = - frac(1)(10)#

#" " " " " " " " " " " " " " 72 a + 450 = - frac(540)(10)#

#" " " " " " " " " " " " " " 72 a + 450 = - 54#

#" " " " " " " " " " " " " " " 72 a = - 504#

#" " " " " " " " " " " " " " " " " a = - 7#

Let's substitute #- 7# in place of #a# in our rational number:

#" " " " " " " " " " " " " " " " frac(a)(9) = - frac(7)(9)#

Therefore, the original rational number is #- frac(7)(9)#.