Simplify \[ \frac{\frac{6}{x-5}+2}{\frac{27}{x-5}+9} \]

To simplify the expression \[ \frac{\frac{6}{x-5}+2}{\frac{27}{x-5}+9}, \] we start by finding a common denominator for the fractions in both the numerator and the denominator. The common denominator is \(x-5\). ### Step 1: Simplify the Numerator The numerator is \[ \frac{6}{x-5} + 2. \] We can express \(2\) with the common denominator: \[ 2 = \frac{2(x-5)}{x-5} = \frac{2x - 10}{x - 5}. \] Now, we can combine these two fractions: \[ \frac{6}{x-5} + 2 = \frac{6 + 2(x-5)}{x-5} = \frac{6 + 2x - 10}{x-5} = \frac{2x - 4}{x-5}. \] ### Step 2: Simplify the Denominator Now, let's simplify the denominator: \[ \frac{27}{x-5} + 9. \] Again, express \(9\) with the common denominator: \[ 9 = \frac{9(x-5)}{x-5} = \frac{9x - 45}{x - 5}. \] Now, combine these fractions as well: \[ \frac{27}{x-5} + 9 = \frac{27 + 9(x-5)}{x-5} = \frac{27 + 9x - 45}{x-5} = \frac{9x - 18}{x-5}. \] ### Step 3: Rewrite the Original Expression We can now rewrite the original expression using these simplified forms of the numerator and denominator: \[ \frac{\frac{2x - 4}{x-5}}{\frac{9x - 18}{x-5}}. \] This simplifies to: \[ \frac{2x - 4}{9x - 18}. \] ### Step 4: Factor Out Common Terms Now, we can factor out the common terms in both the numerator and denominator: The numerator \(2x - 4\) can be factored as: \[ 2(x - 2). \] The denominator \(9x - 18\) can be factored as: \[ 9(x - 2). \] Thus, we have: \[ \frac{2(x - 2)}{9(x - 2)}. \] ### Step 5: Cancel Common Factors Assuming \(x \neq 2\) (since this would make the expression undefined), we can cancel the \(x - 2\) terms: \[ \frac{2}{9}, \quad (x \neq 2). \] ### Final Result Therefore, the simplified form of the original expression is: \[ \boxed{\frac{2}{9}}. \]