Answer:
[tex]\dfrac{7}{9}[/tex]
Step-by-step explanation:
Let x be the denominator.
If the numerator is 2 less than the denominator, then the expression for the numerator is (x - 2):
[tex]\dfrac{x-2}{x}[/tex]
If 3 is added to both the numerator and the denominator, and the answer is 5/6, then:
[tex]\dfrac{x-2+3}{x+3}=\dfrac{5}{6}[/tex]
Now we can solve the equation for x.
Simplify the numerator in the fraction on the left of the equation:
[tex]\dfrac{x+1}{x+3}=\dfrac{5}{6}[/tex]
Cross mutliply:
[tex]6(x+1)=5(x+3)[/tex]
Expand the brackets:
[tex]6 \cdot x +6 \cdot 1 = 5 \cdot x + 5 \cdot 3[/tex]
[tex]6x+6=5x+15[/tex]
Subtract 5x from both sides of the equation:
[tex]6x+6-5x=5x+15-5x[/tex]
[tex]x+6=15[/tex]
Subtract 6 from both sides of the equation:
[tex]x+6-6=15-6[/tex]
[tex]x=9[/tex]
Therefore, the value of x is 9.
Now substitute the found value of x into the original rational expression:
[tex]\dfrac{x-2}{x}=\dfrac{9-2}{9}=\dfrac{7}{9}[/tex]
Therefore, the original fraction is:
[tex]\boxed{\dfrac{7}{9}}[/tex]
