Answer:
The original fraction is equal to  [tex]\frac{5}{6}[/tex]
Step-by-step explanation:
Let
The fraction in the simplest form equal to Â
[tex]\frac{x}{y}[/tex]
we know that
The denominator of a fraction in the simplest form is greater than the numerator by 1
so
[tex]y=x+1[/tex] ----> equation A
If 4 is added to the numerator, and 3 is subtracted from the denominator, then the fraction itself is increased by 2 1/6
Remember that
[tex]2\frac{1}{6}=2+\frac{1}{6}=\frac{13}{6}[/tex]
so
[tex]\frac{x+4}{y-3}=\frac{13}{6}+(\frac{x}{y})[/tex] ----> equation B
substitute equation A in equation B
[tex]\frac{x+4}{x+1-3}=\frac{13}{6}+(\frac{x}{x+1})[/tex]
solve for x
[tex]\frac{x+4}{x-2}=\frac{13}{6}+(\frac{x}{x+1})[/tex]
[tex]\frac{x+4}{x-2}=\frac{13(x+1)+6x}{6(x+1)}[/tex]
[tex]\frac{x+4}{x-2}=\frac{19x+13}{6(x+1)}[/tex]
Multiply in cross  Â
[tex]6(x^2+x+4x+4)=19x^2-38x+13x-26[/tex]
[tex]6(x^2+x+4x+4)=19x^2-25x-26[/tex]
[tex]6x^2+30x+24=19x^2-25x-26[/tex]
[tex]19x^2-6x^2-25x-30x-24-26=0[/tex]
[tex]13x^2-55x-50=0[/tex]
solve the quadratic equation by graphing
The solution is x=5
therefore
Find the value of y
[tex]y=5+1=6[/tex]
The original fraction is equal to  [tex]\frac{5}{6}[/tex]
