Question 1192891
<pre>
You must have copied this problem wrong, or your teacher or the
the textbook writer made a mistake, for there is no solution. 
However, if 1/12 is changed to 1/9, there will be a solution.  

I will work a problem exactly like yours, changing 1/12 to 1/9. 
The problem I will work is:</pre>
The numerator of a POSITIVE PROPER FRACTION REDUCED TO 
LOWEST TERMS is 1 less than the denominator.  When both numerator 
and denominator are increased by 2, the fraction is increased by
1/9.  Find the original fraction.<pre>

Let n = the denominator</pre>
The numerator of a fraction is 1 less than the denominator.<pre> 
Then the numerator = n-1

{{{matrix(1,2,Original,Fraction)}}}{{{""=""}}}{{{(n-1)/n}}}</pre>

when both numerator and denominator are increased by 2,<pre> 

{{{matrix(1,2,Increased,Fraction) = ((n-1)+2)/(n+2)}}}</pre>

the fraction is increased by 1/9.<pre>
{{{matrix(1,2,Increased,Fraction)}}}{{{""=""}}}{{{ matrix(1,2,Original,Fraction)}}}{{{""+""}}}{{{expr(1/9)matrix(1,2,Original,Fraction)}}}

{{{((n-1)+2)/(n+2)}}}{{{""=""}}}{{{(n-1)/n}}}{{{""+""}}}{{{expr(1/9)expr((n-1)/n)}}}

{{{(n-1+2)/(n+2)}}}{{{""=""}}}{{{(n-1)/n}}}{{{""+""}}}{{{(n-1)/(9n)}}}

{{{(n+1)/(n+2)}}}{{{""=""}}}{{{(n-1)/n}}}{{{""+""}}}{{{(n-1)/(9n)}}}

Multiply through by the LCD = 9n(n+2)

{{{9n(n+1)}}}{{{""=""}}}{{{9(n+2)(n-1)}}}{{{""+""}}}{{{(n+2)(n-1)}}}

{{{9n^2+9n}}}{{{""=""}}}{{{9(n^2+n-2)}}}{{{""+""}}}{{{n^2+n-2}}}

{{{9n^2+9n}}}{{{""=""}}}{{{9n^2+9n-18)}}}{{{""+""}}}{{{n^2+n-2}}}

{{{9n^2+9n}}}{{{""=""}}}{{{10n^2+10n-20)}}}

{{{-n^2-n+20}}}{{{""=""}}}{{{0}}}

{{{n^2+n-20}}}{{{""=""}}}{{{0}}}

{{{(n+5)(n-4)}}}{{{""=""}}}{{{0}}}

n+5=0;   n-4=0
  n=-5;    n=4

Ignore the negative answer.

denominator = n = 4
numerator = n-1 = 4-1 = 3

original fraction = {{{3/4}}}

Checking:

The new fraction is {{{(3+2)/(4+2)}}}{{{""=""}}}{{{5/6}}}

{{{1/9}}} of {{{3/4}}} is {{{(1/9)(3/4)}}}{{{""=""}}}{{{3/36}}}{{{""=""}}}{{{1/12}}}

If we add {{{3/4+1/12}}}{{{""=""}}}{{{9/12+1/12}}}{{{""=""}}}{{{10/12}}}{{{""=""}}}{{{5/6}}}.

So the correct answer to the problem I solved is {{{3/4}}}.

Be sure to inform your teacher that the problem as stated 
(with 1/12) has no solution.

Edwin</pre>