Question 1967
all "wordy" problems like this...start slowly. Read it a few times, see what they want you to find and define these things, so ...

Let original numerator = x
Let original denominator = y

so original fraction is x/y

(x-1)/(y+2) = 1/2 and also (x-1)/(y-2) = 2/3

That is the conversion of the English done. From now...purely maths

for each equation, get rid of the fractions:

2(x-1) = y+2   and    3(x-1) = 2(y-2)
  2x-2 = y+2   and      3x-3 = 2y-4
  2x-y = 4     and     3x-2y = -1

eqn1 is 2x-y = 4 --> scaled up to 4x-2y = 8 (just by multiplying every term by 2, so that the y term in both equation now matches.
eqn2 is 3x-2y = -1

4x-2y =  8
3x-2y = -1

Subtracting these together gives x = 9. Put this into either eqn1 or eqn2 gives y=14

so original fraction was 9/14.

AND NOW CHECK THAT THIS IS THE CORRECT ANSWER BY CHANGING IT AS STATED TO SEE IF YOU GET THE ANSWERS...

(9-1)/(14+2) --> 8/16 --> 1/2 : CORRECT!
(9-1)/(14-2) --> 8/12 --> 2/3 : CORRECT!

This is the beauty of maths...You now know you got this correct, so no worries. If these didn't work out, you know you did a mistake somewhere, so you can check and find it and correct it etc etc :-)

...Write an essay on the Russian Revolution 1910-1920...such an open-ended question. Maths is better :-)

cheers
Jon