A fraction becomes 2 / 3 if 1 is added to its numerator and 2 is added to its denominator . If 2 is subtracted from the numerator and 5 is added to its denominator, the fraction becomes 1 / 4, If 3 is added to the numerator and 1 is subtracted from the denominator, then fraction becomes:6/53/45/74/3

The problem states that we have a fraction that becomes 2/3 if 1 is added to its numerator and 2 is added to its denominator. This can be written as:

(x+1)/(y+2) = 2/3

The problem also states that if 2 is subtracted from the numerator and 5 is added to its denominator, the fraction becomes 1/4. This can be written as:

(x-2)/(y+5) = 1/4

We have a system of two equations that we can solve to find the values of x and y.

First, we can cross-multiply to get rid of the fractions:

3(x+1) = 2(y+2) --> 3x + 3 = 2y + 4 --> 3x - 2y = 1 ...(1)

4(x-2) = (y+5) --> 4x - 8 = y + 5 --> 4x - y = 13 ...(2)

We can subtract equation (2) from equation (1) to find the value of x:

(3x - 2y) - (4x - y) = 1 - 13 -x - y = -12 x + y = 12 ...(3)

Adding equation (3) to equation (2) gives:

4x - y + x + y = 13 + 12 5x = 25 x = 5

Substituting x = 5 into equation (3) gives:

5 + y = 12 y = 7

So, the original fraction is 5/7.

Finally, if 3 is added to the numerator and 1 is subtracted from the denominator, the fraction becomes:

(5+3)/(7-1) = 8/6 = 4/3

So, the answer is 4/3.