Question 1103256
let x = the numerator
let y = the denominator.


the fraction you are looking for is x/y.


if 2 is added to the numerator and 3 to the denominator of the fraction, its value will be 3/4.


the equation for the fraction becomes (x+2) / (y+3) = 3/4.


If 3 is subtracted from the numerator and 1 from the denominator it's value will be 1/4.


the equation for the fraction becomes (x-3) / (y-1) = 1/4.


you have two equations that needs to be solved simultaneously.


they are:


(x+2)/(y+3) = 3/4.  (first equation)


and:


(x-3)/(y-1) = 1/4.  (second equation)


cross multiply the first equation of (x+2)/(y+3) = 3/4 to get:
4*(x+2) = 3*(y+3)
simplify to get:
4x+8 = 3y + 9
subtract 8 from both sides of this equation and subtract 3y from both sides of this equation to get:
4x - 3y = 1 (third equation)


cross multiply the second equation of (x-3)/(y-1) = 1/4
simplify to get:
4*(x-3) = 1*(y-1)
simplify to get:
4x-12 = y-1
subtract y from both sides of this equation and add 12 to both sides of this equation to get:
4x - y = 11 (fourth equation)


your two equations are now:


4x - 3y = 1 (third equation)
4x - y = 11 (fourth equation)


subtract the first equation from the second equation to get:
2y = 10
solve for y to get:
y = 5


go back to the third and fourth equations.


they are:


4x - 3y = 1 (third equation)
4x - y = 11 (fourth equation)


replace y with 5 in these equations to get:


4x - 15 = 1 (third equation)
4x - 5 = 11 (fourth equation)


solve for x in both equations to get x = 4.


your solution is x = 4 and y = 5.


your original fraction of x/y becomes 4/5.


if this is correct, then both original statements must be true.


first original statement is that 2 added to the numerator plus 3 added to the denominator = 3/4.


4+2 = 6 for the numerator and 5+3 = 8 for the denominator.


new fraction becomes 6/8 which is equal to 3/4.


second original statement is that 3 subtracted from the numerator plus 1 subtracted from the denominator = 1/4.


4-3 = 1 for the numerator and 5-1 = 4 for the denominator.


new fraction becomes 1/4.


both problem statements are true when x = 4 and y = 5.


solution is that the original fraction = 4/5.