Question 321962
let x = the numerator
let y = the denominator


If you decrease its numerator by 2, then its numerator becomes x-2.
If you increase its denominator by 5, then its denominator becomes y + 5.


If you double its numerator, then its numerator becomes 2*x.
If  you decrease its denominator by 3, then its denominator becomes y - 3.


You have two equations:


The first equation is:


(x-2)/(y+5) = 1/3


The second equation is:


(2*x)/(y-3) = 8/5


You need to solve these 2 equations simultaneously to find your answer.


You can do this in several ways.


I'll solve each equation for x and then set both equations equal to each other.


This way I will have reduced the number of unknown variable to 1.


First equation is:


(x-2)/(y+5) = 1/3


Multiply both sides of this equation by (y+5) to get:


(x-2) = (1/3) * (y+5)


Add 2 to each side of this equation to get:


x = (1/3)*(y+5) + 2 (first equation solved for x)


Second equation is:


(2*x)/(y-3) = 8/5


Multiply both sides of this equation by (y-3) to get:


2*x = (8/5) * (y-3)


Divide both sides of this equation by 2 to get:


x = (8/5) * (y-3) * (1/2) (second equation solved for x)


You have x equal to 2 equations.


Since x = x, then set these equations equal to each other to get:


(1/3)*(y+5) + 2 = (8/5) * (y-3) * (1/2)


Simplify each side of this equation to get:


(y+5)/3 + 2 = (8*(y-3)/(5*2).


Simplify further to get:


(y+5)/3 + 2 = (8*(y-3)/10


Subtract (y+5)/3 from both sides of this equation to get:


2 = (8*(y-3)/10 - (y+5)/3


Multiply both sides of this equation by 30 to get:


60 = 24*(y-3) - 10*(y+5)


Simplify to get:


60 = 24*y - 72 - 10*y - 50


Combine like terms to get:


60 = 14*y - 122


Add 122 to both sides of this equation to get:


182 = 14*y


Divide both sides of this equation by 14 to get:


y = 13.


Since you know what y is, you can now solve for x by substituting in each of the equations that you previously solved for x in terms of y.


First equation is:


x = (1/3)*(y+5) + 2


Substitute 13 for y in this equation to get:


x = (1/3)*(13+5) + 2 which becomes:


x = (1/3)*(18) + 2 which becomes:


x = 6 + 2 which becomes:


x = 8


You now have:


x = 8
y = 13


You need to confirm that these numbers for x and y are good.


Your second equation that solved for x is:


x = (8/5) * (y-3) * (1/2)


Substitute 13 for y and 8 for x in this equation to get:


8 = (8/5) * (13-3) * (1/2) which becomes:


8 = (8/5) * (10) * (1/2) which becomes:


8 = (80/5) * (1/2) which becomes:


8 = (80/10) which becomes:


8 = 8 which is true confirming that the value of 8 for x and 13 for y are good.


Substitute in your original equation to confirm that these values for x and y give you what you want.


First original equation is:


(x-2)/(y+5) = 1/3


Substitute 8 for x and 13 for y to get:


(8-2) / (13+5) = 1/3 which becomes:


6 / 18 = 1/3 which is true.


Second original equation is:


(2*x)/(y-3) = 8/5


Substitute 8 for x and 13 for y to get:


(2*8) / (13-3) = 8/5 which becmes:


16 / 10 = 8/5 which is also true.


The values for x and y are good.


Your original fraction is x/y which is equal to 8/13.


Based on the original problem statements:


A fraction is equivalent to 1/3 if its numerator decreases by 2 and its denominator increases by 5. 


(8-2) / ( 13+5) = 6/18 = 1/3 (good)


When its numerator is doubled and 3 is subtracted from its denominator, the fraction becomes 8/5/ Find the original fraction 


(8*2) / (13-3) = 16/10 = 8/5 (good)