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Question 321962: A fraction is equivalent to 1/3 if its numerator decreases by 2 and its denominator increases by 5. When its numerator is doubled and 3 is subtracted from its denominator, the fraction becomes 8/5/ Find the original fraction
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 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)
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