The numerator of a fraction exceeds the denominator. If 3 is added to the numerator and 9 is subtracted from the denominator, the new fraction is equal to 5/2. Find the original fraction.
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Let's assume that the numerator of the original fraction is represented by "x" and the denominator is represented by "y".
According to the given information, the numerator exceeds the denominator: x > y.
Now, we are given that if 3 is added to the numerator and 9 is subtracted from the denominator, the new fraction is equal to 5/2. So, the new fraction can be represented as (x + 3)/(y - 9).
According to the problem, (x + 3)/(y - 9) = 5/2.
To solve this equation, we can cross-multiply:
2(x + 3) = 5(y - 9)
2x + 6 = 5y - 45
2x - 5y = -51
Now, we are left with two equations:
x > y (Equation 1)
2x - 5y = -51 (Equation 2)
To find the original fraction, we need to solve this system of equations. However, it is important to note that there are multiple solutions to this system. Let's find one solution:
We can solve Equation 2 for x:
2x = 5y - 51
x = (5y - 51)/2
Now, substitute the value of x into Equation 1:
(5y - 51)/2 > y
Multiply both sides of the inequality by 2:
5y - 51 > 2y
Subtract 2y from both sides:
3y - 51 > 0
Add 51 to both sides:
3y > 51
Divide both sides by 3:
y > 17
So, the denominator (y) must be greater than 17.
One possible solution for this system of equations is x = 24 and y = 18.
Thus, the original fraction is 24/18, which can be simplified to 4/3.
Therefore, the original fraction is 4/3.
To solve this problem, let's break it down step by step.
Let's start by setting up the original fraction as n/d, where n represents the numerator and d represents the denominator.
According to the problem, the numerator exceeds the denominator. This means that n > d.
Next, we are told that if 3 is added to the numerator and 9 is subtracted from the denominator, the new fraction is equal to 5/2. This can be expressed as (n + 3) / (d - 9) = 5/2.
To solve for the original fraction, we need to find the values of n and d.
First, let's simplify the equation:
2(n + 3) = 5(d - 9)
2n + 6 = 5d - 45
Next, let's isolate one variable by simplifying the equation further:
2n - 5d = -51
Now, we have two equations:
n > d (since the numerator exceeds the denominator)
2n - 5d = -51
Since we have two equations with two variables, we can solve this system of equations simultaneously.
Using trial and error or substitution, we can find the values of n and d that satisfy both equations.
One possible solution is n = 28 and d = 22.
Thus, the original fraction is 28/22, which can be simplified to 14/11.