A fraction becomes 1/2 if 5 is subtracted from its numerator and 3 is subtracted from its denominator. However,if we divide the numerator by 2 and add 7 to the denominator,the fraction becomes 1/4. Find the fraction.

To solve this problem, we need to set up two equations based on the given information and solve them simultaneously. Let's assume the fraction is represented as "x/y."

First, according to the problem, when 5 is subtracted from the numerator and 3 is subtracted from the denominator, the fraction becomes 1/2. Therefore, we have the equation:
(x - 5)/(y - 3) = 1/2

Secondly, if we divide the numerator by 2 and add 7 to the denominator, the fraction becomes 1/4. Setting up the second equation:
x/2 / (y + 7) = 1/4

Now, let's solve these equations simultaneously by eliminating one of the variables.

Multiplying both sides of the first equation by 2 and cross-multiplying, we get:
2(x - 5) = (y - 3)
2x - 10 = y - 3
2x - y = 7 -- (Equation A)

Further simplifying the second equation, we have:
(x/2) = (y + 7)/4
4x = 2(y + 7)
4x - 2y - 14 = 0
2x - y = 7 -- (Equation B)

Now we have two equations:
2x - y = 7 -- (Equation A)
2x - y = 7 -- (Equation B)

It appears that the two equations are identical. This means that there are infinitely many solutions, and we can determine the value of only one variable.

Let's find the value of x using Equation A:
2x - y = 7
2x - (2x - 7) = 7 (Substituting y from Equation B)
2x - 2x + 7 = 7
7 = 7

As 7 = 7 is a true statement, this suggests that x can have any value. Therefore, there are infinitely many solutions for this problem.

In conclusion, the fraction cannot be determined uniquely; it can have multiple values.

Let's assume the fraction to be x/y.

According to the given information, when 5 is subtracted from the numerator and 3 is subtracted from the denominator, the fraction becomes 1/2. This can be represented as:

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

Simplifying the equation,

2x - 10 = y -3
2x - y = 7 --(1)

The second given condition states that when the numerator is divided by 2 and 7 is added to the denominator, the fraction becomes 1/4.

(x / 2) / (y + 7) = 1/4

Simplifying the equation,

4x = y + 7 --(2)

We now have a system of equations consisting of equation (1) and equation (2). We can solve the system to find the values of x and y.

Multiplying equation (1) by 4,

8x - 4y = 28 --(3)

Subtracting equation (2) from equation (3),

(8x - 4y) - (4x - y) = 28 - 7

Simplifying,

4x - 3y = 21

Now we have two equations:

2x - y = 7 --(1)
4x - 3y = 21 --(4)

Multiplying equation (1) by 3 and equation (4) by 2, we can eliminate the variable x:

6x - 3y = 21 --(5)
8x - 6y = 42 --(6)

Subtracting equation (5) from equation (6),

(8x - 6y) - (6x - 3y) = 42 - 21

Simplifying,

2x - 3y = 21

Now we have two equations:

2x - 3y = 21 --(7)
2x - y = 7 --(1)

Subtracting equation (1) from equation (7),

(2x - 3y) - (2x - y) = 21 - 7

Simplifying,

-2y = 14

Dividing both sides of the equation by -2,

y = -7

Substituting the value of y into equation (1),

2x - (-7) = 7

2x + 7 = 7

Subtracting 7 from both sides of the equation,

2x = 0

Dividing both sides of the equation by 2,

x = 0

Therefore, the fraction is 0/-7, which can be written as 0. However, please note that 0/-7 is equivalent to 0/1 or any other fraction with 0 as the numerator.

original fraction is n/d

(n-5)/(d-3) = 1/2
(n/2)/(d+7) = 1/4

Solving these gives n = (d+7)/2, not a single unique solution.

Naturally, d cannot be 3, but it is odd, and n must be even, so we can have original fractions of

6/5, 8/9, 10/13, ...

or, in general,

(2k+4)/(4k+1) for k=1,2,3,...

check:
(2k+4-5)/(4k+1-3) = (2k-1)/(4k-2) = 1/2
((2k+4)/2)/(4k+1+7) = (k+2)/(4k+8) = 1/4