**Solution:**

Let the required fraction be x/y.

**Step 1:**

According to the problem, when both the numerator and denominator are increased by 2, the fraction becomes 1/2.

So, (x+2)/(y+2) = 1/2

Multiplying both sides by 2(y+2), we get:

2x + 4 = y + 2

2x - y = -2 ------Equation 1

**Step 2:**

According to the problem, when both the numerator and denominator are increased by 12, the fraction becomes 3/4.

So, (x+12)/(y+12) = 3/4

Multiplying both sides by 4(y+12), we get:

4x + 48 = 3y + 36

4x - 3y = -12 ------Equation 2

**Step 3:**

Now, we have two equations:

2x - y = -2 ------Equation 1

4x - 3y = -12 ------Equation 2

We can solve these equations simultaneously to find the values of x and y.

Multiplying Equation 1 by 3 and Equation 2 by 2, we get:

6x - 3y = -6 ------Equation 3

8x - 6y = -24 ------Equation 4

Subtracting Equation 3 from Equation 4, we get:

2x - 3y = -18

2x = 3y - 18

x = (3y - 18)/2

**Step 4:**

Substituting this value of x in Equation 1, we get:

2((3y - 18)/2) - y = -2

3y - 18 - y = -2

2y = 16

y = 8

**Step 5:**

Substituting the values of x and y in the original fraction x/y, we get:

x/y = (3y - 18)/2y = (3(8) - 18)/2(8) = -3/8

**Step 6:**

Therefore, the required fraction is -3/8.

**Explanation:**

We can solve the problem using the following steps:

- First, we need to set up two equations using the given information.
- The first equation comes from the fact that when both the numerator and denominator are increased by 2, the fraction becomes 1/2.
- The second equation comes from the fact that when both the numerator and denominator are increased by 12, the fraction becomes 3/4.
- Next, we can solve these equations simultaneously to find the values of x and y.
- Finally, we can substitute these values in the original fraction x/y to get the required answer.
- It is important to pay attention to the signs of the fractions while solving the equations, as they can affect the final answer.