A fraction becomes `2/3` if 1 is added to both its numerator and denominator. The same fraction becomes `1/2` if 1 is subtracted from both the numerator and denominator. Such a fraction is:

Answer

To solve the problem step by step, let's denote the fraction as \( \frac{N}{D} \), where \( N \) is the numerator and \( D \) is the denominator.

### Step 1: Set up the equations based on the problem statement.

From the first part of the problem, we know that if 1 is added to both the numerator and the denominator, the fraction becomes \( \frac{2}{3} \). This gives us the equation:

\[ \frac{N + 1}{D + 1} = \frac{2}{3} \]

Cross-multiplying gives:

\[ 3(N + 1) = 2(D + 1) \]

Expanding this, we get:

\[ 3N + 3 = 2D + 2 \]

Rearranging it, we can form our first equation:

\[ 3N - 2D = -1 \quad \text{(Equation 1)} \]

### Step 2: Set up the second equation.

From the second part of the problem, if 1 is subtracted from both the numerator and the denominator, the fraction becomes \( \frac{1}{2} \). This gives us the equation:

\[ \frac{N - 1}{D - 1} = \frac{1}{2} \]

Cross-multiplying gives:

\[ 2(N - 1) = 1(D - 1) \]

Expanding this, we get:

\[ 2N - 2 = D - 1 \]

Rearranging it gives us our second equation:

\[ 2N - D = 1 \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations.

Now we have a system of equations:

1. \( 3N - 2D = -1 \) (Equation 1)
2. \( 2N - D = 1 \) (Equation 2)

From Equation 2, we can express \( D \) in terms of \( N \):

\[ D = 2N - 1 \]

### Step 4: Substitute \( D \) in Equation 1.

Now, substitute \( D \) in Equation 1:

\[ 3N - 2(2N - 1) = -1 \]

Expanding this gives:

\[ 3N - 4N + 2 = -1 \]

Combining like terms results in:

\[ -N + 2 = -1 \]

Subtracting 2 from both sides gives:

\[ -N = -3 \]

Thus, we find:

\[ N = 3 \]

### Step 5: Find \( D \) using the value of \( N \).

Now, substitute \( N \) back into the equation for \( D \):

\[ D = 2(3) - 1 = 6 - 1 = 5 \]

### Step 6: Write the final fraction.

Now that we have both \( N \) and \( D \):

\[ \text{The fraction is } \frac{N}{D} = \frac{3}{5} \]

### Final Answer:
The fraction is \( \frac{3}{5} \).

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