यदि किसी भिन्न का अंश, हर से 2 कम है और यदि हर में 1 जोडा जाए तो यह `(1)/(2)` हो जाता है तब वह भिन्न होगी ?

Options

A. `4/5`
B. `6/5`
C. `3/5`
D. `2/5`

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Define the fraction
Let the fraction be represented as \( \frac{p}{q} \), where \( p \) is the numerator and \( q \) is the denominator.

**Hint:** Start by defining the variables for the numerator and denominator of the fraction.

### Step 2: Set up the equation based on the problem statement
According to the problem, the numerator \( p \) is 2 less than the denominator \( q \). Therefore, we can write:
\[ p = q - 2 \]

**Hint:** Use the information given in the problem to express one variable in terms of the other.

### Step 3: Set up the second equation
The problem states that if 1 is added to both the numerator and the denominator, the fraction becomes \( \frac{1}{2} \). This gives us the equation:
\[ \frac{p + 1}{q + 1} = \frac{1}{2} \]

**Hint:** Write down the equation that represents the condition after adding 1 to both the numerator and the denominator.

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 2(p + 1) = 1(q + 1) \]
This simplifies to:
\[ 2p + 2 = q + 1 \]

**Hint:** Cross-multiplication is a useful technique to eliminate fractions in equations.

### Step 5: Substitute \( p \) from Step 2 into the equation
Substituting \( p = q - 2 \) into the equation \( 2p + 2 = q + 1 \):
\[ 2(q - 2) + 2 = q + 1 \]
This simplifies to:
\[ 2q - 4 + 2 = q + 1 \]
\[ 2q - 2 = q + 1 \]

**Hint:** Always substitute known values into equations to simplify them further.

### Step 6: Solve for \( q \)
Now, rearranging the equation:
\[ 2q - q = 1 + 2 \]
\[ q = 3 \]

**Hint:** Combine like terms to isolate the variable you are solving for.

### Step 7: Find \( p \) using the value of \( q \)
Now that we have \( q = 3 \), we can find \( p \):
\[ p = q - 2 = 3 - 2 = 1 \]

**Hint:** Use the value of one variable to find the other variable.

### Step 8: Write the final fraction
The fraction is:
\[ \frac{p}{q} = \frac{1}{3} \]

**Hint:** Ensure you express the final answer clearly as a fraction.

### Final Answer
The fraction is \( \frac{1}{3} \).