The numerator of a fraction is 3 less than its denominator. If 7 is added in the numerator and 2 is subtracted from the denominator, the fraction becomes 2. Find the fracion.

Options

A. a)`5/8`
B. b)`8/11`
C. c)`7/10`
D. d)`3/13`

Answer

To solve the problem step by step, let's define the variables and set up the equations based on the information given.

### Step 1: Define the Variables
Let the denominator of the fraction be \( x \). According to the problem, the numerator is 3 less than the denominator. Therefore, we can express the numerator as:
\[ \text{Numerator} = x - 3 \]

### Step 2: Set Up the Equation
The problem states that if 7 is added to the numerator and 2 is subtracted from the denominator, the fraction becomes 2. We can write this as:
\[ \frac{(x - 3) + 7}{x - 2} = 2 \]

### Step 3: Simplify the Equation
Now simplify the numerator:
\[ \frac{x + 4}{x - 2} = 2 \]

### Step 4: Cross Multiply
To eliminate the fraction, we will cross multiply:
\[ x + 4 = 2(x - 2) \]

### Step 5: Expand and Rearrange
Now, expand the right side of the equation:
\[ x + 4 = 2x - 4 \]
Next, rearrange the equation to isolate \( x \):
\[ 4 + 4 = 2x - x \]
\[ 8 = x \]

### Step 6: Find the Numerator
Now that we have the value of \( x \) (the denominator), we can find the numerator:
\[ \text{Numerator} = x - 3 = 8 - 3 = 5 \]

### Step 7: Write the Fraction
Now we can write the fraction:
\[ \text{Fraction} = \frac{\text{Numerator}}{\text{Denominator}} = \frac{5}{8} \]

### Final Answer
The fraction is:
\[ \frac{5}{8} \]