The denominator of a fraction is 3 more than its numerator. If the numerator is increased by 7 and the denominator is decreased by 2, we obtain 2. The sum of numerator and denominator of the fraction is

Options

A. 5
B. 13
C. 17
D. 19

Answer

To solve the problem step by step, let's denote the numerator of the fraction as \( x \). According to the problem, the denominator is 3 more than the numerator, so we can express the denominator as \( x + 3 \).

### Step 1: Define the variables
Let:
- Numerator = \( x \)
- Denominator = \( x + 3 \)

### Step 2: Set up the equation based on the problem statement
The problem states that if the numerator is increased by 7 and the denominator is decreased by 2, the resulting fraction equals 2. We can express this mathematically as follows:

\[ \frac{x + 7}{(x + 3) - 2} = 2 \]

This simplifies to:

\[ \frac{x + 7}{x + 1} = 2 \]

### Step 3: Cross-multiply to eliminate the fraction
To solve for \( x \), we cross-multiply:

\[ x + 7 = 2(x + 1) \]

### Step 4: Expand and simplify the equation
Expanding the right-hand side gives us:

\[ x + 7 = 2x + 2 \]

Now, we can rearrange the equation:

\[ x + 7 - 2 = 2x \]
\[ x + 5 = 2x \]

### Step 5: Solve for \( x \)
Subtract \( x \) from both sides:

\[ 5 = 2x - x \]
\[ 5 = x \]

### Step 6: Find the numerator and denominator
Now that we have \( x = 5 \), we can find the numerator and denominator:

- Numerator = \( x = 5 \)
- Denominator = \( x + 3 = 5 + 3 = 8 \)

### Step 7: Calculate the sum of the numerator and denominator
Finally, we find the sum of the numerator and denominator:

\[ \text{Sum} = 5 + 8 = 13 \]

### Final Answer
The sum of the numerator and denominator of the fraction is **13**.

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