In a fraction, if numerator is increased by 2 and denominator is increased by 2, it becomes `(3)/(4)` and if numerator is decreased by 3 and denominator is decreased by 6, it becomes `(4)/(3)`. Find the sum of the numerator and denominator

Options

A. 13
B. 18
C. 20
D. 14

Explanation

To solve the problem step by step, we will define the numerator and denominator of the fraction and set up equations based on the given conditions.

### Step 1: Define the Variables
Let the numerator of the fraction be \( x \) and the denominator be \( y \).

### Step 2: Set Up the First Equation
According to the first condition, if the numerator is increased by 2 and the denominator is increased by 2, the fraction becomes \( \frac{3}{4} \). This can be expressed as:
\[ \frac{x + 2}{y + 2} = \frac{3}{4} \]
Cross-multiplying gives:
\[ 4(x + 2) = 3(y + 2) \]
Expanding this, we have:
\[ 4x + 8 = 3y + 6 \]
Rearranging the equation leads to:
\[ 4x - 3y = -2 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
According to the second condition, if the numerator is decreased by 3 and the denominator is decreased by 6, the fraction becomes \( \frac{4}{3} \). This can be expressed as:
\[ \frac{x - 3}{y - 6} = \frac{4}{3} \]
Cross-multiplying gives:
\[ 3(x - 3) = 4(y - 6) \]
Expanding this, we have:
\[ 3x - 9 = 4y - 24 \]
Rearranging the equation leads to:
\[ 3x - 4y = -15 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
Now we have a system of two equations:
1. \( 4x - 3y = -2 \)
2. \( 3x - 4y = -15 \)

We can solve these equations using the elimination method. Let's multiply Equation 1 by 3 and Equation 2 by 4 to align the coefficients of \( x \):
\[ 12x - 9y = -6 \quad \text{(Equation 3)} \]
\[ 12x - 16y = -60 \quad \text{(Equation 4)} \]

Now, we subtract Equation 3 from Equation 4:
\[ (12x - 16y) - (12x - 9y) = -60 - (-6) \]
This simplifies to:
\[ -16y + 9y = -60 + 6 \]
\[ -7y = -54 \]
Dividing both sides by -7 gives:
\[ y = \frac{54}{7} \]

### Step 5: Substitute \( y \) Back to Find \( x \)
Now we can substitute \( y \) back into one of the original equations to find \( x \). Using Equation 1:
\[ 4x - 3\left(\frac{54}{7}\right) = -2 \]
This simplifies to:
\[ 4x - \frac{162}{7} = -2 \]
Multiplying through by 7 to eliminate the fraction:
\[ 28x - 162 = -14 \]
Adding 162 to both sides gives:
\[ 28x = 148 \]
Dividing by 28:
\[ x = \frac{148}{28} = \frac{37}{7} \]

### Step 6: Find the Sum of \( x \) and \( y \)
Now we can find the sum of the numerator and denominator:
\[ x + y = \frac{37}{7} + \frac{54}{7} = \frac{37 + 54}{7} = \frac{91}{7} = 13 \]

### Final Answer
The sum of the numerator and denominator is \( 13 \).

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