If the numerator ad the denominator of a fraction is increased by 3 and 4 repectively, the fraction becomes `(4)/(5)` and if the numerator and denominator of the same fraction are increased by 7 and 3 respectively, the fraction becomes `(4)/(3)`. What is the original fraction ?

Options

A. `(4)/(5)`
B. `(2)/(3)`
C. `(5)/(6)`
D. `(6)/(7)`

Answer

To find the original fraction given the conditions in the problem, we can follow these steps:

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

### Step 2: Set Up the First Equation
According to the first condition, if the numerator is increased by 3 and the denominator by 4, the fraction becomes \( \frac{4}{5} \):
\[ \frac{x + 3}{y + 4} = \frac{4}{5} \]
Cross-multiplying gives:
\[ 5(x + 3) = 4(y + 4) \]
Expanding this leads to:
\[ 5x + 15 = 4y + 16 \]
Rearranging gives us the first equation:
\[ 5x - 4y = 1 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
According to the second condition, if the numerator is increased by 7 and the denominator by 3, the fraction becomes \( \frac{4}{3} \):
\[ \frac{x + 7}{y + 3} = \frac{4}{3} \]
Cross-multiplying gives:
\[ 3(x + 7) = 4(y + 3) \]
Expanding this leads to:
\[ 3x + 21 = 4y + 12 \]
Rearranging gives us the second equation:
\[ 3x - 4y = -9 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
Now we have the system of equations:
1. \( 5x - 4y = 1 \)
2. \( 3x - 4y = -9 \)

We can eliminate \( y \) by subtracting Equation 2 from Equation 1:
\[ (5x - 4y) - (3x - 4y) = 1 - (-9) \]
This simplifies to:
\[ 2x = 10 \]
Dividing both sides by 2 gives:
\[ x = 5 \]

### Step 5: Substitute to Find \( y \)
Now substitute \( x = 5 \) back into Equation 1:
\[ 5(5) - 4y = 1 \]
This simplifies to:
\[ 25 - 4y = 1 \]
Rearranging gives:
\[ 4y = 24 \]
Dividing both sides by 4 gives:
\[ y = 6 \]

### Step 6: Write the Original Fraction
Now that we have \( x = 5 \) and \( y = 6 \), the original fraction is:
\[ \frac{x}{y} = \frac{5}{6} \]

### Final Answer
The original fraction is \( \frac{5}{6} \).

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