When the numerator and the denominator of a fraction are increased by 1 and 2 respectively, the fraction becomes `2/3` and when the numerator and the denominator of the same fraction are increased by 2 and 3 respectively, the fraction becomes `5/7`. What is the original fraction?

Options

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

Answer

To solve the problem, we will denote the original fraction as \( \frac{x}{y} \).

### Step 1: Set up the equations based on the problem statement.

1. According to the first condition, when the numerator is increased by 1 and the denominator by 2, the fraction becomes \( \frac{2}{3} \):
\[ \frac{x + 1}{y + 2} = \frac{2}{3} \]
Cross-multiplying gives:
\[ 3(x + 1) = 2(y + 2) \]
Expanding this, we get:
\[ 3x + 3 = 2y + 4 \]
Rearranging gives us the first equation:
\[ 3x - 2y = 1 \quad \text{(Equation 1)} \]

2. According to the second condition, when the numerator is increased by 2 and the denominator by 3, the fraction becomes \( \frac{5}{7} \):
\[ \frac{x + 2}{y + 3} = \frac{5}{7} \]
Cross-multiplying gives:
\[ 7(x + 2) = 5(y + 3) \]
Expanding this, we get:
\[ 7x + 14 = 5y + 15 \]
Rearranging gives us the second equation:
\[ 7x - 5y = 1 \quad \text{(Equation 2)} \]

### Step 2: Solve the system of equations.

Now we have the following system of equations:
1. \( 3x - 2y = 1 \) (Equation 1)
2. \( 7x - 5y = 1 \) (Equation 2)

We can solve these equations using the method of elimination or substitution. Here, we will use elimination.

First, we will multiply Equation 1 by 5 and Equation 2 by 2 to align the coefficients of \( y \):
\[ 5(3x - 2y) = 5(1) \implies 15x - 10y = 5 \quad \text{(Equation 3)} \]
\[ 2(7x - 5y) = 2(1) \implies 14x - 10y = 2 \quad \text{(Equation 4)} \]

Now we will subtract Equation 4 from Equation 3:
\[ (15x - 10y) - (14x - 10y) = 5 - 2 \]
This simplifies to:
\[ x = 3 \]

### Step 3: Substitute \( x \) back to find \( y \).

Now that we have \( x = 3 \), we can substitute this value back into either Equation 1 or Equation 2 to find \( y \). We will use Equation 1:
\[ 3(3) - 2y = 1 \]
This simplifies to:
\[ 9 - 2y = 1 \]
Rearranging gives:
\[ -2y = 1 - 9 \implies -2y = -8 \implies y = 4 \]

### Step 4: Write the original fraction.

Now we have \( x = 3 \) and \( y = 4 \). Thus, the original fraction is:
\[ \frac{x}{y} = \frac{3}{4} \]

### Final Answer:
The original fraction is \( \frac{3}{4} \).

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