If the numerator of a certain fraction is increased by 2 and the denominator is increased by 1, then the resulting fraction is equal to `1//2`. If however, the numerator is increased by 1 and denominator is decreased by 2, then the resulting fraction is equal to `3/5`. Find the original fraction.

Options

A. `2/7`
B. `3/5`
C. `1/7`
D. `2/5`

Answer

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

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

1. According to the first condition given in the problem, when the numerator is increased by 2 and the denominator is increased by 1, the resulting fraction equals \( \frac{1}{2} \). This can be expressed as:
\[ \frac{x + 2}{y + 1} = \frac{1}{2} \]
Cross-multiplying gives us:
\[ 2(x + 2) = 1(y + 1) \]
Simplifying this, we get:
\[ 2x + 4 = y + 1 \quad \text{(Equation 1)} \]
Rearranging gives:
\[ 2x - y = -3 \]

2. According to the second condition, when the numerator is increased by 1 and the denominator is decreased by 2, the resulting fraction equals \( \frac{3}{5} \). This can be expressed as:
\[ \frac{x + 1}{y - 2} = \frac{3}{5} \]
Cross-multiplying gives us:
\[ 5(x + 1) = 3(y - 2) \]
Simplifying this, we get:
\[ 5x + 5 = 3y - 6 \quad \text{(Equation 2)} \]
Rearranging gives:
\[ 5x - 3y = -11 \]

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

We now have the following system of equations:
1. \( 2x - y = -3 \) (Equation 1)
2. \( 5x - 3y = -11 \) (Equation 2)

To eliminate \( y \), we can multiply Equation 1 by 3:
\[ 3(2x - y) = 3(-3) \]
This gives us:
\[ 6x - 3y = -9 \quad \text{(Equation 3)} \]

Now we can subtract Equation 2 from Equation 3:
\[ (6x - 3y) - (5x - 3y) = -9 - (-11) \]
This simplifies to:
\[ 6x - 5x = -9 + 11 \]
So,
\[ x = 2 \]

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

Now that we have \( x = 2 \), we can substitute this value back into Equation 1 to find \( y \):
\[ 2(2) - y = -3 \]
This simplifies to:
\[ 4 - y = -3 \]
Rearranging gives:
\[ -y = -3 - 4 \]
So,
\[ -y = -7 \quad \Rightarrow \quad y = 7 \]

### Step 4: Write the original fraction.

Now that we have both \( x \) and \( y \):
\[ x = 2, \quad y = 7 \]
The original fraction is:
\[ \frac{x}{y} = \frac{2}{7} \]

### Final Answer:
The original fraction is \( \frac{2}{7} \).

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