If 3 is added to both the numerator and the denominator, the fraction becomes `10/11`. When 4 is subtracted from both the numerator and the denominator of the same fraction, it becomes `3/4` What is the value of the fraction?

Options

A. `3/5`
B. `7/8`
C. `3/4`
D. `6/13`

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Define the variables
Let the numerator of the fraction be \( N \) and the denominator be \( D \).

### Step 2: Set up the first equation
According to the problem, if 3 is added to both the numerator and the denominator, the fraction becomes \( \frac{10}{11} \). This can be expressed as:
\[ \frac{N + 3}{D + 3} = \frac{10}{11} \]
Cross-multiplying gives:
\[ 11(N + 3) = 10(D + 3) \]
Expanding this, we have:
\[ 11N + 33 = 10D + 30 \]
Rearranging leads to:
\[ 11N - 10D = -3 \quad \text{(Equation 1)} \]

### Step 3: Set up the second equation
Next, the problem states that if 4 is subtracted from both the numerator and the denominator, the fraction becomes \( \frac{3}{4} \). This can be expressed as:
\[ \frac{N - 4}{D - 4} = \frac{3}{4} \]
Cross-multiplying gives:
\[ 4(N - 4) = 3(D - 4) \]
Expanding this, we have:
\[ 4N - 16 = 3D - 12 \]
Rearranging leads to:
\[ 4N - 3D = 4 \quad \text{(Equation 2)} \]

### Step 4: Solve the system of equations
Now we have two equations:
1. \( 11N - 10D = -3 \)
2. \( 4N - 3D = 4 \)

To eliminate one variable, we can multiply Equation 2 by 11:
\[ 44N - 33D = 44 \quad \text{(Equation 3)} \]
Now we can multiply Equation 1 by 4:
\[ 44N - 40D = -12 \quad \text{(Equation 4)} \]

### Step 5: Subtract the equations
Now, we subtract Equation 4 from Equation 3:
\[ (44N - 33D) - (44N - 40D) = 44 - (-12) \]
This simplifies to:
\[ 7D = 56 \]
Thus,
\[ D = 8 \]

### Step 6: Substitute back to find \( N \)
Now that we have \( D \), we can substitute \( D = 8 \) back into either Equation 1 or Equation 2. Let's use Equation 2:
\[ 4N - 3(8) = 4 \]
This simplifies to:
\[ 4N - 24 = 4 \]
Adding 24 to both sides gives:
\[ 4N = 28 \]
Thus,
\[ N = 7 \]

### Step 7: Write the final fraction
Now we have both \( N \) and \( D \):
\[ N = 7, \quad D = 8 \]
So the value of the fraction is:
\[ \frac{N}{D} = \frac{7}{8} \]

### Final Answer
The value of the fraction is \( \frac{7}{8} \).

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