The sum of the numerator and the denominator of a fraction is 11. If 1 is added to the numerator and 2 is subtracted from the denominator it becomes `2/5`. The fraction is :

Options

A. `5/6`
B. `3/8`
C. `4/7`
D. `1/10`

Answer

To solve the problem step by step, let's denote the numerator of the fraction as \( p \) and the denominator as \( q \).

### Step 1: Set up the equations
We know from the problem statement:
1. The sum of the numerator and the denominator is 11.
\[ p + q = 11 \quad \text{(Equation 1)} \]

2. If 1 is added to the numerator and 2 is subtracted from the denominator, the fraction becomes \( \frac{2}{5} \).
\[ \frac{p + 1}{q - 2} = \frac{2}{5} \quad \text{(Equation 2)} \]

### Step 2: Cross-multiply to eliminate the fraction
From Equation 2, we can cross-multiply:
\[ 5(p + 1) = 2(q - 2) \]
Expanding both sides gives:
\[ 5p + 5 = 2q - 4 \]
Rearranging this, we get:
\[ 5p - 2q = -9 \quad \text{(Equation 3)} \]

### Step 3: Solve the system of equations
Now we have two equations:
1. \( p + q = 11 \) (Equation 1)
2. \( 5p - 2q = -9 \) (Equation 3)

From Equation 1, we can express \( q \) in terms of \( p \):
\[ q = 11 - p \]

### Step 4: Substitute \( q \) in Equation 3
Substituting \( q \) in Equation 3:
\[ 5p - 2(11 - p) = -9 \]
Expanding this gives:
\[ 5p - 22 + 2p = -9 \]
Combining like terms:
\[ 7p - 22 = -9 \]
Adding 22 to both sides:
\[ 7p = 13 \]
Dividing by 7:
\[ p = \frac{13}{7} \]

### Step 5: Find \( q \)
Now substitute \( p \) back into Equation 1 to find \( q \):
\[ q = 11 - \frac{13}{7} \]
To perform the subtraction, convert 11 into a fraction with a denominator of 7:
\[ q = \frac{77}{7} - \frac{13}{7} = \frac{64}{7} \]

### Step 6: Write the fraction
Now we can write the fraction:
\[ \text{Fraction} = \frac{p}{q} = \frac{\frac{13}{7}}{\frac{64}{7}} = \frac{13}{64} \]

### Final Answer:
The fraction is \( \frac{13}{64} \).

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