A fraction becomes 1/6 when 4 is subtracted from its numerator and 1 is added to its denominator. If 2 and 1 are respectively added to its numerator and the denominator, it becomes 1/3. Then, the LCM of the numerator and denominator of the said fraction, must be

Options

A. a) 14
B. b) 350
C. c) 5
D. d) 70

Answer

To solve the problem step-by-step, we will first set up the equations based on the conditions given in the question.

### Step 1: Let the fraction be represented as \( \frac{x}{y} \)

Let \( x \) be the numerator and \( y \) be the denominator of the fraction.

### Step 2: Set up the first equation based on the first condition

According to the first condition, the fraction becomes \( \frac{1}{6} \) when 4 is subtracted from the numerator and 1 is added to the denominator. This gives us the equation:

\[ \frac{x - 4}{y + 1} = \frac{1}{6} \]

Cross-multiplying gives:

\[ 6(x - 4) = 1(y + 1) \]

Expanding this, we have:

\[ 6x - 24 = y + 1 \]

Rearranging gives us:

\[ 6x - y = 25 \quad \text{(Equation 1)} \]

### Step 3: Set up the second equation based on the second condition

According to the second condition, if 2 is added to the numerator and 1 is added to the denominator, the fraction becomes \( \frac{1}{3} \). This gives us the equation:

\[ \frac{x + 2}{y + 1} = \frac{1}{3} \]

Cross-multiplying gives:

\[ 3(x + 2) = 1(y + 1) \]

Expanding this, we have:

\[ 3x + 6 = y + 1 \]

Rearranging gives us:

\[ 3x - y = -5 \quad \text{(Equation 2)} \]

### Step 4: Solve the system of equations

Now we have two equations:

1. \( 6x - y = 25 \) (Equation 1)
2. \( 3x - y = -5 \) (Equation 2)

We can eliminate \( y \) by subtracting Equation 2 from Equation 1:

\[ (6x - y) - (3x - y) = 25 - (-5) \]

This simplifies to:

\[ 3x = 30 \]

Dividing both sides by 3 gives:

\[ x = 10 \]

### Step 5: Substitute \( x \) back to find \( y \)

Now, substitute \( x = 10 \) back into either equation to find \( y \). We'll use Equation 2:

\[ 3(10) - y = -5 \]

This simplifies to:

\[ 30 - y = -5 \]

Rearranging gives:

\[ y = 30 + 5 = 35 \]

### Step 6: Find the LCM of \( x \) and \( y \)

Now we have \( x = 10 \) and \( y = 35 \). We need to find the LCM of 10 and 35.

The prime factorization of 10 is \( 2 \times 5 \) and for 35 is \( 5 \times 7 \).

The LCM is found by taking the highest power of each prime:

\[ \text{LCM}(10, 35) = 2^1 \times 5^1 \times 7^1 = 70 \]

### Final Answer

The LCM of the numerator and denominator of the fraction is \( \boxed{70} \).