The denominator of a rational number is greater than its numerator by 4. If 4 is subtracted from the numerator and 2 is added to its denominator, the new number becomes `1/6`. Find the original number.
(a)`1/6`
(b)`(6)/(10)`
(c)`(10)/(6)`
(d)6

Options

A. `1/6`
B. `(6)/(10)`
C. `(10)/(6)`
D. 6

Answer

To solve the problem step-by-step, we will define the variables, set up the equations based on the conditions given, and then solve for the original rational number.

### Step 1: Define the Variables
Let the numerator of the rational number be \( x \).
Since the denominator is greater than the numerator by 4, we can express the denominator as \( x + 4 \).

### Step 2: Set Up the Equation
According to the problem, if we subtract 4 from the numerator and add 2 to the denominator, the new fraction becomes \( \frac{1}{6} \).
This gives us the equation:
\[ \frac{x - 4}{(x + 4) + 2} = \frac{1}{6} \]

### Step 3: Simplify the Equation
The denominator simplifies to \( x + 6 \). Thus, the equation becomes:
\[ \frac{x - 4}{x + 6} = \frac{1}{6} \]

### Step 4: Cross-Multiply
To eliminate the fractions, we can cross-multiply:
\[ 6(x - 4) = 1(x + 6) \]

### Step 5: Expand Both Sides
Expanding both sides gives:
\[ 6x - 24 = x + 6 \]

### Step 6: Rearrange the Equation
Now, we will move all terms involving \( x \) to one side and constant terms to the other:
\[ 6x - x = 6 + 24 \]
This simplifies to:
\[ 5x = 30 \]

### Step 7: Solve for \( x \)
Now, divide both sides by 5:
\[ x = 6 \]

### Step 8: Find the Denominator
Now that we have the numerator, we can find the denominator:
\[ \text{Denominator} = x + 4 = 6 + 4 = 10 \]

### Step 9: Write the Original Rational Number
Thus, the original rational number is:
\[ \frac{x}{x + 4} = \frac{6}{10} \]

### Conclusion
The original number is \( \frac{6}{10} \), which corresponds to option (b).

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