the denominator of a rational number is greater than its numerator by `3`. If `3` is subtracted from the numerator and `2` is added to its denominator, the new number becomes `1/5`. Find the original number. Check your solution.

Options

A. `5/8`
B. `3/8`
C. `7/8`
D. `9/8`

Answer

To solve the problem step by step, we will define the variables and set up the equations based on the information provided.

### Step 1: Define the variables
Let the numerator of the rational number be \( x \).
Then, the denominator will be \( x + 3 \) (since the denominator is greater than the numerator by 3).

### Step 2: Set up the equation
According to the problem, if 3 is subtracted from the numerator and 2 is added to the denominator, the new fraction becomes \( \frac{1}{5} \).
This gives us the equation:
\[ \frac{x - 3}{(x + 3) + 2} = \frac{1}{5} \]
This simplifies to:
\[ \frac{x - 3}{x + 5} = \frac{1}{5} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 5(x - 3) = 1(x + 5) \]

### Step 4: Expand both sides
Expanding both sides results in:
\[ 5x - 15 = x + 5 \]

### Step 5: Rearrange the equation
Now, we will move all terms involving \( x \) to one side and constant terms to the other side:
\[ 5x - x = 5 + 15 \]
This simplifies to:
\[ 4x = 20 \]

### Step 6: Solve for \( x \)
Dividing both sides by 4 gives:
\[ x = 5 \]

### Step 7: Find the original rational number
Now that we have the numerator, we can find the denominator:
\[ \text{Denominator} = x + 3 = 5 + 3 = 8 \]
Thus, the original rational number is:
\[ \frac{5}{8} \]

### Step 8: Check the solution
To verify, we will substitute \( x = 5 \) back into the conditions of the problem:
1. Subtracting 3 from the numerator:
\[ 5 - 3 = 2 \]
2. Adding 2 to the denominator:
\[ 8 + 2 = 10 \]
3. The new fraction becomes:
\[ \frac{2}{10} = \frac{1}{5} \]
This matches the condition given in the problem, confirming our solution is correct.

### Final Answer
The original rational number is \( \frac{5}{8} \).

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