The numerator of a fraction is 5 less than its denominator. If 3 is added to the numerator and denominator both, the fraction becomes `(4)/(5)`. Find the original fraction.

Answer

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

### Step 1: Define the Variables
Let the denominator of the fraction be \( x \). According to the question, the numerator is 5 less than the denominator. Therefore, we can express the numerator as:
\[ y = x - 5 \]

### Step 2: Set Up the Equation
The problem states that if 3 is added to both the numerator and the denominator, the fraction becomes \( \frac{4}{5} \). We can write this as:
\[ \frac{y + 3}{x + 3} = \frac{4}{5} \]

### Step 3: Substitute the Expression for \( y \)
Now, substitute \( y \) from Step 1 into the equation:
\[ \frac{(x - 5) + 3}{x + 3} = \frac{4}{5} \]
This simplifies to:
\[ \frac{x - 2}{x + 3} = \frac{4}{5} \]

### Step 4: Cross-Multiply
To eliminate the fraction, we cross-multiply:
\[ 5(x - 2) = 4(x + 3) \]

### Step 5: Expand Both Sides
Expanding both sides gives:
\[ 5x - 10 = 4x + 12 \]

### Step 6: Rearrange the Equation
Now, we will rearrange the equation to isolate \( x \):
\[ 5x - 4x = 12 + 10 \]
This simplifies to:
\[ x = 22 \]

### Step 7: Find the Numerator
Now that we have \( x \), we can find \( y \):
\[ y = x - 5 = 22 - 5 = 17 \]

### Step 8: Write the Original Fraction
The original fraction is:
\[ \frac{y}{x} = \frac{17}{22} \]

### Final Answer
The original fraction is \( \frac{17}{22} \).

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