In a fraction `(x)/(y)` which is reduced to its lowest form where x and y are positive integers, if two is subtracted from numberator and three is added to the denominator , the fraction becomes `(3)/(5)` . What is the largest possible value of the orginal fraction ?

Answer

To solve the problem, we need to find the largest possible value of the fraction \( \frac{x}{y} \) given the conditions stated. Let's break it down step by step.

### Step 1: Set up the equation based on the problem statement
We know that when 2 is subtracted from the numerator \( x \) and 3 is added to the denominator \( y \), the fraction becomes \( \frac{3}{5} \). Therefore, we can set up the equation:

\[ \frac{x - 2}{y + 3} = \frac{3}{5} \]

### Step 2: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:

\[ 5(x - 2) = 3(y + 3) \]

### Step 3: Expand both sides
Expanding both sides of the equation results in:

\[ 5x - 10 = 3y + 9 \]

### Step 4: Rearrange the equation
Rearranging the equation to isolate terms gives:

\[ 5x - 3y = 19 \]

### Step 5: Solve for integer values of \( x \) and \( y \)
Now, we need to find positive integer solutions for \( x \) and \( y \). We can express \( y \) in terms of \( x \):

\[ 3y = 5x - 19 \implies y = \frac{5x - 19}{3} \]

For \( y \) to be an integer, \( 5x - 19 \) must be divisible by 3. We can check values of \( x \) to find suitable \( y \).

### Step 6: Check values of \( x \)
Let's check integer values of \( x \):

1. **For \( x = 5 \)**:
\[ y = \frac{5(5) - 19}{3} = \frac{25 - 19}{3} = \frac{6}{3} = 2 \]
So, \( (x, y) = (5, 2) \).

2. **For \( x = 6 \)**:
\[ y = \frac{5(6) - 19}{3} = \frac{30 - 19}{3} = \frac{11}{3} \quad \text{(not an integer)} \]

3. **For \( x = 7 \)**:
\[ y = \frac{5(7) - 19}{3} = \frac{35 - 19}{3} = \frac{16}{3} \quad \text{(not an integer)} \]

4. **For \( x = 8 \)**:
\[ y = \frac{5(8) - 19}{3} = \frac{40 - 19}{3} = \frac{21}{3} = 7 \]
So, \( (x, y) = (8, 7) \).

5. **For \( x = 9 \)**:
\[ y = \frac{5(9) - 19}{3} = \frac{45 - 19}{3} = \frac{26}{3} \quad \text{(not an integer)} \]

6. **For \( x = 10 \)**:
\[ y = \frac{5(10) - 19}{3} = \frac{50 - 19}{3} = \frac{31}{3} \quad \text{(not an integer)} \]

7. **For \( x = 11 \)**:
\[ y = \frac{5(11) - 19}{3} = \frac{55 - 19}{3} = \frac{36}{3} = 12 \]
So, \( (x, y) = (11, 12) \).

### Step 7: Calculate the fractions
Now we calculate the fractions for the integer pairs we found:

1. For \( (5, 2) \):
\[ \frac{x}{y} = \frac{5}{2} = 2.5 \]

2. For \( (8, 7) \):
\[ \frac{x}{y} = \frac{8}{7} \approx 1.14 \]

3. For \( (11, 12) \):
\[ \frac{x}{y} = \frac{11}{12} \approx 0.92 \]

### Conclusion
The largest possible value of the original fraction \( \frac{x}{y} \) is:

\[ \frac{5}{2} \]