When numberator of a fraction is increased by 10 % and denominator decreased by 20%
find the original fraction ?

Answer

To solve the problem step by step, we need to find the original fraction when the numerator is increased by 10% and the denominator is decreased by 20%. Let's denote the original fraction as \( \frac{x}{y} \), where \( x \) is the numerator and \( y \) is the denominator.

### Step 1: Understand the changes in the numerator and denominator
- The numerator \( x \) is increased by 10%. This means the new numerator becomes:
\[ x + 0.10x = 1.10x \]

- The denominator \( y \) is decreased by 20%. This means the new denominator becomes:
\[ y - 0.20y = 0.80y \]

### Step 2: Set up the equation
After these changes, the new fraction can be expressed as:
\[ \frac{1.10x}{0.80y} \]

### Step 3: Simplify the new fraction
To simplify the new fraction:
\[ \frac{1.10x}{0.80y} = \frac{1.10}{0.80} \cdot \frac{x}{y} \]
Calculating \( \frac{1.10}{0.80} \):
\[ \frac{1.10}{0.80} = \frac{11}{8} \]
Thus, the new fraction can be expressed as:
\[ \frac{11x}{8y} \]

### Step 4: Find the original fraction
Since we need to find the original fraction \( \frac{x}{y} \), we can set:
\[ \frac{11x}{8y} = k \quad \text{(where k is some constant)} \]
This implies that:
\[ \frac{x}{y} = \frac{8k}{11} \]

### Step 5: Conclusion
The original fraction can be expressed in terms of \( k \):
\[ \frac{x}{y} = \frac{8}{11} \]
Thus, the original fraction is:
\[ \frac{8}{11} \]