If the numerator of a fraction is increased by `20%` and the denominator be diminished by `10%`, the value of the fraction is `16 /21` . The original fraction is-

Options

A. `3/5`
B. `4/7`
C. `2/3`
D. `5/7`

Answer

To find the original fraction, 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 to the fraction
According to the problem:
- The numerator is increased by 20%, which means the new numerator becomes \( x + 0.2x = 1.2x \).
- The denominator is diminished by 10%, which means the new denominator becomes \( y - 0.1y = 0.9y \).

### Step 2: Set up the equation
After these changes, the new fraction is given as:
\[ \frac{1.2x}{0.9y} = \frac{16}{21} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 1.2x \cdot 21 = 16 \cdot 0.9y \]
This simplifies to:
\[ 25.2x = 14.4y \]

### Step 4: Simplify the equation
To simplify, divide both sides by \( 0.6 \):
\[ \frac{25.2}{0.6}x = \frac{14.4}{0.6}y \]
Calculating the divisions:
\[ 42x = 24y \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ \frac{x}{y} = \frac{24}{42} \]
This can be simplified further:
\[ \frac{x}{y} = \frac{4}{7} \]

### Step 6: Conclusion
Thus, the original fraction is:
\[ \frac{4}{7} \]