If the numberator of a fraction is increased by `50%` of its denominator be decreased by `10%` the value of fraction becomes `15//16`. Find the original fraction.

Options

A. `(19)/(6)`
B. `(16)/(9)`
C. `(11)/(6)`
D. `(9)/(16)`

Answer

To find the original fraction based on the given conditions, let's denote the numerator of the fraction as \( x \) and the denominator as \( y \). The fraction can be represented as \( \frac{x}{y} \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
The problem states that if the numerator \( x \) is increased by \( 50\% \) of the denominator \( y \), and the denominator \( y \) is decreased by \( 10\% \), the new value of the fraction becomes \( \frac{15}{16} \).

2. **Calculate the New Numerator**:
- The increase of \( 50\% \) of \( y \) is calculated as:
\[ 50\% \text{ of } y = 0.5y \]
- Therefore, the new numerator becomes:
\[ x + 0.5y = 1.5x \]

3. **Calculate the New Denominator**:
- The decrease of \( 10\% \) of \( y \) is calculated as:
\[ 10\% \text{ of } y = 0.1y \]
- Therefore, the new denominator becomes:
\[ y - 0.1y = 0.9y \]

4. **Set Up the Equation**:
- According to the problem, the new fraction is equal to \( \frac{15}{16} \):
\[ \frac{1.5x}{0.9y} = \frac{15}{16} \]

5. **Cross-Multiply to Solve for \( x \) and \( y \)**:
- Cross-multiplying gives:
\[ 1.5x \cdot 16 = 15 \cdot 0.9y \]
- Simplifying this:
\[ 24x = 13.5y \]

6. **Rearranging the Equation**:
- Dividing both sides by \( y \) gives:
\[ \frac{x}{y} = \frac{13.5}{24} \]

7. **Simplifying the Fraction**:
- To simplify \( \frac{13.5}{24} \), we can multiply the numerator and denominator by \( 10 \) to eliminate the decimal:
\[ \frac{135}{240} \]
- Now, simplifying \( \frac{135}{240} \) by finding the GCD (which is 15):
\[ \frac{135 \div 15}{240 \div 15} = \frac{9}{16} \]

8. **Conclusion**:
- Therefore, the original fraction is:
\[ \frac{x}{y} = \frac{9}{16} \]

### Final Answer:
The original fraction is \( \frac{9}{16} \).

---