If the numerator of certain fraction is increased by 200% and the denominator is increased by 150% the new fraction thus formed is `(9)/(10)` . What is the original fraction ?

Options

A. `(3)/(4)`
B. `(1)/(4)`
C. `(3)/(4)`
D. `(2)/(5)`

Explanation

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the problem
We need to find the original fraction \( \frac{x}{y} \) given that the numerator is increased by 200% and the denominator is increased by 150%, resulting in a new fraction of \( \frac{9}{10} \).

### Step 2: Express the changes in the numerator and denominator
- If the numerator \( x \) is increased by 200%, it becomes:
\[ x + 200\% \text{ of } x = x + 2x = 3x \]
- If the denominator \( y \) is increased by 150%, it becomes:
\[ y + 150\% \text{ of } y = y + 1.5y = 2.5y \]

### Step 3: Set up the equation for the new fraction
The new fraction formed after these changes is given as:
\[ \frac{3x}{2.5y} = \frac{9}{10} \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 3x \cdot 10 = 9 \cdot 2.5y \]
This simplifies to:
\[ 30x = 22.5y \]

### Step 5: Simplify the equation
To simplify, we can divide both sides by 7.5:
\[ \frac{30x}{7.5} = \frac{22.5y}{7.5} \]
This further simplifies to:
\[ 4x = 3y \]

### Step 6: Express \( y \) in terms of \( x \)
From the equation \( 4x = 3y \), we can express \( y \) in terms of \( x \):
\[ y = \frac{4}{3}x \]

### Step 7: Write the original fraction
The original fraction \( \frac{x}{y} \) can now be expressed as:
\[ \frac{x}{y} = \frac{x}{\frac{4}{3}x} = \frac{3}{4} \]

### Step 8: Conclusion
Thus, the original fraction is:
\[ \frac{3}{4} \]