A fraction is such that if double of the numberator and the triple of denominator is changed by `40%` increase and `30%` decrease respectively. Then we get `14%` of `16//21` find the original fraction?

Options

A. `(24)/(2)`
B. `(25)/(2)`
C. `(2)/(25)`
D. `(22)/(2)`

Answer

To solve the problem step by step, let's denote the numerator of the fraction as \( n \) and the denominator as \( d \). The fraction can be represented as \( \frac{n}{d} \).

### Step 1: Understand the changes to the numerator and denominator
According to the problem:
- The double of the numerator is \( 2n \).
- The triple of the denominator is \( 3d \).
- The numerator is increased by \( 40\% \), which means the new numerator becomes:
\[ 2n + 0.4(2n) = 2n(1 + 0.4) = 2n \times 1.4 = 2.8n \]
- The denominator is decreased by \( 30\% \), which means the new denominator becomes:
\[ 3d - 0.3(3d) = 3d(1 - 0.3) = 3d \times 0.7 = 2.1d \]

### Step 2: Set up the equation based on the problem statement
The problem states that after these changes, the fraction becomes \( 14\% \) of \( \frac{16}{21} \). First, we need to calculate \( 14\% \) of \( \frac{16}{21} \):
\[ 14\% = \frac{14}{100} = \frac{7}{50} \]
Thus,
\[ \frac{7}{50} \times \frac{16}{21} = \frac{7 \times 16}{50 \times 21} = \frac{112}{1050} \]

### Step 3: Write the equation
Now we can set up the equation based on the modified fraction:
\[ \frac{2.8n}{2.1d} = \frac{112}{1050} \]

### Step 4: Cross-multiply to solve for \( n \) and \( d \)
Cross-multiplying gives:
\[ 2.8n \times 1050 = 112 \times 2.1d \]
This simplifies to:
\[ 2940n = 235.2d \]

### Step 5: Simplify the equation
Dividing both sides by \( 2.4 \) to simplify:
\[ 1225n = 98d \]
Thus,
\[ \frac{n}{d} = \frac{98}{1225} \]

### Step 6: Simplify the fraction
To simplify \( \frac{98}{1225} \):
- The GCD of \( 98 \) and \( 1225 \) is \( 49 \).
- Dividing both the numerator and denominator by \( 49 \):
\[ \frac{98 \div 49}{1225 \div 49} = \frac{2}{25} \]

### Conclusion
The original fraction is:
\[ \frac{n}{d} = \frac{2}{25} \]