If the numerator of a fraction is increased by 15% and denominator is decreased by 20%, then the fraction, so obtained, is `17/65`. What is the original fraction?

Options

A. `281/1495`
B. `278/1495`
C. `267/1495`
D. `272/1495`

Answer

To solve the problem step by step, let's denote the original fraction as \( \frac{n}{d} \), where \( n \) is the numerator and \( d \) is the denominator.

### Step 1: Express the changes in the numerator and denominator
The problem states that the numerator is increased by 15% and the denominator is decreased by 20%.

- The new numerator after the increase is:
\[ n + 0.15n = 1.15n \]

- The new denominator after the decrease is:
\[ d - 0.20d = 0.80d \]

### Step 2: Set up the equation based on the new fraction
According to the problem, the new fraction equals \( \frac{17}{65} \). Therefore, we can write the equation:
\[ \frac{1.15n}{0.80d} = \frac{17}{65} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 1.15n \cdot 65 = 17 \cdot 0.80d \]

### Step 4: Simplify the equation
Calculating the left side:
\[ 1.15 \cdot 65 = 74.75 \]
So the equation becomes:
\[ 74.75n = 13.6d \]

### Step 5: Rearrange the equation to express \( \frac{n}{d} \)
We can rearrange the equation to find the ratio \( \frac{n}{d} \):
\[ \frac{n}{d} = \frac{13.6}{74.75} \]

### Step 6: Simplify the fraction
Now we simplify \( \frac{13.6}{74.75} \):
To simplify, we can multiply both the numerator and denominator by 100 to eliminate the decimals:
\[ \frac{1360}{7475} \]
Now we can divide both by 5:
\[ \frac{272}{1495} \]

### Step 7: Final result
Thus, the original fraction \( \frac{n}{d} \) is:
\[ \frac{272}{1495} \]