If the numerator of a fraction be increased by `12%` and its denominator decreased by `2%` the value of the fraction becomes `(6/7).` The original fraction is-

Options

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

Answer

To solve the problem step-by-step, we need to find the original fraction given the conditions stated in the question.

### Step 1: Define the variables
Let the original numerator be \( n \) and the original denominator be \( d \). Thus, the original fraction can be represented as \( \frac{n}{d} \).

**Hint:** Define the variables clearly to represent the numerator and denominator of the fraction.

### Step 2: Express the changes in the numerator and denominator
According to the problem, the numerator is increased by \( 12\% \), which can be expressed as:
\[ \text{New numerator} = n + 0.12n = 1.12n \]
The denominator is decreased by \( 2\% \), which can be expressed as:
\[ \text{New denominator} = d - 0.02d = 0.98d \]

**Hint:** Translate the percentage changes into mathematical expressions.

### Step 3: Set up the equation
The problem states that the new fraction becomes \( \frac{6}{7} \). Therefore, we can set up the equation:
\[ \frac{1.12n}{0.98d} = \frac{6}{7} \]

**Hint:** Use the information given to form an equation that relates the new fraction to the specified value.

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 1.12n \cdot 7 = 6 \cdot 0.98d \]
This simplifies to:
\[ 7.84n = 5.88d \]

**Hint:** Cross-multiplication is a useful technique to simplify equations involving fractions.

### Step 5: Rearrange the equation
Rearranging the equation gives:
\[ \frac{n}{d} = \frac{5.88}{7.84} \]

**Hint:** Isolate the fraction \( \frac{n}{d} \) to find the original fraction.

### Step 6: Simplify the fraction
Now, we simplify \( \frac{5.88}{7.84} \):
\[ \frac{5.88 \div 5.88}{7.84 \div 5.88} = \frac{1}{1.3333} = \frac{3}{4} \]

**Hint:** Divide both the numerator and denominator by the same number to simplify the fraction.

### Step 7: Conclusion
Thus, the original fraction is:
\[ \frac{n}{d} = \frac{3}{4} \]

**Final Answer:** The original fraction is \( \frac{3}{4} \).