Finding the Required Fraction

Video Explanation

Question

If 2 is added to the numerator of a fraction, it becomes \( \frac{1}{2} \). If 1 is subtracted from the denominator, it becomes \( \frac{1}{3} \). Find the fraction.

Solution

Step 1: Let the Variables

Let the numerator = \(x\)

Let the denominator = \(y\)

Step 2: Form the Equations

First condition:

\[ \frac{x + 2}{y} = \frac{1}{2} \]

Cross multiply:

\[ 2(x + 2) = y \]

\[ 2x + 4 = y \quad (1) \]

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Second condition:

\[ \frac{x}{y – 1} = \frac{1}{3} \]

Cross multiply:

\[ 3x = y – 1 \quad (2) \]

Step 3: Solve the Equations

From equation (1):

\[ y = 2x + 4 \]

Substitute into equation (2):

\[ 3x = (2x + 4) – 1 \]

\[ 3x = 2x + 3 \]

\[ x = 3 \]

Step 4: Find the Value of y

\[ y = 2(3) + 4 \]

\[ y = 10 \]

Conclusion

Required fraction:

\[ \boxed{\frac{3}{10}} \]

Final Answer (For Exam)

The required fraction is 3/10.