Finding the Required Fraction

Video Explanation

Question

If 1 is added to the numerator and 1 is subtracted from the denominator, the fraction becomes 1. It becomes \( \frac{1}{2} \) if 1 is added only to the denominator. Find the fraction.

Solution

Step 1: Let the Variables

Let the numerator = \(x\)

Let the denominator = \(y\)

Step 2: Form the Equations

First condition:

\[ \frac{x + 1}{y – 1} = 1 \]

So,

\[ x + 1 = y – 1 \]

\[ x – y = -2 \quad (1) \]

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Second condition:

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

Cross multiply:

\[ 2x = y + 1 \]

\[ 2x – y = 1 \quad (2) \]

Step 3: Solve the Equations

Subtract equation (1) from equation (2):

\[ (2x – y) – (x – y) = 1 – (-2) \]

\[ x = 3 \]

Step 4: Find the Value of y

Substitute \(x = 3\) in equation (1):

\[ 3 – y = -2 \]

\[ -y = -5 \]

\[ y = 5 \]

Conclusion

Required fraction:

\[ \boxed{\frac{3}{5}} \]

Final Answer (For Exam)

The required fraction is 3/5.