Finding the Required Fraction

Video Explanation

Question

A fraction becomes \( \frac{1}{3} \) if 1 is subtracted from both its numerator and denominator. If 1 is added to both the numerator and denominator, it becomes \( \frac{1}{2} \). Find the fraction.

Solution

Step 1: Let the Variables

Let the numerator = \(x\)

Let the denominator = \(y\)

Step 2: Form the Equations

After subtracting 1:

\[ \frac{x – 1}{y – 1} = \frac{1}{3} \]

Cross multiply:

\[ 3(x – 1) = y – 1 \]

\[ 3x – 3 = y – 1 \]

\[ 3x – y = 2 \quad (1) \]

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After adding 1:

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

Cross multiply:

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

\[ 2x + 2 = y + 1 \]

\[ 2x – y = -1 \quad (2) \]

Step 3: Solve the Equations

Subtract equation (2) from equation (1):

\[ (3x – y) – (2x – y) = 2 – (-1) \]

\[ x = 3 \]

Step 4: Find the Value of y

Substitute \(x = 3\) in equation (2):

\[ 2(3) – y = -1 \]

\[ 6 – y = -1 \]

\[ -y = -7 \]

\[ y = 7 \]

Conclusion

Required fraction:

\[ \boxed{\frac{3}{7}} \]

Final Answer (For Exam)

The required fraction is 3/7.