Finding the Required Fraction

Video Explanation

Question

A fraction becomes \( \frac{1}{3} \) when 2 is subtracted from the numerator. It becomes \( \frac{1}{2} \) when 1 is subtracted from 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:

\[ \frac{x – 2}{y} = \frac{1}{3} \]

Cross multiply:

\[ 3(x – 2) = y \]

\[ 3x – 6 = y \quad (1) \]

—

Second condition:

\[ \frac{x}{y – 1} = \frac{1}{2} \]

Cross multiply:

\[ 2x = y – 1 \quad (2) \]

Step 3: Solve the Equations

From equation (1):

\[ y = 3x – 6 \]

Substitute into equation (2):

\[ 2x = (3x – 6) – 1 \]

\[ 2x = 3x – 7 \]

\[ x = 7 \]

Step 4: Find the Value of y

\[ y = 3(7) – 6 \]

\[ y = 21 – 6 = 15 \]

Conclusion

Required fraction:

\[ \boxed{\frac{7}{15}} \]

Final Answer (For Exam)

The required fraction is 7/15.