Finding the Required Fraction

Video Explanation

Question

The sum of the numerator and denominator of a fraction is 12. If the denominator is increased by 3, the fraction 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

Sum condition:

\[ x + y = 12 \quad (1) \]

Denominator increased by 3:

\[ \frac{x}{y + 3} = \frac{1}{2} \]

Cross multiply:

\[ 2x = y + 3 \quad (2) \]

Step 3: Solve the Equations

From equation (1):

\[ y = 12 – x \]

Substitute in equation (2):

\[ 2x = (12 – x) + 3 \]

\[ 2x = 15 – x \]

\[ 3x = 15 \]

\[ x = 5 \]

Step 4: Find the Value of y

\[ y = 12 – 5 \]

\[ y = 7 \]

Conclusion

Required fraction:

\[ \boxed{\frac{5}{7}} \]

Final Answer (For Exam)

The required fraction is 5/7.