Finding the Required Fraction

Video Explanation

Question

The sum of the numerator and denominator of a fraction is 18. If the denominator is increased by 2, the fraction reduces to \( \frac{1}{3} \). Find the fraction.

Solution

Step 1: Let the Variables

Let the numerator = \(x\)

Let the denominator = \(y\)

Step 2: Form the Equations

Sum condition:

\[ x + y = 18 \quad (1) \]

Denominator increased by 2:

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

Cross multiply:

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

Step 3: Solve the Equations

From equation (1):

\[ y = 18 – x \]

Substitute in equation (2):

\[ 3x = (18 – x) + 2 \]

\[ 3x = 20 – x \]

\[ 4x = 20 \]

\[ x = 5 \]

Step 4: Find the Value of y

\[ y = 18 – 5 \]

\[ y = 13 \]

Conclusion

Required fraction:

\[ \boxed{\frac{5}{13}} \]

Final Answer (For Exam)

The required fraction is 5/13.