Finding the Required Fraction

Video Explanation

Question

When 3 is added to the denominator and 2 is subtracted from the numerator, the fraction becomes \( \frac{1}{4} \). When 6 is added to the numerator and the denominator is multiplied by 3, it becomes \( \frac{2}{3} \). Find the fraction.

Solution

Step 1: Let the Variables

Let the numerator = \(x\)

Let the denominator = \(y\)

Step 2: Form the First Equation

\[ \frac{x – 2}{y + 3} = \frac{1}{4} \]

Cross multiply:

\[ 4(x – 2) = y + 3 \]

\[ 4x – 8 = y + 3 \]

\[ 4x – y = 11 \quad (1) \]

Step 3: Form the Second Equation

\[ \frac{x + 6}{3y} = \frac{2}{3} \]

Cross multiply:

\[ 3(x + 6) = 2(3y) \]

\[ 3x + 18 = 6y \]

\[ x – 2y = -6 \quad (2) \]

Step 4: Solve the Equations

Multiply equation (2) by 4:

\[ 4x – 8y = -24 \quad (3) \]

Subtract equation (1) from (3):

\[ (4x – 8y) – (4x – y) = -24 – 11 \]

\[ -7y = -35 \]

\[ y = 5 \]

Step 5: Find the Value of x

Substitute \(y = 5\) in equation (2):

\[ x – 2(5) = -6 \]

\[ x – 10 = -6 \]

\[ x = 4 \]

Conclusion

Required fraction:

\[ \boxed{\frac{4}{5}} \]

Final Answer (For Exam)

The required fraction is 4/5.