Finding the Required Fraction

Video Explanation

Question

If the numerator of a fraction is multiplied by 2 and the denominator is reduced by 5, the fraction becomes \( \frac{6}{5} \). If the denominator is doubled and the numerator is increased by 8, the fraction becomes \( \frac{2}{5} \). Find the fraction.

Solution

Step 1: Let the Variables

Let the numerator = \(x\)

Let the denominator = \(y\)

Step 2: Form the First Equation

\[ \frac{2x}{y – 5} = \frac{6}{5} \]

Cross multiply:

\[ 5(2x) = 6(y – 5) \]

\[ 10x = 6y – 30 \]

\[ 10x – 6y = -30 \]

Divide by 2:

\[ 5x – 3y = -15 \quad (1) \]

Step 3: Form the Second Equation

\[ \frac{x + 8}{2y} = \frac{2}{5} \]

Cross multiply:

\[ 5(x + 8) = 2(2y) \]

\[ 5x + 40 = 4y \]

\[ 5x – 4y = -40 \quad (2) \]

Step 4: Solve the Equations

Subtract equation (1) from equation (2):

\[ (5x – 4y) – (5x – 3y) = -40 – (-15) \]

\[ -y = -25 \]

\[ y = 25 \]

Step 5: Find the Value of x

Substitute \(y = 25\) in equation (1):

\[ 5x – 3(25) = -15 \]

\[ 5x – 75 = -15 \]

\[ 5x = 60 \]

\[ x = 12 \]

Conclusion

Required fraction:

\[ \boxed{\frac{12}{25}} \]

Final Answer (For Exam)

The required fraction is 12/25.