Finding the Required Fraction

Video Explanation

Question

The sum of the numerator and denominator of a fraction is 4 more than twice the numerator. If the numerator and denominator are increased by 3, they are in the ratio 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

Sum condition:

\[ x + y = 2x + 4 \]

\[ y = x + 4 \quad (1) \]

Step 3: Form the Second Equation

After increasing both by 3:

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

Cross multiply:

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

\[ 3x + 9 = 2y + 6 \]

\[ 3x – 2y = -3 \quad (2) \]

Step 4: Solve the Equations

Substitute equation (1) into equation (2):

\[ 3x – 2(x + 4) = -3 \]

\[ 3x – 2x – 8 = -3 \]

\[ x – 8 = -3 \]

\[ x = 5 \]

Step 5: Find the Value of y

\[ y = 5 + 4 \]

\[ y = 9 \]

Conclusion

Required fraction:

\[ \boxed{\frac{5}{9}} \]

Final Answer (For Exam)

The required fraction is 5/9.