Finding the Required Fraction

Video Explanation

Question

A fraction becomes \( \frac{9}{11} \) if 2 is added to both the numerator and denominator. It becomes \( \frac{5}{7} \) if 2 is subtracted from both the numerator and denominator. Find the fraction.

Solution

Step 1: Let the Variables

Let the numerator = \(x\)

Let the denominator = \(y\)

Step 2: Form the Equations

After adding 2:

\[ \frac{x + 2}{y + 2} = \frac{9}{11} \]

Cross multiply:

\[ 11(x + 2) = 9(y + 2) \]

\[ 11x + 22 = 9y + 18 \]

\[ 11x – 9y = -4 \quad (1) \]

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After subtracting 2:

\[ \frac{x – 2}{y – 2} = \frac{5}{7} \]

Cross multiply:

\[ 7(x – 2) = 5(y – 2) \]

\[ 7x – 14 = 5y – 10 \]

\[ 7x – 5y = 4 \quad (2) \]

Step 3: Solve the Equations

Multiply equation (2) by 9:

\[ 63x – 45y = 36 \quad (3) \]

Multiply equation (1) by 5:

\[ 55x – 45y = -20 \quad (4) \]

Subtract (4) from (3):

\[ (63x – 45y) – (55x – 45y) = 36 – (-20) \]

\[ 8x = 56 \]

\[ x = 7 \]

Step 4: Find the Value of y

Substitute \(x = 7\) in equation (2):

\[ 7(7) – 5y = 4 \]

\[ 49 – 5y = 4 \]

\[ -5y = -45 \]

\[ y = 9 \]

Conclusion

Required fraction:

\[ \boxed{\frac{7}{9}} \]

Final Answer (For Exam)

The required fraction is 7/9.