Question

A fraction becomes $\frac{9}{11}$ if 2 is added to both the numerator & the denominator. If 3 is added to both the numerator and denominator, the fraction becomes $\frac{5}{6}$. Find the fraction.

• A. $\frac{7}{9}$
• B. $\frac{2}{3}$
• C. $\frac{7}{11}$
• D. $\frac{8}{9}$

Hint:

For fraction problems: - Assume $\frac{x}{y}$ - Convert conditions into linear equations - Solve systematically

Answer 1

The Correct Option is A

Concept: Let the fraction be: \[ \frac{x}{y} \]

Step 1: Form first equation.
\[ \frac{x+2}{y+2} = \frac{9}{11} \] \[ 11(x+2) = 9(y+2) \] \[ 11x + 22 = 9y + 18 \] \[ 11x - 9y = -4 \quad ...(1) \]

Step 2: Form second equation.
\[ \frac{x+3}{y+3} = \frac{5}{6} \] \[ 6(x+3) = 5(y+3) \] \[ 6x + 18 = 5y + 15 \] \[ 6x - 5y = -3 \quad ...(2) \]

Step 3: Solve equations.
Multiply (2) by 9: \[ 54x - 45y = -27 \] Multiply (1) by 5: \[ 55x - 45y = -20 \] Subtract: \[ x = 7 \]

Step 4: Find $y$.
Substitute in (2): \[ 6(7) - 5y = -3 \] \[ 42 - 5y = -3 \] \[ 5y = 45 \Rightarrow y = 9 \]

Step 5: Final conclusion.
\[ \frac{x}{y} = \frac{7}{9} \]