Question

If the numerator and the denominator of a fraction are each decreased by 3, the fraction becomes 2/3. If both the numerator and the denominator are increased by 7, the fraction becomes 3/4. Find the fraction.

• A. \( \frac{23}{33} \)
• B. \( \frac{13}{33} \)
• C. \( \frac{11}{3} \)
• D. None of these

Hint:

Form two equations from the conditions and solve simultaneously when fractions are altered by adding or subtracting constants.

Answer 1

The Correct Option is A

Step 1: Let the fraction be \( \frac{x}{y}. \)
\(\frac{x - 3}{y - 3} = \frac{2}{3}\). Cross multiply: \(3(x - 3) = 2(y - 3)\).
Step 2: Second condition.
\(\frac{x + 7}{y + 7} = \frac{3}{4}\). Cross multiply: \(4(x + 7) = 3(y + 7)\).
Step 3: Solve equations.
Eq1: \(3x - 9 = 2y - 6 \Rightarrow 3x - 2y = 3.\)
Eq2: \(4x + 28 = 3y + 21 \Rightarrow 4x - 3y = -7.\)
Solve simultaneously: Multiply Eq1 by 3 → \(9x - 6y = 9\). Multiply Eq2 by 2 → \(8x - 6y = -14\). Subtract: \(x = 23\).
Substitute in Eq1: \(3(23) - 2y = 3 \Rightarrow 69 - 3 = 2y \Rightarrow y = 33.\)
Step 4: Conclusion.
The fraction is 23/33.