Question

A fraction becomes \( \frac{1}{3} \) when 1 is added to the numerator and it becomes \( \frac{1}{4} \) when 1 is subtracted from its denominator. Find the fraction.

Hint:

When given conditions about changes to the numerator or denominator of a fraction, form equations and solve them step by step.

Answer 1

Let the fraction be \( \frac{x}{y} \), where \( x \) is the numerator and \( y \) is the denominator. We are given two conditions: 1. When 1 is added to the numerator, the fraction becomes \( \frac{1}{3} \): \[ \frac{x + 1}{y} = \frac{1}{3}. \] 2. When 1 is subtracted from the denominator, the fraction becomes \( \frac{1}{4} \): \[ \frac{x}{y - 1} = \frac{1}{4}. \] Step 1: Solve the first equation. From the first equation: \[ \frac{x + 1}{y} = \frac{1}{3} \quad \implies \quad x + 1 = \frac{y}{3} \quad \implies \quad x = \frac{y}{3} - 1. \] Step 2: Solve the second equation. From the second equation: \[ \frac{x}{y - 1} = \frac{1}{4} \quad \implies \quad x = \frac{y - 1}{4}. \] Step 3: Set the two expressions for \( x \) equal. Now, equate the two expressions for \( x \): \[ \frac{y}{3} - 1 = \frac{y - 1}{4}. \] Multiply both sides by 12 to eliminate the denominators: \[ 12 \left( \frac{y}{3} - 1 \right) = 12 \left( \frac{y - 1}{4} \right). \] Simplifying: \[ 4y - 12 = 3(y - 1). \] Expanding: \[ 4y - 12 = 3y - 3 \quad \implies \quad 4y - 3y = 12 - 3 \quad \implies \quad y = 9. \] Step 4: Find \( x \). Substitute \( y = 9 \) into \( x = \frac{y}{3} - 1 \): \[ x = \frac{9}{3} - 1 = 3 - 1 = 2. \]
Conclusion:
The fraction is \( \frac{2}{9} \).