Question

If 1 is subtracted from the numerator of a fraction, then it becomes \( \frac{1}{3} \), and if 8 is added to its denominator, then it becomes \( \frac{1}{4} \). Find that fraction.

Hint:

To solve problems involving fractions, use substitution to find expressions for unknowns and solve step by step.

Answer 1

Let the original fraction be \( \frac{a}{b} \). We are given two conditions: 1. When 1 is subtracted from the numerator, the fraction becomes \( \frac{1}{3} \), so: \[ \frac{a - 1}{b} = \frac{1}{3} \quad \text{(1)} \] 2. When 8 is added to the denominator, the fraction becomes \( \frac{1}{4} \), so: \[ \frac{a}{b + 8} = \frac{1}{4} \quad \text{(2)} \]
Step 1: Solve equation (1).
From equation (1), we have: \[ a - 1 = \frac{b}{3} \] Thus, \[ a = \frac{b}{3} + 1 \]
Step 2: Solve equation (2).
From equation (2), we have: \[ a = \frac{b + 8}{4} \]
Step 3: Set the two expressions for \( a \) equal to each other.
\[ \frac{b}{3} + 1 = \frac{b + 8}{4} \] Multiplying both sides by 12 to eliminate the fractions: \[ 4b + 12 = 3(b + 8) \] Simplifying: \[ 4b + 12 = 3b + 24 \] \[ 4b - 3b = 24 - 12 \] \[ b = 12 \]
Step 4: Find \( a \).
Substitute \( b = 12 \) into \( a = \frac{b}{3} + 1 \): \[ a = \frac{12}{3} + 1 = 4 + 1 = 5 \]
Conclusion:
The fraction is \( \frac{5}{12} \).