Question

On adding 1 to the numerator of a fraction it becomes \( \frac{1}{2} \) and on adding 1 to the denominator it becomes \( \frac{1}{3} \). Write the equation for this statement.

Hint:

To solve such problems, write equations based on the given conditions and solve them step by step.

Answer 1

Let the fraction be \( \frac{x}{y} \).
According to the first condition, when 1 is added to the numerator, the fraction becomes \( \frac{1}{2} \): \[ \frac{x+1}{y} = \frac{1}{2} \] Cross-multiply: \[ 2(x+1) = y \] \[ 2x + 2 = y \] Thus, \( y = 2x + 2 \). \quad \cdots \text{(1)}
According to the second condition, when 1 is added to the denominator, the fraction becomes \( \frac{1}{3} \): \[ \frac{x}{y+1} = \frac{1}{3} \] Cross-multiply: \[ 3x = y + 1 \] \[ y = 3x - 1 \] Thus, \( y = 3x - 1 \). \quad \cdots \text{(2)}
Now, solving equations (1) and (2):
From equation (1), \( y = 2x + 2 \), and from equation (2), \( y = 3x - 1 \): \[ 2x + 2 = 3x - 1 \] \[ 2x - 3x = -1 - 2 \] \[ -x = -3 \] \[ x = 3 \] Substitute \( x = 3 \) in equation (1): \[ y = 2(3) + 2 = 6 + 2 = 8 \] Thus, the fraction is \( \frac{x}{y} = \frac{3}{8} \).
Correct Answer: The fraction is \( \frac{3}{8} \).