2.1 If the denominator and the numerator of a fraction are each increased by 1 , the fraction equals \( \frac{2}{3} \). If the numerator is decreased by 1 and the Denominator is unchanged, the fraction is equal to \( \frac{1}{2} \). Find the fraction.

Let the fraction be \( \frac{x}{y} \). From the first condition, we have: \[ \frac{x + 1}{y + 1} = \frac{2}{3} \] Cross multiplying gives: \[ 3(x + 1) = 2(y + 1) \] Expanding gives: \[ 3x + 3 = 2y + 2 \quad \Rightarrow \quad 3x - 2y = -1 \quad \text{(Equation 1)} \] From the second condition, we have: \[ \frac{x - 1}{y} = \frac{1}{2} \] Cross multiplying gives: \[ 2(x - 1) = 1y \] Expanding gives: \[ 2x - 2 = y \quad \Rightarrow \quad y = 2x - 2 \quad \text{(Equation 2)} \] Now, we can substitute Equation 2 into Equation 1: \[ 3x - 2(2x - 2) = -1 \] Expanding gives: \[ 3x - 4x + 4 = -1 \quad \Rightarrow \quad -x + 4 = -1 \] So, we find: \[ -x = -5 \quad \Rightarrow \quad x = 5 \] Now using \( x = 5 \) to find \( y \): \[ y = 2(5) - 2 = 10 - 2 = 8 \] Thus, the fraction is: \[ \frac{x}{y} = \frac{5}{8} \] To verify: 1. Increasing both by 1 gives \( \frac{6}{9} = \frac{2}{3} \) (correct). 2. Decreasing the numerator by 1 gives \( \frac{4}{8} = \frac{1}{2} \) (correct). Therefore, the fraction is \( \frac{5}{8} \).