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 original fraction be \(\frac{n}{d}\). 1. According to the first condition, when both the numerator and denominator are increased by 1, we have: \[ \frac{n+1}{d+1}=\frac{2}{3} \] 2. According to the second condition, when the numerator is decreased by 1 (with the denominator unchanged), we have: \[ \frac{n-1}{d}=\frac{1}{2} \] **Step 1. Solve the second equation for \(d\):** Multiply both sides of \(\frac{n-1}{d}=\frac{1}{2}\) by \(d\): \[ n-1=\frac{d}{2} \] Then multiply both sides by 2: \[ 2n-2=d \] Thus, \[ d=2n-2 \] **Step 2. Substitute \(d=2n-2\) into the first equation:** Replace \(d\) in \(\frac{n+1}{d+1}=\frac{2}{3}\): \[ \frac{n+1}{(2n-2)+1}=\frac{n+1}{2n-1}=\frac{2}{3} \] **Step 3. Solve for \(n\):** Cross-multiply to eliminate the fraction: \[ 3(n+1)=2(2n-1) \] Simplify both sides: \[ 3n+3=4n-2 \] Subtract \(3n\) from both sides: \[ 3= n-2 \] Add 2 to both sides: \[ n=5 \] **Step 4. Determine \(d\):** Substitute \(n=5\) into \(d=2n-2\): \[ d=2(5)-2=8 \] The original fraction is therefore: \[ \frac{n}{d}=\frac{5}{8} \]