X-2/16=3/4 1/2-y=2/8 2+1/2+3+1/3+4+1/4+5+1/5+6+1/6

Let's solve the equations and simplify the expression step-by-step: 1. **Solve the first equation:** \[ \frac{x - 2}{16} = \frac{3}{4} \] Multiply both sides by 16: \[ x - 2 = 12 \] Add 2 to both sides: \[ x = 14 \] 2. **Solve the second equation:** \[ \frac{1}{2} - y = \frac{2}{8} \] Simplify \(\frac{2}{8}\) to \(\frac{1}{4}\): \[ \frac{1}{2} - y = \frac{1}{4} \] Subtract \(\frac{1}{2}\) from both sides: \[ -y = \frac{1}{4} - \frac{1}{2} \] The common denominator is 4, so rewrite \(\frac{1}{2}\) as \(\frac{2}{4}\): \[ -y = \frac{1}{4} - \frac{2}{4} = -\frac{1}{4} \] Multiply both sides by -1: \[ y = \frac{1}{4} \] 3. **Simplify the expression:** \[ 2 + \frac{1}{2} + 3 + \frac{1}{3} + 4 + \frac{1}{4} + 5 + \frac{1}{5} + 6 + \frac{1}{6} \] First, combine the whole numbers: \[ 2 + 3 + 4 + 5 + 6 = 20 \] Now, add the fractions: \[ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} \] To add these fractions, we need a common denominator. The least common multiple (LCM) of \(2, 3, 4, 5, 6\) is 60. Now convert each fraction: \[ \frac{1}{2} = \frac{30}{60}, \quad \frac{1}{3} = \frac{20}{60}, \quad \frac{1}{4} = \frac{15}{60}, \quad \frac{1}{5} = \frac{12}{60}, \quad \frac{1}{6} = \frac{10}{60} \] Now add them: \[ \frac{30}{60} + \frac{20}{60} + \frac{15}{60} + \frac{12}{60} + \frac{10}{60} = \frac{97}{60} \] Finally, combine the whole number sum with the fraction: \[ 20 + \frac{97}{60} = \frac{1200}{60} + \frac{97}{60} = \frac{1297}{60} \] Thus, the final answers are: - \(x = 14\) - \(y = \frac{1}{4}\) - The simplified expression is \(\frac{1297}{60}\) or approximately \(21.62\).