What is the value of \( \frac{11 / 2-1 \frac{2}{3}+1 \frac{1}{3}}{1 \frac{1}{3}+1 / 6 \div 1 / 2} \)

To solve the expression \[ \frac{11 / 2 - 1 \frac{2}{3} + 1 \frac{1}{3}}{1 \frac{1}{3} + 1 / 6 \div 1 / 2}, \] let's simplify both the numerator and the denominator step by step. ### Step 1: Simplify the Numerator 1. Convert all mixed numbers to improper fractions: - \( 11/2 \) is already a fraction. - \( 1 \frac{2}{3} = \frac{3 \cdot 1 + 2}{3} = \frac{5}{3} \). - \( 1 \frac{1}{3} = \frac{3 \cdot 1 + 1}{3} = \frac{4}{3} \). Now, the numerator becomes: \[ \frac{11}{2} - \frac{5}{3} + \frac{4}{3}. \] 2. Combine \( -\frac{5}{3} + \frac{4}{3} \) first: \[ -\frac{5}{3} + \frac{4}{3} = -\frac{1}{3}. \] 3. Now combine this result with \( \frac{11}{2} \): \[ \frac{11}{2} - \frac{1}{3}. \] 4. To do this, find a common denominator (which is 6): - Convert \( \frac{11}{2} \) to sixths: \[ \frac{11}{2} = \frac{11 \times 3}{2 \times 3} = \frac{33}{6}. \] - Convert \( -\frac{1}{3} \) to sixths: \[ -\frac{1}{3} = -\frac{1 \times 2}{3 \times 2} = -\frac{2}{6}. \] 5. Now combine them: \[ \frac{33}{6} - \frac{2}{6} = \frac{33 - 2}{6} = \frac{31}{6}. \] ### Step 2: Simplify the Denominator 1. Start with \( 1 \frac{1}{3} + 1 / 6 \div 1 / 2 \): - Convert \( 1 \frac{1}{3} = \frac{4}{3} \). 2. Evaluate \( 1/6 \div 1/2 \): - Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{1}{6} \div \frac{1}{2} = \frac{1}{6} \times \frac{2}{1} = \frac{2}{6} = \frac{1}{3}. \] 3. Now, add this to \( \frac{4}{3} \): \[ \frac{4}{3} + \frac{1}{3} = \frac{4 + 1}{3} = \frac{5}{3}. \] ### Step 3: Combine Numerator and Denominator Now our expression is: \[ \frac{\frac{31}{6}}{\frac{5}{3}}. \] 4. To simplify this, multiply by the reciprocal of the denominator: \[ \frac{31}{6} \times \frac{3}{5} = \frac{31 \times 3}{6 \times 5} = \frac{93}{30}. \] 5. Simplify \( \frac{93}{30} \): - The greatest common divisor of 93 and 30 is 3. Dividing both by 3 gives: \[ \frac{93 \div 3}{30 \div 3} = \frac{31}{10}. \] ### Final Answer Thus, the value of the expression is \[ \boxed{\frac{31}{10}}. \]