15. Brian worked for \( \sigma \frac{2}{3} \) hours yesterday and \( 6 \frac{3}{4} \) hours today. How many hours in all did he work yesterday and today? Give your answer as a mixed number in simplest form.

To find the total hours Brian worked yesterday and today, we need to add the two time durations together. 1. **Convert the mixed numbers to improper fractions:** - For yesterday's work: \[ \sigma \frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3} \] - For today's work: \[ 6 \frac{3}{4} = \frac{6 \times 4 + 3}{4} = \frac{24 + 3}{4} = \frac{27}{4} \] 2. **Find a common denominator to add the fractions:** - The denominators are 3 and 4. The least common multiple (LCM) of 3 and 4 is 12. 3. **Convert each fraction to have the common denominator:** - For \( \frac{8}{3} \): \[ \frac{8}{3} = \frac{8 \times 4}{3 \times 4} = \frac{32}{12} \] - For \( \frac{27}{4} \): \[ \frac{27}{4} = \frac{27 \times 3}{4 \times 3} = \frac{81}{12} \] 4. **Add the two fractions:** \[ \frac{32}{12} + \frac{81}{12} = \frac{32 + 81}{12} = \frac{113}{12} \] 5. **Convert the improper fraction back to a mixed number:** - Divide 113 by 12: - \( 113 \div 12 = 9 \) remainder \( 5 \) - Thus, \( \frac{113}{12} = 9 \frac{5}{12} \) Therefore, the total hours Brian worked yesterday and today is \( 9 \frac{5}{12} \) hours.