1. **Simplify the expression:** $5(-8 + 12) + 7 + 6^2 \times (4 \div 2)$ - First, simplify inside the parentheses: $-8 + 12 = 4$ - Then multiply: $5 \times 4 = 20$ - Calculate the exponent: $6^2 = 36$ - Divide inside the parentheses: $4 \div 2 = 2$ - Multiply: $36 \times 2 = 72$ - Add all parts: $20 + 7 + 72 = 99$ 2. **Simplify the expression:** $\left(4 \frac{2}{3} \times 2 \frac{1}{6}\right) \div \frac{7}{3}$ - Convert mixed numbers to improper fractions: - $4 \frac{2}{3} = \frac{14}{3}$ - $2 \frac{1}{6} = \frac{13}{6}$ - Multiply the fractions: $\frac{14}{3} \times \frac{13}{6} = \frac{182}{18}$ - Simplify $\frac{182}{18}$ by dividing numerator and denominator by 2: $\frac{91}{9}$ - Divide by $\frac{7}{3}$ is the same as multiplying by its reciprocal: $\frac{91}{9} \times \frac{3}{7} = \frac{273}{63}$ - Simplify $\frac{273}{63}$ by dividing numerator and denominator by 9: $\frac{273 \div 9}{63 \div 9} = \frac{30.333\ldots}{7}$ but since 273 and 63 are divisible by 9: $\frac{273}{63} = \frac{91}{21} = \frac{13}{3}$ after dividing numerator and denominator by 7 3. **Bookings in the restaurant:** - Total tables: 12 - Time slots: 5 (7 P.M. to 11 P.M.) **i. Fraction of tables booked for each time:** - 7 P.M.: 4 booked out of 12, fraction $\frac{4}{12} = \frac{1}{3}$ - 8 P.M.: 12 booked out of 12, fraction $\frac{12}{12} = 1$ - 9 P.M.: 9 booked out of 12, fraction $\frac{9}{12} = \frac{3}{4}$ - 10 P.M.: 8 booked out of 12, fraction $\frac{8}{12} = \frac{2}{3}$ - 11 P.M.: 7 booked out of 12, fraction $\frac{7}{12}$ **ii. Fraction of total possible bookings booked for the whole evening:** - Total possible bookings: $5 \times 12 = 60$ - Count total booked (sum of all X's): - 7 P.M.: 4 - 8 P.M.: 12 - 9 P.M.: 9 - 10 P.M.: 8 - 11 P.M.: 7 - Total booked: $4 + 12 + 9 + 8 + 7 = 40$ - Fraction booked: $\frac{40}{60} = \frac{2}{3}$ 4. **True or False:** - $-4 \in \mathbb{Z}$ (Is -4 an integer?) - Since integers $\mathbb{Z}$ include all whole numbers and their negatives, $-4$ is an integer. - **Answer: True**