1. **Problem 6: Simplify $4 + \frac{3}{7}$** We need to add a whole number and a fraction. 2. The formula for adding a whole number and a fraction is simply: $$\text{sum} = \text{whole number} + \frac{\text{numerator}}{\text{denominator}}$$ 3. Substitute the values: $$4 + \frac{3}{7}$$ 4. To add, write 4 as a fraction with denominator 7: $$4 = \frac{4 \times 7}{7} = \frac{28}{7}$$ 5. Now add the fractions: $$\frac{28}{7} + \frac{3}{7} = \frac{28 + 3}{7} = \frac{31}{7}$$ 6. The simplified answer is: $$\boxed{\frac{31}{7}}$$ --- 7. **Problem 7 (part i): Solve for $c$ given $c = \frac{1}{2}(u + d)$, $u=11$, $d=4$** 8. Substitute values: $$c = \frac{1}{2}(11 + 4)$$ 9. Simplify inside parentheses: $$c = \frac{1}{2}(15)$$ 10. Multiply: $$c = \frac{15}{2} = 7.5$$ 11. So, $c = \boxed{7.5}$ --- 12. **Problem 7 (part ii): Solve for $c$ given $c = \frac{a}{(a + b)^2}$, $a=2$, $d=4$, and $b = \sqrt{a} + d^2$** 13. First find $b$: $$b = \sqrt{2} + 4^2 = \sqrt{2} + 16$$ 14. Calculate $a + b$: $$a + b = 2 + (\sqrt{2} + 16) = 18 + \sqrt{2}$$ 15. Square $a + b$: $$ (18 + \sqrt{2})^2 = 18^2 + 2 \times 18 \times \sqrt{2} + (\sqrt{2})^2 = 324 + 36\sqrt{2} + 2 = 326 + 36\sqrt{2} $$ 16. Now calculate $c$: $$c = \frac{2}{326 + 36\sqrt{2}}$$ 17. This is the simplified exact form for $c$. --- 18. **Problem 7 (part iii): Solve for $r$ in $A = P(1 + \frac{r}{t})^{ct}$ given $A=1000$, $P=800$** 19. The formula is compound interest. To solve for $r$, more information about $c$ and $t$ is needed. Since not given, we cannot solve further. --- 20. **Problem 8 (a): Cost for 4-hour job with $30$ upfront and $46$ per hour** 21. Formula: $$\text{Cost} = 30 + 46 \times \text{hours}$$ 22. Substitute hours = 4: $$30 + 46 \times 4 = 30 + 184 = 214$$ 23. Cost is $\boxed{214}$ --- 24. **Problem 8 (b): Cost for 10-hour job** 25. Substitute hours = 10: $$30 + 46 \times 10 = 30 + 460 = 490$$ 26. Cost is $\boxed{490}$ --- 27. **Problem 8 (c): Cost for 24 days averaging 6 hours per day** 28. Total hours: $$24 \times 6 = 144$$ 29. Cost: $$30 + 46 \times 144 = 30 + 6624 = 6654$$ 30. Cost is $\boxed{6654}$ --- 31. **Problem 8 (d): Find hours worked if total cost is 1000** 32. Let $h$ be hours worked: $$30 + 46h = 1000$$ 33. Subtract 30: $$46h = 1000 - 30 = 970$$ 34. Divide both sides by 46: $$h = \frac{970}{46}$$ 35. Simplify fraction: $$h = \frac{\cancel{970}}{\cancel{46}} = 21.0869565...$$ 36. Rounded to nearest half hour: $$h \approx 21.0$$ 37. Hours worked is $\boxed{21}$ --- 38. **Problem 9: Perimeter of square is 68 cm, find side length** 39. Formula for perimeter of square: $$P = 4s$$ 40. Substitute $P=68$: $$68 = 4s$$ 41. Divide both sides by 4: $$s = \frac{68}{4} = 17$$ 42. Side length is $\boxed{17}$ cm --- 43. **Problem 10: Sum of two consecutive numbers is 35, find the numbers** 44. Let the first number be $x$, then the next consecutive number is $x+1$. 45. Equation: $$x + (x + 1) = 35$$ 46. Simplify: $$2x + 1 = 35$$ 47. Subtract 1: $$2x = 34$$ 48. Divide by 2: $$x = 17$$ 49. The two numbers are $17$ and $18$ 50. Final answer: $\boxed{17 \text{ and } 18}$