1. **Problem:** Heidi and Micah have 51 dimes and nickels. The total value is 4.10. Find how many more dimes than nickels there are. 2. **Define variables:** Let $d$ = number of dimes, $n$ = number of nickels. 3. **Equations:** - Total coins: $$d + n = 51$$ - Total value in cents: $$10d + 5n = 410$$ 4. **Solve system:** From first, $$d = 51 - n$$. Substitute into second: $$10(51 - n) + 5n = 410$$ $$510 - 10n + 5n = 410$$ $$510 - 5n = 410$$ $$-5n = 410 - 510$$ $$-5n = -100$$ $$n = \frac{\cancel{-100}}{\cancel{-5}} = 20$$ 5. Find $d$: $$d = 51 - 20 = 31$$ 6. Difference: $$d - n = 31 - 20 = 11$$ --- 7. **Problem:** Business averages 650.50 for 10 days, then 874.75 for 20 days. Find overall average for 30 days. 8. **Calculate total transactions:** $$10 \times 650.50 = 6505$$ $$20 \times 874.75 = 17495$$ 9. **Total for 30 days:** $$6505 + 17495 = 24000$$ 10. **Overall average:** $$\frac{24000}{30} = 800$$ --- 11. **Problem:** Ratio of laughers to stolid is 2 to 17, total 7600. Find how many did not laugh. 12. **Define variables:** Let $L$ = laughers, $S$ = stolid. 13. **Ratio:** $$\frac{L}{S} = \frac{2}{17}$$ 14. **Total:** $$L + S = 7600$$ 15. Express $L$ in terms of $S$: $$L = \frac{2}{17}S$$ 16. Substitute: $$\frac{2}{17}S + S = 7600$$ $$\frac{2}{17}S + \frac{17}{17}S = 7600$$ $$\frac{19}{17}S = 7600$$ 17. Solve for $S$: $$S = \frac{7600 \times 17}{19} = 6800$$ 18. Find $L$: $$L = 7600 - 6800 = 800$$ 19. Number who did not laugh = $S = 6800$. --- 20. **Simplify:** $$(0.0016 \times 10^{-7})(3000 \times 10^{5})$$ 21. Multiply coefficients: $$0.0016 \times 3000 = 4.8$$ 22. Multiply powers of 10: $$10^{-7} \times 10^{5} = 10^{-2}$$ 23. Combine: $$4.8 \times 10^{-2} = 4.8 \times 0.01 = 0.048$$ --- 24. **Simplify:** $$(0.003 \times 10^{-5})(700 \times 10^{14})$$ 25. Multiply coefficients: $$0.003 \times 700 = 2.1$$ 26. Multiply powers of 10: $$10^{-5} \times 10^{14} = 10^{9}$$ 27. Combine: $$2.1 \times 10^{9} = 2100000000$$ --- 28. **Find equations of lines A and B:** Given no explicit data, assume lines from problem 9: $$y = x + 2$$ $$y = -x$$ 29. These are the equations of lines A and B. --- Final answers: 1. There are 11 more dimes than nickels. 3. Overall average is 800. 4. Number who did not laugh is 6800. 5. Simplified value is 0.048. 6. Simplified value is 2100000000. 7. Lines A and B: $y = x + 2$ and $y = -x$.