1. Problem: The number of bacteria increased by 280 percent overnight. If there were 30,000 bacteria yesterday, how many bacteria were present this morning? Formula: New amount = Original amount + (Percent increase \times Original amount) Calculation: $$\text{New amount} = 30000 + \frac{280}{100} \times 30000$$ $$= 30000 + 2.8 \times 30000$$ $$= 30000 + 84000$$ $$= 114000$$ Answer: There were 114,000 bacteria this morning. 2. Problem: When Charles inspected the troops that survived, he found that 3600 were still alive. If 40 percent died in the fight, how many troops did he begin with? Formula: Survivors = (1 - Percent died) \times Original number Calculation: $$3600 = (1 - 0.40) \times \text{Original number}$$ $$3600 = 0.60 \times \text{Original number}$$ $$\text{Original number} = \frac{3600}{0.60}$$ $$= \frac{3600}{\cancel{0.60}} \times \frac{\cancel{1}}{\cancel{0.60}} = 6000$$ Answer: Charles began with 6000 troops. 3. Problem: Edna and Mabel climbed 40 percent of the mountains in the whole country. If they climbed 184 mountains, how many mountains were in the country? Formula: Part = Percent \times Whole Calculation: $$184 = 0.40 \times \text{Whole}$$ $$\text{Whole} = \frac{184}{0.40}$$ $$= \frac{184}{\cancel{0.40}} \times \frac{\cancel{1}}{\cancel{0.40}} = 460$$ Answer: There are 460 mountains in the country. 4. Simplify $\sqrt{72}$: $$\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$$ 5. Simplify $3\sqrt{75}$: $$3\sqrt{75} = 3 \times \sqrt{25 \times 3} = 3 \times 5 \sqrt{3} = 15\sqrt{3}$$ 6. Simplify $4\sqrt{324}$: $$4\sqrt{324} = 4 \times \sqrt{18^2} = 4 \times 18 = 72$$ 7. Multiply $(4x - 3)(12x + 2)$: $$= 4x \times 12x + 4x \times 2 - 3 \times 12x - 3 \times 2$$ $$= 48x^2 + 8x - 36x - 6$$ $$= 48x^2 - 28x - 6$$ 8. Graph $y = -2$: Horizontal line crossing y-axis at -2. 9. Graph $y = -2x - 5$: Line with slope -2 crossing y-axis at -5. 10. Graph $y = -2x + 5$: Line with slope -2 crossing y-axis at 5. 11. Graph on number line: $x > 4$ with domain integers means all integers greater than 4. 13. Solve system by substitution: $$\begin{cases} 3x - 2y = 15 \\ 5x + y = 12 \end{cases}$$ From second equation: $$y = 12 - 5x$$ Substitute into first: $$3x - 2(12 - 5x) = 15$$ $$3x - 24 + 10x = 15$$ $$13x - 24 = 15$$ $$13x = 39$$ $$x = 3$$ Then: $$y = 12 - 5(3) = 12 - 15 = -3$$ 14. Solve system: $$\begin{cases} y + 2x = 12 \\ x + 2y = 12 \end{cases}$$ From first: $$y = 12 - 2x$$ Substitute into second: $$x + 2(12 - 2x) = 12$$ $$x + 24 - 4x = 12$$ $$-3x + 24 = 12$$ $$-3x = -12$$ $$x = 4$$ Then: $$y = 12 - 2(4) = 12 - 8 = 4$$ 16. Simplify $\frac{x}{y}$ with $\frac{y}{x}$ crossed and replaced with $yx$: $$\frac{x}{y} \times yx = x^2$$ 17. Simplify $\frac{x}{x}$ with $\frac{x}{y}$ crossed and replaced with $xy$: $$\frac{x}{x} = 1$$ 18. Simplify $\frac{1}{y}$: stays as $\frac{1}{y}$. 19. Add $\frac{4}{a^2 x} + \frac{7}{x(x + a)}$: Common denominator is $a^2 x (x + a)$. $$\frac{4(x + a)}{a^2 x (x + a)} + \frac{7 a^2}{a^2 x (x + a)} = \frac{4(x + a) + 7 a^2}{a^2 x (x + a)}$$ 20. Add $2 + \frac{3}{y}$: Rewrite 2 as $\frac{2y}{y}$: $$\frac{2y}{y} + \frac{3}{y} = \frac{2y + 3}{y}$$ 21. Add $1 + \frac{x}{y}$: Rewrite 1 as $\frac{y}{y}$: $$\frac{y}{y} + \frac{x}{y} = \frac{y + x}{y}$$ 22. Factor $40x^y m^2 z - 20x^3 y^5 m^2 z + 20 x^2 p^2 m$: Common factor is $20 x^2 m$: $$20 x^2 m (2 x^{y-2} m z - x y^5 m z + p^2)$$ 23. Simplify $\frac{4x + 4x^2}{4x}$: $$= \frac{4x(1 + x)}{4x} = \cancel{\frac{\cancel{4x}(1 + x)}{\cancel{4x}}} = 1 + x$$ 24. Simplify $k p^{-2} k (p^0)^2 / k p k (p^{-2})^2$: $$= \frac{k^2 p^{-2} \times 1}{k^2 p (p^{-4})} = \frac{k^2 p^{-2}}{k^2 p^{-3}} = p^{-2 - (-3)} = p^{1} = p$$ 25. Simplify $\left(\frac{3 m^2}{y^{-4}}\right)^2 \times \frac{m}{y}$: $$= \left(3 m^2 y^4\right)^2 \times \frac{m}{y} = 9 m^4 y^8 \times \frac{m}{y} = 9 m^{5} y^{7}$$ 26. Simplify $\frac{2 p^2 x^{-4} (x)(x^2)}{y^{-4} (p^2)^{-2} x}$: $$= \frac{2 p^2 x^{-4 + 1 + 2}}{y^{-4} p^{-4} x^{1}} = \frac{2 p^2 x^{-1}}{y^{-4} p^{-4} x} = 2 p^{2 - (-4)} x^{-1 - 1} y^{4} = 2 p^{6} x^{-2} y^{4}$$ 27. Simplify $- | -3^0 | - 3^0 (-2)(-3)(-2 - 3)$: $$- | -1 | - 1 \times (-2) \times (-3) \times (-5) = -1 - (-30) = -1 + 30 = 29$$ 28. Simplify by adding like terms: $$- \frac{3 x^2 y^{-2}}{x^{-2} y^{-2}} - 2 x^y y y^{-1} + 4 x^3 x y y^{-1} - \frac{2 x^2}{x^{-2}}$$ First term: $$= - 3 x^{2 - (-2)} y^{-2 - (-2)} = -3 x^{4} y^{0} = -3 x^{4}$$ Second term: $$- 2 x^{y} y^{1 - 1} = - 2 x^{y}$$ Third term: $$4 x^{3 + 1} y^{1 - 1} = 4 x^{4}$$ Fourth term: $$- 2 x^{2 - (-2)} = - 2 x^{4}$$ Sum: $$(-3 x^{4}) + (-2 x^{y}) + 4 x^{4} - 2 x^{4} = (-3 + 4 - 2) x^{4} - 2 x^{y} = (-1) x^{4} - 2 x^{y} = - x^{4} - 2 x^{y}$$ 29. Expand: $$- \frac{x^{-2}}{y^{4}} (x^{2} y^{4} - \frac{3 x^{-2}}{y^{4}}) = - \frac{x^{-2}}{y^{4}} x^{2} y^{4} + \frac{x^{-2}}{y^{4}} \times \frac{3 x^{-2}}{y^{4}}$$ $$= - x^{0} y^{0} + 3 x^{-4} y^{-8} = -1 + 3 x^{-4} y^{-8}$$ 30. Evaluate $x - (x^{2})^{0} (x - y) - |x - y|$ for $x = -2$, $y = -3$: $$(x^{2})^{0} = 1$$ $$x - 1 (x - y) - |x - y| = x - (x - y) - |x - y|$$ $$= -2 - (-2 + 3) - |-2 + 3| = -2 - 1 - 1 = -4$$ Final answers: 1) 114000 2) 6000 3) 460 4) $6\sqrt{2}$ 5) $15\sqrt{3}$ 6) 72 7) $48x^2 - 28x - 6$ 13) $(x,y) = (3,-3)$ 14) $(x,y) = (4,4)$ 16) $x^2$ 17) 1 18) $\frac{1}{y}$ 19) $\frac{4(x + a) + 7 a^2}{a^2 x (x + a)}$ 20) $\frac{2y + 3}{y}$ 21) $\frac{y + x}{y}$ 22) $20 x^2 m (2 x^{y-2} m z - x y^5 m z + p^2)$ 23) $1 + x$ 24) $p$ 25) $9 m^{5} y^{7}$ 26) $2 p^{6} x^{-2} y^{4}$ 27) 29 28) $- x^{4} - 2 x^{y}$ 29) $-1 + 3 x^{-4} y^{-8}$ 30) $-4$