1. Problem: Patricians and plutocrats are in the ratio 2:13, total 315. Find plutocrats. Formula: If ratio is $a:b$ and total is $T$, then part corresponding to $b$ is $\frac{b}{a+b} \times T$. Calculation: $$\text{Plutocrats} = \frac{13}{2+13} \times 315 = \frac{13}{15} \times 315$$ $$= 13 \times \cancel{21} = 273$$ 2. Problem: Girls' team outlifted boys' by 140%. Boys lifted 1400 pounds. Find girls' lift. Formula: Increase by 140% means girls lifted $1400 + 1.4 \times 1400 = 2.4 \times 1400$. Calculation: $$\text{Girls} = 2.4 \times 1400 = 3360$$ 3. Problem: 15% of flowers are roses, 120 roses. Find total flowers. Formula: $\text{Total} = \frac{\text{Part}}{\text{Percent}} = \frac{120}{0.15}$. Calculation: $$\text{Total} = \frac{120}{0.15} = 800$$ 4. Problem: $7 \frac{2}{5}$ of what number is $1 \frac{7}{10}$? Convert mixed numbers: $$7 \frac{2}{5} = \frac{37}{5}, \quad 1 \frac{7}{10} = \frac{17}{10}$$ Equation: $$\frac{37}{5} x = \frac{17}{10}$$ Solve for $x$: $$x = \frac{17}{10} \times \frac{5}{37} = \frac{85}{370} = \frac{17}{74}$$ 5. Problem: What fraction of $14 \frac{1}{4}$ is $\frac{3}{8}$? Convert: $$14 \frac{1}{4} = \frac{57}{4}$$ Fraction: $$\text{Fraction} = \frac{3/8}{57/4} = \frac{3}{8} \times \frac{4}{57} = \frac{12}{456} = \frac{1}{38}$$ 6. Solve: $7 \frac{2}{5} x + 5 \frac{1}{3} = \frac{1}{15}$ Convert: $$7 \frac{2}{5} = \frac{37}{5}, \quad 5 \frac{1}{3} = \frac{16}{3}$$ Equation: $$\frac{37}{5} x + \frac{16}{3} = \frac{1}{15}$$ Subtract $\frac{16}{3}$: $$\frac{37}{5} x = \frac{1}{15} - \frac{16}{3} = \frac{1}{15} - \frac{80}{15} = -\frac{79}{15}$$ Solve for $x$: $$x = -\frac{79}{15} \times \frac{5}{37} = -\frac{395}{555} = -\frac{79}{111}$$ 7. Solve: $-[-(-2p)] - 3(-3p + 15) = -(-4)(p - 12)$ Simplify left: $$-[-(-2p)] = -[2p] = -2p$$ $$-3(-3p + 15) = 9p - 45$$ Left side: $$-2p + 9p - 45 = 7p - 45$$ Right side: $$-(-4)(p - 12) = 4(p - 12) = 4p - 48$$ Equation: $$7p - 45 = 4p - 48$$ Solve: $$7p - 4p = -48 + 45$$ $$3p = -3$$ $$p = -1$$ 8. Graph $x \neq -2$ on number line means all real numbers except $-2$. 9. Evaluate $a^{-3}(a^{-2} - 2a)$ for $a = -2$ Calculate powers: $$a^{-3} = (-2)^{-3} = -\frac{1}{8}$$ $$a^{-2} = (-2)^{-2} = \frac{1}{4}$$ Expression: $$-\frac{1}{8} \left( \frac{1}{4} - 2(-2) \right) = -\frac{1}{8} \left( \frac{1}{4} + 4 \right) = -\frac{1}{8} \times \frac{17}{4} = -\frac{17}{32}$$ 12. Solve system: $$4x + y = -5$$ $$2x - y = -1$$ Add equations: $$(4x + y) + (2x - y) = -5 + (-1)$$ $$6x = -6$$ $$x = -1$$ Substitute $x$: $$4(-1) + y = -5$$ $$-4 + y = -5$$ $$y = -1$$ 13. Solve system: $$x - 3y = -7$$ $$3x + y = -1$$ Multiply second by 3: $$9x + 3y = -3$$ Add to first: $$(x - 3y) + (9x + 3y) = -7 + (-3)$$ $$10x = -10$$ $$x = -1$$ Substitute $x$: $$-1 - 3y = -7$$ $$-3y = -6$$ $$y = 2$$ 14. Solve system: $$4x - y = -7$$ $$2x + 2y = 4$$ Multiply first by 2: $$8x - 2y = -14$$ Add to second: $$(8x - 2y) + (2x + 2y) = -14 + 4$$ $$10x = -10$$ $$x = -1$$ Substitute $x$: $$4(-1) - y = -7$$ $$-4 - y = -7$$ $$-y = -3$$ $$y = 3$$ 15. Multiply $(4x - 3)(x + 2)$ Use distributive property: $$4x \times x + 4x \times 2 - 3 \times x - 3 \times 2 = 4x^2 + 8x - 3x - 6 = 4x^2 + 5x - 6$$ 16. Square $(4x + 3)^2$ Formula: $(a+b)^2 = a^2 + 2ab + b^2$ Calculate: $$16x^2 + 24x + 9$$ 17. Graph $y = -3x$ Line through origin with slope $-3$, downward from left to right. 18. Graph $y = 3x$ Line through origin with slope $3$, upward from left to right. 19. Graph $4 + 3x - y = 0$ Rewrite: $$y = 3x + 4$$ Line crosses y-axis at 4, slope 3. 20. Simplify $\frac{a/b}{c + x}$ Rewrite: $$\frac{a/b}{c + x} = \frac{a}{b(c + x)}$$ 21. Same as 20. 22. Simplify $\frac{1}{\frac{1}{a + b}}$ Invert denominator: $$= a + b$$ 23. Add $\frac{4}{xyc} - \frac{5m}{xy(c + 1)} - \frac{3k}{xy^2}$ Common denominator is $xyc(c+1)y$ but expression is simplified as is. 24. Add $k + \frac{1}{k}$ Cannot combine further without common denominator. 25. Add $my + \frac{p}{y}$ Common denominator $y$: $$\frac{my^2 + p}{y}$$ 26. Factor $20x^2 m^3 k^6 - 10x m^4 k^4 + 30x^5 m^4 k^6$ Find GCF: $$10x m^3 k^4$$ Factor out: $$10x m^3 k^4 (2x - m + 3x^4 m k^2)$$ 27. Simplify $\frac{(x^2 y^0)^2 y^0 k^2}{(2x^2 k^5)^{-4} y}$ Simplify powers: $$y^0 = 1$$ $$(x^2)^2 = x^4$$ Denominator: $$(2x^2 k^5)^{-4} = \frac{1}{(2x^2 k^5)^4} = \frac{1}{16 x^8 k^{20}}$$ Expression: $$\frac{x^4 k^2}{\frac{1}{16 x^8 k^{20}} y} = x^4 k^2 \times 16 x^8 k^{20} \times \frac{1}{y} = 16 x^{12} k^{22} \times \frac{1}{y} = \frac{16 x^{12} k^{22}}{y}$$ 28. Simplify $\frac{a^0 x^2 x^0}{m^1 y^0 m^{-2}}$ Simplify powers: $$a^0 = 1, x^0 = 1, y^0 = 1$$ Expression: $$\frac{x^2}{m \times m^{-2}} = \frac{x^2}{m^{1-2}} = \frac{x^2}{m^{-1}} = x^2 m$$ 29. Simplify $\left( \frac{p^2 y}{y} \right) \left( \frac{x^{-1} y}{p^2} \right)$ Simplify inside: $$\frac{p^2 y}{y} = p^2$$ Expression: $$p^2 \times \frac{x^{-1} y}{p^2} = x^{-1} y = \frac{y}{x}$$ 30. Simplify: $$-2\{[(-2 - 2) - 3^0(-2 - 1)] - [-2(-3 + 5) - 2]\} - |-2| + \sqrt{-8}$$ Calculate inside: $$(-2 - 2) = -4$$ $$3^0 = 1$$ $$-2 - 1 = -3$$ $$[-4 - 1 \times (-3)] = [-4 + 3] = -1$$ Second bracket: $$-3 + 5 = 2$$ $$-2 \times 2 = -4$$ $$-4 - 2 = -6$$ Expression: $$-2(-1 - (-6)) - 2 + \sqrt{-8} = -2(-1 + 6) - 2 + \sqrt{-8} = -2(5) - 2 + \sqrt{-8} = -10 - 2 + \sqrt{-8} = -12 + \sqrt{-8}$$ Since $\sqrt{-8} = 2i \sqrt{2}$ (imaginary), final answer: $$-12 + 2i \sqrt{2}$$