1. Problem: The patricians and plutocrats are in the ratio 2 to 13, and the total number is 315. Find the number of plutocrats. 2. Formula: If two quantities are in ratio $a:b$ and their total is $T$, then the parts are $\frac{a}{a+b}T$ and $\frac{b}{a+b}T$. 3. Calculation: - Total parts = $2 + 13 = 15$ - Number of plutocrats = $\frac{13}{15} \times 315$ 4. Simplify: $$\frac{13}{15} \times 315 = 13 \times \frac{315}{15}$$ $$= 13 \times \cancel{\frac{315}{15}} = 13 \times 21 = 273$$ 5. Answer: There are **273 plutocrats**. --- 2. Problem: The girls' weightlifting team outlifted the boys' team by 140%. Boys lifted 1400 pounds. Find how many pounds the girls lifted. 3. Formula: Increase by 140% means girls lifted $100\% + 140\% = 240\%$ of boys' lift. 4. Calculation: $$240\% = \frac{240}{100} = 2.4$$ Girls' lift = $2.4 \times 1400 = 3360$ pounds. 5. Answer: The girls lifted **3360 pounds**. --- 3. Problem: 15% of flowers are roses, and there are 120 roses. Find total flowers. 4. Formula: $\text{Roses} = 15\% \times \text{Total flowers}$ 5. Calculation: $$120 = 0.15 \times \text{Total flowers}$$ $$\text{Total flowers} = \frac{120}{0.15} = 800$$ 6. Answer: There are **800 flowers** in all. --- 4. Problem: $7 \frac{2}{5}$ of what number is $1 \frac{7}{10}$? 5. Convert mixed numbers to improper fractions: $$7 \frac{2}{5} = \frac{37}{5}, \quad 1 \frac{7}{10} = \frac{17}{10}$$ 6. Let the number be $x$: $$\frac{37}{5} x = \frac{17}{10}$$ 7. Solve for $x$: $$x = \frac{17}{10} \div \frac{37}{5} = \frac{17}{10} \times \frac{5}{37} = \frac{85}{370}$$ 8. Simplify fraction: $$\frac{85}{370} = \frac{\cancel{85}}{\cancel{370}} = \frac{1}{\cancel{4.35}}$$ Actually, $85 \times 4 = 340$, $85 \times 5 = 425$, so divide numerator and denominator by 5: $$\frac{85}{370} = \frac{17}{74}$$ 9. Answer: The number is **$\frac{17}{74}$**. --- 5. Problem: What fraction of $14 \frac{1}{4}$ is $\frac{3}{8}$? 6. Convert $14 \frac{1}{4}$ to improper fraction: $$14 \frac{1}{4} = \frac{57}{4}$$ 7. Let the fraction be $f$: $$f \times \frac{57}{4} = \frac{3}{8}$$ 8. Solve for $f$: $$f = \frac{3}{8} \div \frac{57}{4} = \frac{3}{8} \times \frac{4}{57} = \frac{12}{456}$$ 9. Simplify: $$\frac{12}{456} = \frac{\cancel{12}}{\cancel{456}} = \frac{1}{38}$$ 10. Answer: The fraction is **$\frac{1}{38}$**. --- 6. Problem: Solve $7 \frac{2}{5} x + 5 \frac{1}{3} = \frac{1}{15}$. 7. Convert mixed numbers: $$7 \frac{2}{5} = \frac{37}{5}, \quad 5 \frac{1}{3} = \frac{16}{3}$$ 8. Equation: $$\frac{37}{5} x + \frac{16}{3} = \frac{1}{15}$$ 9. Subtract $\frac{16}{3}$ from both sides: $$\frac{37}{5} x = \frac{1}{15} - \frac{16}{3}$$ 10. Find common denominator 15: $$\frac{1}{15} - \frac{16}{3} = \frac{1}{15} - \frac{16 \times 5}{15} = \frac{1}{15} - \frac{80}{15} = -\frac{79}{15}$$ 11. Solve for $x$: $$x = \frac{-\frac{79}{15}}{\frac{37}{5}} = -\frac{79}{15} \times \frac{5}{37} = -\frac{395}{555}$$ 12. Simplify numerator and denominator by 5: $$x = -\frac{79}{111}$$ 13. Answer: $x = -\frac{79}{111}$. --- 7. Problem: Solve $-[-(-2p)] - 3(-3p + 15) = -(-4)(p - 12)$. 8. Simplify inside brackets: $$-[-(-2p)] = -[2p] = -2p$$ 9. Expand: $$-3(-3p + 15) = 9p - 45$$ 10. Right side: $$-(-4)(p - 12) = 4(p - 12) = 4p - 48$$ 11. Equation becomes: $$-2p + 9p - 45 = 4p - 48$$ 12. Simplify left: $$7p - 45 = 4p - 48$$ 13. Subtract $4p$ from both sides: $$7p - 4p - 45 = -48$$ $$3p - 45 = -48$$ 14. Add 45 to both sides: $$3p = -48 + 45 = -3$$ 15. Solve for $p$: $$p = \frac{-3}{3} = -1$$ 16. Answer: $p = -1$. --- 8. Problem: Graph $x \neq -2$ on a number line. 9. Explanation: This means all real numbers except $-2$. 10. On number line, mark all points except $-2$ (open circle at $-2$). --- 9. Problem: Evaluate $a^{-3}(a^{-2} - 2a)$ if $a = -2$. 10. Substitute $a = -2$: $$(-2)^{-3} \left((-2)^{-2} - 2(-2)\right)$$ 11. Calculate powers: $$(-2)^{-3} = \frac{1}{(-2)^3} = \frac{1}{-8} = -\frac{1}{8}$$ $$(-2)^{-2} = \frac{1}{(-2)^2} = \frac{1}{4}$$ 12. Calculate inside parentheses: $$\frac{1}{4} - 2(-2) = \frac{1}{4} + 4 = \frac{1}{4} + \frac{16}{4} = \frac{17}{4}$$ 13. Multiply: $$-\frac{1}{8} \times \frac{17}{4} = -\frac{17}{32}$$ 14. Answer: $-\frac{17}{32}$. --- 12. Problem: Solve system $\{4x + y = -5; 2x - y = -1\}$. 13. Add equations: $$(4x + y) + (2x - y) = -5 + (-1)$$ $$6x = -6$$ $$x = -1$$ 14. Substitute $x = -1$ into $4x + y = -5$: $$4(-1) + y = -5$$ $$-4 + y = -5$$ $$y = -1$$ 15. Answer: $x = -1$, $y = -1$. --- 13. Problem: Solve system $\{x - 3y = -7; 3x + y = -1\}$. 14. Multiply second equation by 3: $$9x + 3y = -3$$ 15. Add to first equation multiplied by 1: $$(x - 3y) + (9x + 3y) = -7 + (-3)$$ $$10x = -10$$ $$x = -1$$ 16. Substitute $x = -1$ into $x - 3y = -7$: $$-1 - 3y = -7$$ $$-3y = -6$$ $$y = 2$$ 17. Answer: $x = -1$, $y = 2$. --- 14. Problem: Solve system $\{4x - y = -7; 2x + 2y = 4\}$. 15. Multiply first equation by 2: $$8x - 2y = -14$$ 16. Add to second equation: $$(8x - 2y) + (2x + 2y) = -14 + 4$$ $$10x = -10$$ $$x = -1$$ 17. Substitute $x = -1$ into $4x - y = -7$: $$4(-1) - y = -7$$ $$-4 - y = -7$$ $$-y = -3$$ $$y = 3$$ 18. Answer: $x = -1$, $y = 3$. --- 15. Problem: Multiply $(4x - 3)(x + 2)$. 16. Use distributive property: $$4x \times x + 4x \times 2 - 3 \times x - 3 \times 2$$ $$= 4x^2 + 8x - 3x - 6$$ $$= 4x^2 + 5x - 6$$ 17. Answer: $4x^2 + 5x - 6$. --- 16. Problem: Multiply $(4x + 3)^2$. 17. Use formula $(a + b)^2 = a^2 + 2ab + b^2$: $$= (4x)^2 + 2 \times 4x \times 3 + 3^2$$ $$= 16x^2 + 24x + 9$$ 18. Answer: $16x^2 + 24x + 9$. --- 17. Problem: Graph $y = -3x$. 18. This is a line with slope $-3$ and y-intercept $0$. 19. The graph passes through origin and goes down 3 units for every 1 unit right. --- 18. Problem: Graph $y = 3x$. 19. This is a line with slope $3$ and y-intercept $0$. 20. The graph passes through origin and goes up 3 units for every 1 unit right. --- 19. Problem: Graph $4 + 3x - y = 0$. 20. Rewrite as $y = 3x + 4$. 21. This is a line with slope $3$ and y-intercept $4$. 22. The graph crosses y-axis at 4 and rises 3 units for every 1 unit right. --- 20. Problem: Simplify $\frac{\frac{a}{b}}{c + x}$. 21. Rewrite as: $$\frac{a}{b} \times \frac{1}{c + x} = \frac{a}{b(c + x)}$$ 22. Answer: $\frac{a}{b(c + x)}$. --- 21. Problem: Simplify $\frac{\frac{a}{b}}{c + x}$ (same as 20). 22. Answer: $\frac{a}{b(c + x)}$. --- 22. Problem: Simplify $\frac{1}{a + b}$. 23. This is already simplified. 24. Answer: $\frac{1}{a + b}$. --- 23. Problem: Add $\frac{4}{xyc} - \frac{5m}{xy(c + 1)} - \frac{3k}{xy^2}$. 24. Since denominators differ, no further simplification without common denominator. 25. Answer: Expression remains as is. --- 24. Problem: Add $k + \frac{1}{k}$. 25. Cannot combine further. 26. Answer: $k + \frac{1}{k}$. --- 25. Problem: Add $my + \frac{p}{y}$. 26. Cannot combine further. 27. Answer: $my + \frac{p}{y}$. --- 26. Problem: Factor $20x^2 m^3 k^6 - 10x m k^4 + 30x^5 m k^6$. 27. Find common factors: - Coefficients: 10 - $x$: minimum power is $x^1$ - $m$: minimum power is $m^1$ - $k$: minimum power is $k^4$ 28. Factor out $10 x m k^4$: $$10 x m k^4 (2 x m^2 k^2 - 1 + 3 x^4 k^2)$$ 29. Answer: $10 x m k^4 (2 x m^2 k^2 - 1 + 3 x^4 k^2)$. --- 27. Problem: Simplify $\frac{(x^2 y^0)^2 y^0 k^2}{(2 x^2 k^3)^{-4} y}$. 28. Simplify powers: $$y^0 = 1$$ $$(x^2)^2 = x^{4}$$ 29. Numerator: $$x^{4} \times 1 \times k^{2} = x^{4} k^{2}$$ 30. Denominator: $$(2 x^{2} k^{3})^{-4} y = 2^{-4} x^{-8} k^{-12} y$$ 31. Rewrite denominator: $$\frac{y}{2^{4} x^{8} k^{12}} = \frac{y}{16 x^{8} k^{12}}$$ 32. So entire expression: $$\frac{x^{4} k^{2}}{\frac{y}{16 x^{8} k^{12}}} = x^{4} k^{2} \times \frac{16 x^{8} k^{12}}{y} = \frac{16 x^{12} k^{14}}{y}$$ 33. Answer: $\frac{16 x^{12} k^{14}}{y}$. --- 28. Problem: Simplify $\frac{a^{0} x^{2} x^{0}}{m^{2} y^{0} m^{-2}}$. 29. Simplify powers: $$a^{0} = 1, x^{0} = 1, y^{0} = 1$$ 30. Numerator: $$x^{2}$$ 31. Denominator: $$m^{2} m^{-2} = m^{2 - 2} = m^{0} = 1$$ 32. Expression: $$\frac{x^{2}}{1} = x^{2}$$ 33. Answer: $x^{2}$. --- 29. Problem: Simplify $\frac{(p^{2} x)^{2}}{y} \times (x^{-1} y)^{3} \div p^{1}$. 30. Simplify powers: $$(p^{2} x)^{2} = p^{4} x^{2}$$ $$(x^{-1} y)^{3} = x^{-3} y^{3}$$ 31. Expression: $$\frac{p^{4} x^{2}}{y} \times x^{-3} y^{3} \times p^{-1} = p^{4 - 1} x^{2 - 3} y^{-1 + 3} = p^{3} x^{-1} y^{2}$$ 32. Rewrite: $$p^{3} \frac{y^{2}}{x}$$ 33. Answer: $\frac{p^{3} y^{2}}{x}$. --- 30. Problem: Simplify $$-2\{[(-2 - 2) - 3^{0}(-2 - 1)] - [-2(-3 + 5) - 2]\} - |-2| + \sqrt{-8}$$ 31. Calculate inside brackets: $$(-2 - 2) = -4$$ $$3^{0} = 1$$ $$(-2 - 1) = -3$$ 32. First bracket: $$-4 - 1 \times (-3) = -4 + 3 = -1$$ 33. Second bracket: $$-2(-3 + 5) - 2 = -2(2) - 2 = -4 - 2 = -6$$ 34. Expression inside curly braces: $$[-1] - [-6] = -1 + 6 = 5$$ 35. Multiply by $-2$: $$-2 \times 5 = -10$$ 36. Calculate $|-2| = 2$ 37. Calculate $\sqrt{-8}$ is imaginary, write as: $$\sqrt{-8} = \sqrt{8} i = 2 \sqrt{2} i$$ 38. Final expression: $$-10 - 2 + 2 \sqrt{2} i = -12 + 2 \sqrt{2} i$$ 39. Answer: $-12 + 2 \sqrt{2} i$.