1. Solve the equation $\frac{1}{2}(2f - 3) + \frac{1}{3}(f - 4) = 0$. Start by distributing the fractions: $$\frac{1}{2} \times 2f - \frac{1}{2} \times 3 + \frac{1}{3} \times f - \frac{1}{3} \times 4 = 0$$ Simplify: $$f - \frac{3}{2} + \frac{f}{3} - \frac{4}{3} = 0$$ Combine like terms: $$f + \frac{f}{3} = \frac{3}{2} + \frac{4}{3}$$ Find common denominator for the right side: $$\frac{3}{2} = \frac{9}{6}, \quad \frac{4}{3} = \frac{8}{6}$$ So, $$f + \frac{f}{3} = \frac{17}{6}$$ Combine left side: $$\frac{3f}{3} + \frac{f}{3} = \frac{4f}{3}$$ Set equation: $$\frac{4f}{3} = \frac{17}{6}$$ Multiply both sides by 3: $$4f = \frac{17}{6} \times 3 = \frac{51}{6} = \frac{17}{2}$$ Divide both sides by 4: $$f = \frac{17}{2} \times \frac{1}{4} = \frac{17}{8} = 2 \frac{1}{8}$$ 2. Solve $0 = \frac{3}{2}(2m - 1) - \frac{3}{4}(m + 5)$. Distribute: $$0 = 3m - \frac{3}{2} - \frac{3m}{4} - \frac{15}{4}$$ Combine like terms: $$0 = 3m - \frac{3m}{4} - \frac{3}{2} - \frac{15}{4}$$ Convert constants to common denominator 4: $$- \frac{3}{2} = - \frac{6}{4}$$ So, $$0 = \left(3m - \frac{3m}{4}\right) - \frac{6}{4} - \frac{15}{4} = \frac{12m}{4} - \frac{3m}{4} - \frac{21}{4} = \frac{9m}{4} - \frac{21}{4}$$ Multiply both sides by 4: $$0 = 9m - 21$$ Add 21: $$9m = 21$$ Divide by 9: $$m = \frac{21}{9} = \frac{7}{3} = 2 \frac{1}{3}$$ 3. Solve $\frac{1}{3}(3a - 2) - \frac{1}{4}(5a + 1) + \frac{1}{5}(2a - 7) = 0$. Distribute: $$a - \frac{2}{3} - \frac{5a}{4} - \frac{1}{4} + \frac{2a}{5} - \frac{7}{5} = 0$$ Group like terms: $$a - \frac{5a}{4} + \frac{2a}{5} = \frac{2}{3} + \frac{1}{4} + \frac{7}{5}$$ Find common denominator for constants: 60 $$\frac{2}{3} = \frac{40}{60}, \quad \frac{1}{4} = \frac{15}{60}, \quad \frac{7}{5} = \frac{84}{60}$$ Sum constants: $$\frac{40 + 15 + 84}{60} = \frac{139}{60}$$ Combine a terms with common denominator 20: $$\frac{20a}{20} - \frac{25a}{20} + \frac{8a}{20} = \frac{3a}{20}$$ Set equation: $$\frac{3a}{20} = \frac{139}{60}$$ Multiply both sides by 20: $$3a = \frac{139}{60} \times 20 = \frac{139}{3}$$ Divide by 3: $$a = \frac{139}{9} = 15 \frac{4}{9}$$ 4. Solve $-\frac{1}{8}(3x - 4) + \frac{1}{5}(4x - 1) = \frac{1}{15} - \frac{1}{3}(x + 5)$. Distribute: $$-\frac{3x}{8} + \frac{1}{2} + \frac{4x}{5} - \frac{1}{5} = \frac{1}{15} - \frac{x}{3} - \frac{5}{3}$$ Simplify constants: $$\frac{1}{2} - \frac{1}{5} = \frac{5}{10} - \frac{2}{10} = \frac{3}{10}$$ Right side constants: $$\frac{1}{15} - \frac{5}{3} = \frac{1}{15} - \frac{25}{15} = -\frac{24}{15} = -\frac{8}{5}$$ Rewrite equation: $$-\frac{3x}{8} + \frac{4x}{5} + \frac{3}{10} = -\frac{x}{3} - \frac{8}{5}$$ Bring all x terms to left and constants to right: $$-\frac{3x}{8} + \frac{4x}{5} + \frac{x}{3} = -\frac{8}{5} - \frac{3}{10}$$ Find common denominator for x terms: 120 $$-\frac{45x}{120} + \frac{96x}{120} + \frac{40x}{120} = \frac{91x}{120}$$ Constants right side common denominator 10: $$-\frac{8}{5} = -\frac{16}{10}$$ So, $$-\frac{16}{10} - \frac{3}{10} = -\frac{19}{10}$$ Equation: $$\frac{91x}{120} = -\frac{19}{10}$$ Multiply both sides by 120: $$91x = -\frac{19}{10} \times 120 = -228$$ Divide by 91: $$x = -\frac{228}{91} = -2 \frac{46}{91}$$ 5. Solve $\frac{t}{3} - \frac{t}{5} = 2$. Find common denominator 15: $$\frac{5t}{15} - \frac{3t}{15} = 2$$ Simplify: $$\frac{2t}{15} = 2$$ Multiply both sides by 15: $$2t = 30$$ Divide by 2: $$t = 15$$ 6. Solve $\frac{V}{5} + \frac{V}{3} = \frac{47}{30} - \frac{V}{4}$. Bring all V terms to left: $$\frac{V}{5} + \frac{V}{3} + \frac{V}{4} = \frac{47}{30}$$ Find common denominator 60: $$\frac{12V}{60} + \frac{20V}{60} + \frac{15V}{60} = \frac{47}{30}$$ Sum left side: $$\frac{47V}{60} = \frac{47}{30}$$ Multiply both sides by 60: $$47V = \frac{47}{30} \times 60 = 94$$ Divide by 47: $$V = 2$$ 7. Solve $\frac{m}{3} + 3 - \frac{m}{6} = 1 - \frac{m}{3}$. Bring m terms to left and constants to right: $$\frac{m}{3} - \frac{m}{6} + \frac{m}{3} = 1 - 3$$ Simplify constants: $$1 - 3 = -2$$ Find common denominator 6 for m terms: $$\frac{2m}{6} - \frac{m}{6} + \frac{2m}{6} = \frac{3m}{6} = \frac{m}{2}$$ Equation: $$\frac{m}{2} = -2$$ Multiply both sides by 2: $$m = -4$$ 8. Solve $\frac{5}{x} = 10$. Multiply both sides by x: $$5 = 10x$$ Divide by 10: $$x = \frac{5}{10} = \frac{1}{2}$$ 9. Solve $2t + \frac{1}{2} = \frac{4}{7}t - \frac{3}{10}$. Bring t terms to left and constants to right: $$2t - \frac{4}{7}t = - \frac{3}{10} - \frac{1}{2}$$ Find common denominator 70 for t terms: $$\frac{14t}{7} - \frac{4t}{7} = \frac{10t}{7}$$ Constants right side common denominator 10: $$- \frac{3}{10} - \frac{5}{10} = - \frac{8}{10} = - \frac{4}{5}$$ Equation: $$\frac{10t}{7} = - \frac{4}{5}$$ Multiply both sides by 7: $$10t = - \frac{28}{5}$$ Divide by 10: $$t = - \frac{28}{50} = - \frac{14}{25} = -0.56$$ 10. Solve $\frac{3}{w} = \frac{9}{5}$. Cross multiply: $$3 \times 5 = 9 \times w$$ $$15 = 9w$$ Divide by 9: $$w = \frac{15}{9} = \frac{5}{3} = 1 \frac{2}{3}$$ 11. Solve $\frac{5x}{9} - \frac{2x}{5} = \frac{1}{15}$. Find common denominator 45 for x terms: $$\frac{25x}{45} - \frac{18x}{45} = \frac{1}{15}$$ Simplify left side: $$\frac{7x}{45} = \frac{1}{15}$$ Multiply both sides by 45: $$7x = 3$$ Divide by 7: $$x = \frac{3}{7}$$ 12. Solve $\frac{1}{3}a + \frac{1}{4}a = \frac{3}{20}$. Find common denominator 12 for left side: $$\frac{4a}{12} + \frac{3a}{12} = \frac{3}{20}$$ Simplify left side: $$\frac{7a}{12} = \frac{3}{20}$$ Multiply both sides by 12: $$7a = \frac{36}{20} = \frac{9}{5}$$ Divide by 7: $$a = \frac{9}{35} = 0.2571$$ 13. Solve $\frac{c + 3}{4} = 2 + \frac{c - 3}{5} \times x$. Assuming $x=1$ (since $x$ is not defined), Multiply both sides by 20: $$5(c + 3) = 40 + 4(c - 3)$$ Distribute: $$5c + 15 = 40 + 4c - 12$$ Simplify right side: $$5c + 15 = 4c + 28$$ Subtract 4c: $$c + 15 = 28$$ Subtract 15: $$c = 13$$ 14. Solve $\frac{3}{2} + \frac{3y}{20} = -\frac{y - 6}{12} + \frac{2y}{15}$. Distribute: $$\frac{3}{2} + \frac{3y}{20} = -\frac{y}{12} + \frac{6}{12} + \frac{2y}{15}$$ Simplify constants: $$\frac{6}{12} = \frac{1}{2}$$ Bring y terms to left and constants to right: $$\frac{3y}{20} + \frac{y}{12} - \frac{2y}{15} = \frac{1}{2} - \frac{3}{2} = -1$$ Find common denominator 60 for y terms: $$\frac{9y}{60} + \frac{5y}{60} - \frac{8y}{60} = \frac{6y}{60} = \frac{y}{10}$$ Equation: $$\frac{y}{10} = -1$$ Multiply both sides by 10: $$y = -10$$ 15. Solve $\frac{5}{4} - \frac{d}{5} = \frac{d}{20} + \frac{7}{20} \times x$. Assuming $x=1$, Multiply both sides by 20: $$25 - 4d = d + 7$$ Bring d terms to left and constants to right: $$-4d - d = 7 - 25$$ $$-5d = -18$$ Divide by -5: $$d = \frac{18}{5} = 3.6$$ 16. Solve $4 - a = \frac{2a + 6}{5}$. Multiply both sides by 5: $$20 - 5a = 2a + 6$$ Bring a terms to left and constants to right: $$-5a - 2a = 6 - 20$$ $$-7a = -14$$ Divide by -7: $$a = 2$$ 17. Solve $\frac{f - 2}{f - 3} = 4$. Multiply both sides by $f - 3$: $$f - 2 = 4(f - 3)$$ Distribute: $$f - 2 = 4f - 12$$ Bring f terms to left and constants to right: $$f - 4f = -12 + 2$$ $$-3f = -10$$ Divide by -3: $$f = \frac{10}{3} = 3 \frac{1}{3}$$ 18. Solve $\frac{3}{y - 2} = \frac{4}{y + 4}$. Cross multiply: $$3(y + 4) = 4(y - 2)$$ Distribute: $$3y + 12 = 4y - 8$$ Bring y terms to left and constants to right: $$3y - 4y = -8 - 12$$ $$-y = -20$$ Multiply both sides by -1: $$y = 20$$ 19. Solve $\frac{2}{a - 5} = \frac{3}{a - 1}$. Cross multiply: $$2(a - 1) = 3(a - 5)$$ Distribute: $$2a - 2 = 3a - 15$$ Bring a terms to left and constants to right: $$2a - 3a = -15 + 2$$ $$-a = -13$$ Multiply both sides by -1: $$a = 13$$ 20. Solve $\frac{4}{3b + 2} = \frac{7}{5(b - 3)} \times x$. Assuming $x=1$, Cross multiply: $$4 \times 5(b - 3) = 7(3b + 2)$$ Distribute: $$20b - 60 = 21b + 14$$ Bring b terms to left and constants to right: $$20b - 21b = 14 + 60$$ $$-b = 74$$ Multiply both sides by -1: $$b = -74$$ 21. Solve $\frac{3}{4} - \frac{1}{2a - 1} = \frac{1}{2}$. Bring constants to right: $$- \frac{1}{2a - 1} = \frac{1}{2} - \frac{3}{4} = - \frac{1}{4}$$ Multiply both sides by $-(2a - 1)$: $$1 = \frac{1}{4} (2a - 1)$$ Multiply both sides by 4: $$4 = 2a - 1$$ Add 1: $$5 = 2a$$ Divide by 2: $$a = \frac{5}{2} = 2.5$$