1. **Statement:** Simplify and evaluate the expressions A, B, C, D as given, and also simplify algebraic expressions involving variables a, b, c. --- ### Expression A Given: $$A = 1 - \left(\frac{3}{4} - \frac{2}{3}\right) - \left[1 - \left(\frac{5}{3} - \frac{1}{4}\right)\right] - \left[1 - \left(\frac{4}{3} \times \frac{3}{4}\right)\right]$$ Step 1: Compute inside parentheses: $$\frac{3}{4} - \frac{2}{3} = \frac{9}{12} - \frac{8}{12} = \frac{1}{12}$$ Step 2: $$\frac{5}{3} - \frac{1}{4} = \frac{20}{12} - \frac{3}{12} = \frac{17}{12}$$ So, $$1 - \left(\frac{5}{3} - \frac{1}{4}\right) = 1 - \frac{17}{12} = \frac{12}{12} - \frac{17}{12} = -\frac{5}{12}$$ Step 3: $$\frac{4}{3} \times \frac{3}{4} = 1$$ Hence, $$1 - \left(\frac{4}{3} \times \frac{3}{4}\right) = 1 - 1 = 0$$ Step 4: Substitute back: $$A = 1 - \frac{1}{12} - \left(-\frac{5}{12}\right) - 0 = 1 - \frac{1}{12} + \frac{5}{12} = 1 + \frac{4}{12} = 1 + \frac{1}{3} = \frac{4}{3}$$ --- ### Expression B Given: $$B = \left[2 - \left(\frac{5}{7} - \frac{4}{3}\right)\right] \times \left(1 - \frac{8}{5}\right)$$ Step 1 inside parentheses: $$\frac{5}{7} - \frac{4}{3} = \frac{15}{21} - \frac{28}{21} = -\frac{13}{21}$$ Step 2: $$2 - \left(-\frac{13}{21}\right) = 2 + \frac{13}{21} = \frac{42}{21} + \frac{13}{21} = \frac{55}{21}$$ Step 3: $$1 - \frac{8}{5} = \frac{5}{5} - \frac{8}{5} = -\frac{3}{5}$$ Step 4: $$B = \frac{55}{21} \times -\frac{3}{5} = -\frac{165}{105} = -\frac{11}{7}$$ --- ### Expression C Given equality: $$\frac{3}{4} \times \frac{5}{3} \times \frac{4}{3} = \frac{5}{7} \times \frac{2}{3} \times \left(-\frac{7}{10}\right)$$ Calculate left side: $$\frac{3}{4} \times \frac{5}{3} = \frac{5}{4}$$ Then: $$\frac{5}{4} \times \frac{4}{3} = \frac{5}{3}$$ Calculate right side: $$\frac{5}{7} \times \frac{2}{3} = \frac{10}{21}$$ Then: $$\frac{10}{21} \times \left(-\frac{7}{10}\right) = -\frac{7}{21} = -\frac{1}{3}$$ Since left side $= \frac{5}{3}$ and right side $= -\frac{1}{3}$, the equality does not hold. --- ### Expression D Given: $$D = \frac{1 + \frac{1}{3}}{1 - \frac{1}{3}} \times \frac{1 + \frac{1}{2}}{1 - \frac{1}{2}}$$ Simplify numerators and denominators: $$1 + \frac{1}{3} = \frac{4}{3}$$ $$1 - \frac{1}{3} = \frac{2}{3}$$ $$1 + \frac{1}{2} = \frac{3}{2}$$ $$1 - \frac{1}{2} = \frac{1}{2}$$ Substitute: $$D = \frac{\frac{4}{3}}{\frac{2}{3}} \times \frac{\frac{3}{2}}{\frac{1}{2}} = \frac{4}{3} \times \frac{3}{2} \times \frac{3}{2} \times 2 = 2 \times 3 = 6$$ --- ### Additional simplifications \(A = \frac{3 \times 2 \times 18 \times 16}{2 \times 8 \times 12} - \frac{3 \times 5 \times 6}{2 \times 5 \times 12}\) Numerator 1: $3 \times 2 \times 18 \times 16 = 1728$ Denominator 1: $2 \times 8 \times 12 = 192$ Fraction 1: $\frac{1728}{192} = 9$ Numerator 2: $3 \times 5 \times 6 = 90$ Denominator 2: $2 \times 5 \times 12=120$ Fraction 2: $\frac{90}{120} = \frac{3}{4}$ So, $$A = 9 - \frac{3}{4} = \frac{36}{4} - \frac{3}{4} = \frac{33}{4} = 8.25$$ \(B = \frac{5 \times 4 \times 13}{10 \times 61} - \frac{5 \times 6 \times 30}{25 \times 26}\) Numerator 1: $5 \times 4 \times 13 = 260$ Denominator 1: $10 \times 61=610$ Fraction 1: $\frac{260}{610} = \frac{26}{61}$ Numerator 2: $5 \times 6 \times 30=900$ Denominator 2: $25 \times 26=650$ Fraction 2: $\frac{900}{650} = \frac{18}{13}$ So, $$B = \frac{26}{61} - \frac{18}{13} = \frac{26 \times 13}{61 \times 13} - \frac{18 \times 61}{13 \times 61} = \frac{338}{793} - \frac{1098}{793} = -\frac{760}{793}$$ --- ### Algebraic expressions simplifications: 1. $$A = 3a - 4b + c + 18a - 3b - 7c = (3a + 18a) + (-4b -3b) + (c - 7c) = 21a - 7b - 6c$$ 2. $$B = 5ab + 3ac - 2bc + 4ac - 3bc - 18ac = 5ab + (3ac +4ac -18ac) + (-2bc -3bc) = 5ab -11ac -5bc$$ 3. $$C = 11a^2 - 7a + 3ab - 13a^2 - 11ab + a = (11a^2 - 13a^2) + (-7a + a) + (3ab - 11ab) = -2a^2 - 6a - 8ab$$ 4. $$D = (a + b - c) - [(a - c) - (a - b)]$$ Inside bracket: $$(a - c) - (a - b) = a - c - a + b = b - c$$ Thus, $$D = (a + b - c) - (b - c) = a + b - c - b + c = a$$ 5. $$E = 2(3a - 5) - 4(4a + 3) = 6a - 10 - 16a - 12 = (6a - 16a) + (-10 - 12) = -10a - 22$$ 6. $$F = 3(a + b - c) - 2(a - b + c) + b - a - 2c$$ Expand: $$3a + 3b - 3c - 2a + 2b - 2c + b - a - 2c$$ Combine like terms: $$(3a - 2a - a) + (3b + 2b + b) + (-3c - 2c - 2c) = 0a + 6b - 7c = 6b - 7c$$ 7. Given formula: $$c = \frac{ab}{a + b}$$ 8. Simplify expression: $$\frac{a - c}{b - c}$$ Substitute $$c = \frac{ab}{a + b}$$ Numerator: $$a - \frac{ab}{a + b} = \frac{a(a + b) - ab}{a + b} = \frac{a^2 + ab - ab}{a + b} = \frac{a^2}{a + b}$$ Denominator: $$b - \frac{ab}{a + b} = \frac{b(a + b) - ab}{a + b} = \frac{ab + b^2 - ab}{a + b} = \frac{b^2}{a + b}$$ Thus: $$\frac{a - c}{b - c} = \frac{\frac{a^2}{a + b}}{\frac{b^2}{a + b}} = \frac{a^2}{a + b} \times \frac{a + b}{b^2} = \frac{a^2}{b^2} = \left(\frac{a}{b}\right)^2$$ --- **Final simplified results:** $$\boxed{A = \frac{4}{3}, \quad B = -\frac{11}{7}, \quad C \text{ equality false}, \quad D = 6}$$ $$\boxed{A = 21a - 7b - 6c}$$ $$\boxed{B = 5ab - 11ac - 5bc}$$ $$\boxed{C = -2a^{2} - 6a - 8ab}$$ $$\boxed{D = a}$$ $$\boxed{E = -10a - 22}$$ $$\boxed{F = 6b - 7c}$$ $$\boxed{\frac{a-c}{b-c} = \left(\frac{a}{b}\right)^2}$$