1. $\frac{\frac{2}{3} + \frac{3}{4}}{\frac{1}{5}} = \frac{\frac{8}{12} + \frac{9}{12}}{\frac{1}{5}} = \frac{\frac{17}{12}}{\frac{1}{5}} = \frac{17}{12} \times 5 = \frac{85}{12}$
2. $\frac{\frac{3}{4} - \frac{1}{3}}{\frac{1}{2} + \frac{1}{6}} = \frac{\frac{9}{12} - \frac{4}{12}}{\frac{3}{6} + \frac{1}{6}} = \frac{\frac{5}{12}}{\frac{4}{6}} = \frac{5}{12} \times \frac{6}{4} = \frac{30}{48} = \frac{5}{8}$
3. $\frac{\frac{1}{2} + \frac{1}{3} + \frac{1}{4}}{3 - \frac{4}{5}} = \frac{\frac{6}{12} + \frac{4}{12} + \frac{3}{12}}{\frac{15}{5} - \frac{4}{5}} = \frac{\frac{13}{12}}{\frac{11}{5}} = \frac{13}{12} \times \frac{5}{11} = \frac{65}{132}$
4. $\frac{\frac{4}{5}}{\frac{1}{5} + \frac{2}{3}} = \frac{\frac{4}{5}}{\frac{3}{15} + \frac{10}{15}} = \frac{\frac{4}{5}}{\frac{13}{15}} = \frac{4}{5} \times \frac{15}{13} = \frac{60}{65} = \frac{12}{13}$
5. $\frac{5 - \frac{2}{5}}{6 + \frac{1}{3}} = \frac{\frac{25}{5} - \frac{2}{5}}{\frac{18}{3} + \frac{1}{3}} = \frac{\frac{23}{5}}{\frac{19}{3}} = \frac{23}{5} \times \frac{3}{19} = \frac{69}{95}$
6. $\frac{\frac{1}{2} + \frac{3}{5}}{\frac{2}{5} + \frac{1}{3}} = \frac{\frac{5}{10} + \frac{6}{10}}{\frac{6}{15} + \frac{5}{15}} = \frac{\frac{11}{10}}{\frac{11}{15}} = \frac{11}{10} \times \frac{15}{11} = \frac{165}{110} = \frac{33}{22} = \frac{3}{2} = 1.5$ (Note: The answer given is 101/22, which is incorrect. The correct answer is 3/2.)
7. $\frac{x + \frac{y}{4}}{x - \frac{1}{3}y} = \frac{\frac{4x + y}{4}}{\frac{3x - y}{3}} = \frac{4x + y}{4} \times \frac{3}{3x - y} = \frac{3(4x + y)}{4(3x - y)}$
8. $\frac{\frac{a}{b} + \frac{x}{y}}{\frac{a}{z} - \frac{c}{x}} = \frac{\frac{ay + bx}{by}}{\frac{ax - cz}{zx}} = \frac{ay + bx}{by} \times \frac{zx}{ax - cz} = \frac{zx(ay + bx)}{by(ax - cz)}$
9. $\frac{1 + \frac{x}{y}}{1 - \frac{x^2}{y^2}} = \frac{\frac{y + x}{y}}{\frac{y^2 - x^2}{y^2}} = \frac{y + x}{y} \times \frac{y^2}{y^2 - x^2} = \frac{(y + x)y^2}{y(y^2 - x^2)} = \frac{y(y + x)}{y^2 - x^2} = \frac{y(y + x)}{(y - x)(y + x)} = \frac{y}{y - x}$
10. $\frac{x + \frac{a}{c}}{x + \frac{b}{d}} = \frac{\frac{cx + a}{c}}{\frac{dx + b}{d}} = \frac{cx + a}{c} \times \frac{d}{dx + b} = \frac{d(cx + a)}{c(dx + b)} = \frac{d(a + cx)}{c(b + dx)}$
11. $\frac{\frac{a^2}{4} + \frac{x}{3}}{\frac{3}{x} + \frac{2d}{3ac}} = \frac{\frac{3a^2 + 4x}{12}}{\frac{9 + 2d}{3ac}} = \frac{3a^2 + 4x}{12} \times \frac{3ac}{9 + 2d} = \frac{3ac(3a^2 + 4x)}{12(9 + 2d)} = \frac{ac(3a^2 + 4x)}{4(9 + 2d)}$ (Note: The answer given is 5(3a² + x)/3(20 + x), which is incorrect.)
12. $\frac{3a^2 - 3y^2}{\frac{a + y}{3}} = (3a^2 - 3y^2) \times \frac{3}{a + y} = \frac{3(3a^2 - 3y^2)}{a + y} = \frac{9(a^2 - y^2)}{a + y} = \frac{9(a - y)(a + y)}{a + y} = 9(a - y)$
13. $\frac{x + \frac{2d}{3ac}}{x + \frac{3d}{2ac}} = \frac{\frac{3acx + 2d}{3ac}}{\frac{2acx + 3d}{2ac}} = \frac{3acx + 2d}{3ac} \times \frac{2ac}{2acx + 3d} = \frac{2(3acx + 2d)}{3(2acx + 3d)}$
14. $\frac{4a^2 - 4x^2}{\frac{a + x}{a - x}} = (4a^2 - 4x^2) \times \frac{a - x}{a + x} = \frac{4(a^2 - x^2)(a - x)}{a + x} = \frac{4(a - x)(a + x)(a - x)}{a + x} = 4(a - x)^2$
Parent Tip: Review the logic above to help your child master the concept of complex fraction worksheet.
