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Bài giải
\(\frac{2}{3}+\frac{3}{4}+\frac{4}{5}=\frac{40}{60}+\frac{45}{60}+\frac{48}{60}=\frac{133}{60}\)
\(\frac{8}{5}+\frac{7}{6}+\frac{10}{9}+\frac{1}{2}=\frac{144}{90}+\frac{105}{90}+\frac{100}{90}+\frac{45}{90}=\frac{394}{90}\)
\(\frac{15}{17}-\frac{11}{13}+\frac{3}{26}=\frac{390}{442}+\frac{374}{442}+\frac{51}{442}=\frac{815}{442}\)
\(\frac{9}{12}\text{ x }\frac{4}{3}\text{ : }\frac{8}{5}=\frac{9}{12}\text{ x }\frac{4}{3}\text{ x }\frac{5}{8}=\frac{9\text{ x }4\text{ x }5}{12\text{ x }3\text{ x }8}=\frac{5}{8}\)
\(\frac{4}{5}\text{ x }\frac{15}{8}\text{ : }\frac{5}{7}=\frac{4}{5}\text{ x }\frac{15}{8}\text{ x }\frac{7}{5}=\frac{4\text{ x }15\text{ x }7}{5\text{ x }8\text{ x }5}=\frac{21}{10}\)
\(\frac{2}{3}+\frac{3}{4}+\frac{4}{5}=\frac{40}{60}+\frac{45}{60}+\frac{48}{60}=\frac{133}{60}\)
\(\frac{8}{5}+\frac{7}{6}+\frac{10}{9}+\frac{1}{2}=\frac{144}{90}+\frac{105}{90}+\frac{100}{90}+\frac{45}{90}=\frac{197}{45}\)
\(\frac{15}{17}-\frac{11}{13}+\frac{1}{26}=\frac{390}{442}+\frac{374}{442}+\frac{51}{442}=\frac{815}{442}\)
\(\frac{9}{12}\times\frac{4}{3}:\frac{8}{5}=1:\frac{8}{5}=\frac{5}{8}\)
\(\frac{4}{5}\times\frac{15}{8}:\frac{5}{7}=\frac{3}{2}:\frac{5}{7}=\frac{21}{10}\)
\(\frac{5}{9}\times\frac{12}{7}=\frac{5\times12}{9\times7}=\frac{5\times4\times3}{3\times3\times7}=\frac{20}{21}\)
\(\frac{6}{5}:\frac{8}{3}=\frac{6}{5}\times\frac{8}{3}=\frac{2\times3\times8}{5\times3}=\frac{16}{5}\)
\(\frac{9}{20}\times\frac{5}{12}=\frac{3\times3\times5}{4\times5\times4\times3}=\frac{3}{16}\)
\(\frac{15}{16}:\frac{25}{24}=\frac{15}{16}\times\frac{24}{25}=\frac{3\times5\times4\times2\times3}{4\times2\times2\times5\times5}=\frac{9}{10}\)
\(14\times\frac{5}{21}=\frac{2\times7\times5}{3\times7}=\frac{10}{3}\)
\(10:\frac{5}{3}=10\times\frac{3}{5}=\frac{2\times5\times3}{5}=6\)
\(\frac{5}{3}:10=\frac{5}{3}\times\frac{1}{10}=\frac{5}{3\times2\times5}=\frac{1}{6}\)
\(\frac{5}{12}\times\frac{2}{3}+\frac{2}{3}\times\frac{7}{12}+\frac{1}{3}\)
\(=\frac{2}{3}\times\left(\frac{5}{12}+\frac{7}{12}\right)+\frac{1}{3}\)
\(=\frac{2}{3}\times1+\frac{1}{3}\)
\(=\frac{2}{3}+\frac{1}{3}\)
\(=1\)
\(\frac{5}{12}\times\frac{2}{3}+\frac{2}{3}\times\frac{7}{12}+\frac{1}{3}=\frac{2}{3}\times\left(\frac{5}{12}+\frac{7}{12}\right)+\frac{1}{3}\)
\(=\frac{2}{3}\times\frac{12}{12}+\frac{1}{3}=\frac{2}{3}\times1+\frac{1}{3}=\frac{2}{3}+\frac{1}{3}=\frac{3}{3}=1\)
Ta có:
\(S=\dfrac{2}{3}+\dfrac{2}{6}+\dfrac{2}{12}+\dfrac{2}{24}+\dfrac{2}{48}+\dfrac{2}{96}+\dfrac{2}{192}+\dfrac{2}{384}\\ =\dfrac{2}{3}+\dfrac{2}{2\times3}+\dfrac{2}{2\times6}+\dfrac{2}{2\times12}+\dfrac{2}{2\times24}+\dfrac{2}{2\times48}+\dfrac{2}{2\times96}+\dfrac{2}{2\times192}\\ =\dfrac{2}{3}+\dfrac{1}{3}+\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{24}+\dfrac{1}{48}+\dfrac{1}{96}+\dfrac{1}{192}\\ \)
\(\dfrac{S}{2}=\dfrac{1}{2}\left(\dfrac{2}{3}+\dfrac{2}{6}+\dfrac{2}{12}+\dfrac{2}{24}+\dfrac{2}{48}+\dfrac{2}{96}+\dfrac{2}{192}+\dfrac{2}{384}\right)\\ =\dfrac{1}{3}+\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{24}+\dfrac{1}{48}+\dfrac{1}{96}+\dfrac{1}{192}+\dfrac{1}{384}\)
\(S-\dfrac{S}{2}=\dfrac{2}{3}+\dfrac{1}{3}+\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{24}+\dfrac{1}{48}+\dfrac{1}{96}+\dfrac{1}{192}-\left(\dfrac{1}{3}+\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{24}+\dfrac{1}{48}+\dfrac{1}{96}+\dfrac{1}{192}+\dfrac{1}{384}\right)\\ =\dfrac{2}{3}-\dfrac{1}{384}=\dfrac{2\times128-1}{384}\\ =\dfrac{85}{128}\\ \Rightarrow S=\dfrac{85}{128}\times2=\dfrac{85}{64}\)
\(A=\dfrac{2}{3}+\dfrac{2}{6}+\dfrac{2}{12}+\dfrac{2}{24}+\dfrac{2}{48}+\dfrac{2}{96}+\dfrac{2}{192}+\dfrac{2}{384}\)
\(A.2=\dfrac{4}{3}+\dfrac{2}{3}+\dfrac{2}{6}+\dfrac{2}{12}+\dfrac{2}{24}+\dfrac{2}{48}+\dfrac{2}{96}+\dfrac{2}{192}\)
\(A=A.2-A=\dfrac{4}{3}-\dfrac{2}{384}=\dfrac{127}{96}\)