Solve {29/10-[(2+8/15-3/20+frac{5/4)-frac{7}{15}-(frac{12}{25}-frac{2}{15})]}:(frac{19}{20}-frac{7}{10}+frac{1}{2})}{[frac{83}{60}-frac{3}{5}*(frac{1}{8}+frac{5}{6}-frac{4}{9})]-frac{3}{40}}

$$\frac { \{ \frac { 29 } { 10 } - [ ( 2 + \frac { 8 } { 15 } - \frac { 3 } { 20 } + \frac { 5 } { 4 } ) - \frac { 7 } { 15 } - ( \frac { 12 } { 25 } - \frac { 2 } { 15 } ) ] \} : ( \frac { 19 } { 20 } - \frac { 7 } { 10 } + \frac { 1 } { 2 } ) } { [ \frac { 83 } { 60 } - \frac { 3 } { 5 } \times ( \frac { 1 } { 8 } + \frac { 5 } { 6 } - \frac { 4 } { 9 } ) ] - \frac { 3 } { 40 } }$$

Evaluate

$$\frac{8}{75}\approx 0.106666667$$

Solution Steps

Convert 2 to fraction \frac{30}{15}.

$$\frac{\frac{\frac{29}{10}-\left(\frac{30}{15}+\frac{8}{15}-\frac{3}{20}+\frac{5}{4}-\frac{7}{15}-\left(\frac{12}{25}-\frac{2}{15}\right)\right)}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Since \frac{30}{15} and \frac{8}{15} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\frac{29}{10}-\left(\frac{30+8}{15}-\frac{3}{20}+\frac{5}{4}-\frac{7}{15}-\left(\frac{12}{25}-\frac{2}{15}\right)\right)}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Add 30 and 8 to get 38.

$$\frac{\frac{\frac{29}{10}-\left(\frac{38}{15}-\frac{3}{20}+\frac{5}{4}-\frac{7}{15}-\left(\frac{12}{25}-\frac{2}{15}\right)\right)}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Least common multiple of 15 and 20 is 60. Convert \frac{38}{15} and \frac{3}{20} to fractions with denominator 60.

$$\frac{\frac{\frac{29}{10}-\left(\frac{152}{60}-\frac{9}{60}+\frac{5}{4}-\frac{7}{15}-\left(\frac{12}{25}-\frac{2}{15}\right)\right)}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Since \frac{152}{60} and \frac{9}{60} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\frac{29}{10}-\left(\frac{152-9}{60}+\frac{5}{4}-\frac{7}{15}-\left(\frac{12}{25}-\frac{2}{15}\right)\right)}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Subtract 9 from 152 to get 143.

$$\frac{\frac{\frac{29}{10}-\left(\frac{143}{60}+\frac{5}{4}-\frac{7}{15}-\left(\frac{12}{25}-\frac{2}{15}\right)\right)}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Least common multiple of 60 and 4 is 60. Convert \frac{143}{60} and \frac{5}{4} to fractions with denominator 60.

$$\frac{\frac{\frac{29}{10}-\left(\frac{143}{60}+\frac{75}{60}-\frac{7}{15}-\left(\frac{12}{25}-\frac{2}{15}\right)\right)}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Since \frac{143}{60} and \frac{75}{60} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\frac{29}{10}-\left(\frac{143+75}{60}-\frac{7}{15}-\left(\frac{12}{25}-\frac{2}{15}\right)\right)}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Add 143 and 75 to get 218.

$$\frac{\frac{\frac{29}{10}-\left(\frac{218}{60}-\frac{7}{15}-\left(\frac{12}{25}-\frac{2}{15}\right)\right)}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Reduce the fraction \frac{218}{60} to lowest terms by extracting and canceling out 2.

$$\frac{\frac{\frac{29}{10}-\left(\frac{109}{30}-\frac{7}{15}-\left(\frac{12}{25}-\frac{2}{15}\right)\right)}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Least common multiple of 30 and 15 is 30. Convert \frac{109}{30} and \frac{7}{15} to fractions with denominator 30.

$$\frac{\frac{\frac{29}{10}-\left(\frac{109}{30}-\frac{14}{30}-\left(\frac{12}{25}-\frac{2}{15}\right)\right)}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Since \frac{109}{30} and \frac{14}{30} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\frac{29}{10}-\left(\frac{109-14}{30}-\left(\frac{12}{25}-\frac{2}{15}\right)\right)}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Subtract 14 from 109 to get 95.

$$\frac{\frac{\frac{29}{10}-\left(\frac{95}{30}-\left(\frac{12}{25}-\frac{2}{15}\right)\right)}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Reduce the fraction \frac{95}{30} to lowest terms by extracting and canceling out 5.

$$\frac{\frac{\frac{29}{10}-\left(\frac{19}{6}-\left(\frac{12}{25}-\frac{2}{15}\right)\right)}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Least common multiple of 25 and 15 is 75. Convert \frac{12}{25} and \frac{2}{15} to fractions with denominator 75.

$$\frac{\frac{\frac{29}{10}-\left(\frac{19}{6}-\left(\frac{36}{75}-\frac{10}{75}\right)\right)}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Since \frac{36}{75} and \frac{10}{75} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\frac{29}{10}-\left(\frac{19}{6}-\frac{36-10}{75}\right)}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Subtract 10 from 36 to get 26.

$$\frac{\frac{\frac{29}{10}-\left(\frac{19}{6}-\frac{26}{75}\right)}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Least common multiple of 6 and 75 is 150. Convert \frac{19}{6} and \frac{26}{75} to fractions with denominator 150.

$$\frac{\frac{\frac{29}{10}-\left(\frac{475}{150}-\frac{52}{150}\right)}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Since \frac{475}{150} and \frac{52}{150} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\frac{29}{10}-\frac{475-52}{150}}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Subtract 52 from 475 to get 423.

$$\frac{\frac{\frac{29}{10}-\frac{423}{150}}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Reduce the fraction \frac{423}{150} to lowest terms by extracting and canceling out 3.

$$\frac{\frac{\frac{29}{10}-\frac{141}{50}}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Least common multiple of 10 and 50 is 50. Convert \frac{29}{10} and \frac{141}{50} to fractions with denominator 50.

$$\frac{\frac{\frac{145}{50}-\frac{141}{50}}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Since \frac{145}{50} and \frac{141}{50} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\frac{145-141}{50}}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Subtract 141 from 145 to get 4.

$$\frac{\frac{\frac{4}{50}}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Reduce the fraction \frac{4}{50} to lowest terms by extracting and canceling out 2.

$$\frac{\frac{\frac{2}{25}}{\frac{19}{20}-\frac{7}{10}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Least common multiple of 20 and 10 is 20. Convert \frac{19}{20} and \frac{7}{10} to fractions with denominator 20.

$$\frac{\frac{\frac{2}{25}}{\frac{19}{20}-\frac{14}{20}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Since \frac{19}{20} and \frac{14}{20} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\frac{2}{25}}{\frac{19-14}{20}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Subtract 14 from 19 to get 5.

$$\frac{\frac{\frac{2}{25}}{\frac{5}{20}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Reduce the fraction \frac{5}{20} to lowest terms by extracting and canceling out 5.

$$\frac{\frac{\frac{2}{25}}{\frac{1}{4}+\frac{1}{2}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Least common multiple of 4 and 2 is 4. Convert \frac{1}{4} and \frac{1}{2} to fractions with denominator 4.

$$\frac{\frac{\frac{2}{25}}{\frac{1}{4}+\frac{2}{4}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Since \frac{1}{4} and \frac{2}{4} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\frac{2}{25}}{\frac{1+2}{4}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Add 1 and 2 to get 3.

$$\frac{\frac{\frac{2}{25}}{\frac{3}{4}}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Divide \frac{2}{25} by \frac{3}{4} by multiplying \frac{2}{25} by the reciprocal of \frac{3}{4}.

$$\frac{\frac{2}{25}\times \frac{4}{3}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Multiply \frac{2}{25} times \frac{4}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{2\times 4}{25\times 3}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Do the multiplications in the fraction \frac{2\times 4}{25\times 3}.

$$\frac{\frac{8}{75}}{\frac{83}{60}-\frac{3}{5}\left(\frac{1}{8}+\frac{5}{6}-\frac{4}{9}\right)-\frac{3}{40}}$$

Least common multiple of 8 and 6 is 24. Convert \frac{1}{8} and \frac{5}{6} to fractions with denominator 24.

$$\frac{\frac{8}{75}}{\frac{83}{60}-\frac{3}{5}\left(\frac{3}{24}+\frac{20}{24}-\frac{4}{9}\right)-\frac{3}{40}}$$

Since \frac{3}{24} and \frac{20}{24} have the same denominator, add them by adding their numerators.

$$\frac{\frac{8}{75}}{\frac{83}{60}-\frac{3}{5}\left(\frac{3+20}{24}-\frac{4}{9}\right)-\frac{3}{40}}$$

Add 3 and 20 to get 23.

$$\frac{\frac{8}{75}}{\frac{83}{60}-\frac{3}{5}\left(\frac{23}{24}-\frac{4}{9}\right)-\frac{3}{40}}$$

Least common multiple of 24 and 9 is 72. Convert \frac{23}{24} and \frac{4}{9} to fractions with denominator 72.

$$\frac{\frac{8}{75}}{\frac{83}{60}-\frac{3}{5}\left(\frac{69}{72}-\frac{32}{72}\right)-\frac{3}{40}}$$

Since \frac{69}{72} and \frac{32}{72} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{8}{75}}{\frac{83}{60}-\frac{3}{5}\times \frac{69-32}{72}-\frac{3}{40}}$$

Subtract 32 from 69 to get 37.

$$\frac{\frac{8}{75}}{\frac{83}{60}-\frac{3}{5}\times \frac{37}{72}-\frac{3}{40}}$$

Multiply \frac{3}{5} times \frac{37}{72} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{8}{75}}{\frac{83}{60}-\frac{3\times 37}{5\times 72}-\frac{3}{40}}$$

Do the multiplications in the fraction \frac{3\times 37}{5\times 72}.

$$\frac{\frac{8}{75}}{\frac{83}{60}-\frac{111}{360}-\frac{3}{40}}$$

Reduce the fraction \frac{111}{360} to lowest terms by extracting and canceling out 3.

$$\frac{\frac{8}{75}}{\frac{83}{60}-\frac{37}{120}-\frac{3}{40}}$$

Least common multiple of 60 and 120 is 120. Convert \frac{83}{60} and \frac{37}{120} to fractions with denominator 120.

$$\frac{\frac{8}{75}}{\frac{166}{120}-\frac{37}{120}-\frac{3}{40}}$$

Since \frac{166}{120} and \frac{37}{120} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{8}{75}}{\frac{166-37}{120}-\frac{3}{40}}$$

Subtract 37 from 166 to get 129.

$$\frac{\frac{8}{75}}{\frac{129}{120}-\frac{3}{40}}$$

Reduce the fraction \frac{129}{120} to lowest terms by extracting and canceling out 3.

$$\frac{\frac{8}{75}}{\frac{43}{40}-\frac{3}{40}}$$

Since \frac{43}{40} and \frac{3}{40} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{8}{75}}{\frac{43-3}{40}}$$

Subtract 3 from 43 to get 40.

$$\frac{\frac{8}{75}}{\frac{40}{40}}$$

Divide 40 by 40 to get 1.

$$\frac{\frac{8}{75}}{1}$$

Anything divided by one gives itself.

$$\frac{8}{75}$$

Factor

$$\frac{2 ^ {3}}{3 \cdot 5 ^ {2}} = 0.10666666666666667$$