Solve {l}{1/2:7+(21/7-134/35)*(33/9-1frac{3}{4})}{(1frac{3}{4}:frac{4}{7}-frac{1}{4}*frac{1}{4}-frac{5}{8}):(3frac{3}{4}+frac{2}{5}:2-frac{7}{12}:frac{14}{15})}

$$\left. \begin{array} { l } { \frac { 1 } { 2 } : 7 + ( 2 \frac { 1 } { 7 } - 1 \frac { 34 } { 35 } ) \cdot ( 3 \frac { 3 } { 9 } - 1 \frac { 3 } { 4 } ) } \\ { ( 1 \frac { 3 } { 4 } : \frac { 4 } { 7 } - \frac { 1 } { 4 } \cdot \frac { 1 } { 4 } - \frac { 5 } { 8 } ) : ( 3 \frac { 3 } { 4 } + \frac { 2 } { 5 } : 2 - \frac { 7 } { 12 } : \frac { 14 } { 15 } ) } \end{array} \right.$$

Sort

$$\frac{12}{35},\ \frac{5}{7}$$

Solution Steps

Express \frac{\frac{1}{2}}{7} as a single fraction.

$$sort(\frac{1}{2\times 7}+\left(\frac{2\times 7+1}{7}-\frac{1\times 35+34}{35}\right)\left(\frac{3\times 9+3}{9}-\frac{1\times 4+3}{4}\right),\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Multiply 2 and 7 to get 14.

$$sort(\frac{1}{14}+\left(\frac{2\times 7+1}{7}-\frac{1\times 35+34}{35}\right)\left(\frac{3\times 9+3}{9}-\frac{1\times 4+3}{4}\right),\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Multiply 2 and 7 to get 14.

$$sort(\frac{1}{14}+\left(\frac{14+1}{7}-\frac{1\times 35+34}{35}\right)\left(\frac{3\times 9+3}{9}-\frac{1\times 4+3}{4}\right),\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Add 14 and 1 to get 15.

$$sort(\frac{1}{14}+\left(\frac{15}{7}-\frac{1\times 35+34}{35}\right)\left(\frac{3\times 9+3}{9}-\frac{1\times 4+3}{4}\right),\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Multiply 1 and 35 to get 35.

$$sort(\frac{1}{14}+\left(\frac{15}{7}-\frac{35+34}{35}\right)\left(\frac{3\times 9+3}{9}-\frac{1\times 4+3}{4}\right),\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Add 35 and 34 to get 69.

$$sort(\frac{1}{14}+\left(\frac{15}{7}-\frac{69}{35}\right)\left(\frac{3\times 9+3}{9}-\frac{1\times 4+3}{4}\right),\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Least common multiple of 7 and 35 is 35. Convert \frac{15}{7} and \frac{69}{35} to fractions with denominator 35.

$$sort(\frac{1}{14}+\left(\frac{75}{35}-\frac{69}{35}\right)\left(\frac{3\times 9+3}{9}-\frac{1\times 4+3}{4}\right),\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Since \frac{75}{35} and \frac{69}{35} have the same denominator, subtract them by subtracting their numerators.

$$sort(\frac{1}{14}+\frac{75-69}{35}\left(\frac{3\times 9+3}{9}-\frac{1\times 4+3}{4}\right),\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Subtract 69 from 75 to get 6.

$$sort(\frac{1}{14}+\frac{6}{35}\left(\frac{3\times 9+3}{9}-\frac{1\times 4+3}{4}\right),\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Multiply 3 and 9 to get 27.

$$sort(\frac{1}{14}+\frac{6}{35}\left(\frac{27+3}{9}-\frac{1\times 4+3}{4}\right),\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Add 27 and 3 to get 30.

$$sort(\frac{1}{14}+\frac{6}{35}\left(\frac{30}{9}-\frac{1\times 4+3}{4}\right),\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Reduce the fraction \frac{30}{9} to lowest terms by extracting and canceling out 3.

$$sort(\frac{1}{14}+\frac{6}{35}\left(\frac{10}{3}-\frac{1\times 4+3}{4}\right),\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Multiply 1 and 4 to get 4.

$$sort(\frac{1}{14}+\frac{6}{35}\left(\frac{10}{3}-\frac{4+3}{4}\right),\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Add 4 and 3 to get 7.

$$sort(\frac{1}{14}+\frac{6}{35}\left(\frac{10}{3}-\frac{7}{4}\right),\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Least common multiple of 3 and 4 is 12. Convert \frac{10}{3} and \frac{7}{4} to fractions with denominator 12.

$$sort(\frac{1}{14}+\frac{6}{35}\left(\frac{40}{12}-\frac{21}{12}\right),\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Since \frac{40}{12} and \frac{21}{12} have the same denominator, subtract them by subtracting their numerators.

$$sort(\frac{1}{14}+\frac{6}{35}\times \frac{40-21}{12},\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Subtract 21 from 40 to get 19.

$$sort(\frac{1}{14}+\frac{6}{35}\times \frac{19}{12},\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Multiply \frac{6}{35} times \frac{19}{12} by multiplying numerator times numerator and denominator times denominator.

$$sort(\frac{1}{14}+\frac{6\times 19}{35\times 12},\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Do the multiplications in the fraction \frac{6\times 19}{35\times 12}.

$$sort(\frac{1}{14}+\frac{114}{420},\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Reduce the fraction \frac{114}{420} to lowest terms by extracting and canceling out 6.

$$sort(\frac{1}{14}+\frac{19}{70},\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Least common multiple of 14 and 70 is 70. Convert \frac{1}{14} and \frac{19}{70} to fractions with denominator 70.

$$sort(\frac{5}{70}+\frac{19}{70},\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Since \frac{5}{70} and \frac{19}{70} have the same denominator, add them by adding their numerators.

$$sort(\frac{5+19}{70},\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Add 5 and 19 to get 24.

$$sort(\frac{24}{70},\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Reduce the fraction \frac{24}{70} to lowest terms by extracting and canceling out 2.

$$sort(\frac{12}{35},\frac{\frac{\frac{1\times 4+3}{4}}{\frac{4}{7}}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Divide \frac{1\times 4+3}{4} by \frac{4}{7} by multiplying \frac{1\times 4+3}{4} by the reciprocal of \frac{4}{7}.

$$sort(\frac{12}{35},\frac{\frac{\left(1\times 4+3\right)\times 7}{4\times 4}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Multiply 1 and 4 to get 4.

$$sort(\frac{12}{35},\frac{\frac{\left(4+3\right)\times 7}{4\times 4}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Add 4 and 3 to get 7.

$$sort(\frac{12}{35},\frac{\frac{7\times 7}{4\times 4}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Multiply 7 and 7 to get 49.

$$sort(\frac{12}{35},\frac{\frac{49}{4\times 4}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Multiply 4 and 4 to get 16.

$$sort(\frac{12}{35},\frac{\frac{49}{16}-\frac{1}{4}\times \frac{1}{4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Multiply \frac{1}{4} times \frac{1}{4} by multiplying numerator times numerator and denominator times denominator.

$$sort(\frac{12}{35},\frac{\frac{49}{16}-\frac{1\times 1}{4\times 4}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Do the multiplications in the fraction \frac{1\times 1}{4\times 4}.

$$sort(\frac{12}{35},\frac{\frac{49}{16}-\frac{1}{16}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Since \frac{49}{16} and \frac{1}{16} have the same denominator, subtract them by subtracting their numerators.

$$sort(\frac{12}{35},\frac{\frac{49-1}{16}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Subtract 1 from 49 to get 48.

$$sort(\frac{12}{35},\frac{\frac{48}{16}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Divide 48 by 16 to get 3.

$$sort(\frac{12}{35},\frac{3-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Convert 3 to fraction \frac{24}{8}.

$$sort(\frac{12}{35},\frac{\frac{24}{8}-\frac{5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Since \frac{24}{8} and \frac{5}{8} have the same denominator, subtract them by subtracting their numerators.

$$sort(\frac{12}{35},\frac{\frac{24-5}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Subtract 5 from 24 to get 19.

$$sort(\frac{12}{35},\frac{\frac{19}{8}}{\frac{3\times 4+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Multiply 3 and 4 to get 12.

$$sort(\frac{12}{35},\frac{\frac{19}{8}}{\frac{12+3}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Add 12 and 3 to get 15.

$$sort(\frac{12}{35},\frac{\frac{19}{8}}{\frac{15}{4}+\frac{\frac{2}{5}}{2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Express \frac{\frac{2}{5}}{2} as a single fraction.

$$sort(\frac{12}{35},\frac{\frac{19}{8}}{\frac{15}{4}+\frac{2}{5\times 2}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Cancel out 2 in both numerator and denominator.

$$sort(\frac{12}{35},\frac{\frac{19}{8}}{\frac{15}{4}+\frac{1}{5}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Least common multiple of 4 and 5 is 20. Convert \frac{15}{4} and \frac{1}{5} to fractions with denominator 20.

$$sort(\frac{12}{35},\frac{\frac{19}{8}}{\frac{75}{20}+\frac{4}{20}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Since \frac{75}{20} and \frac{4}{20} have the same denominator, add them by adding their numerators.

$$sort(\frac{12}{35},\frac{\frac{19}{8}}{\frac{75+4}{20}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Add 75 and 4 to get 79.

$$sort(\frac{12}{35},\frac{\frac{19}{8}}{\frac{79}{20}-\frac{\frac{7}{12}}{\frac{14}{15}}})$$

Divide \frac{7}{12} by \frac{14}{15} by multiplying \frac{7}{12} by the reciprocal of \frac{14}{15}.

$$sort(\frac{12}{35},\frac{\frac{19}{8}}{\frac{79}{20}-\frac{7}{12}\times \frac{15}{14}})$$

Multiply \frac{7}{12} times \frac{15}{14} by multiplying numerator times numerator and denominator times denominator.

$$sort(\frac{12}{35},\frac{\frac{19}{8}}{\frac{79}{20}-\frac{7\times 15}{12\times 14}})$$

Do the multiplications in the fraction \frac{7\times 15}{12\times 14}.

$$sort(\frac{12}{35},\frac{\frac{19}{8}}{\frac{79}{20}-\frac{105}{168}})$$

Reduce the fraction \frac{105}{168} to lowest terms by extracting and canceling out 21.

$$sort(\frac{12}{35},\frac{\frac{19}{8}}{\frac{79}{20}-\frac{5}{8}})$$

Least common multiple of 20 and 8 is 40. Convert \frac{79}{20} and \frac{5}{8} to fractions with denominator 40.

$$sort(\frac{12}{35},\frac{\frac{19}{8}}{\frac{158}{40}-\frac{25}{40}})$$

Since \frac{158}{40} and \frac{25}{40} have the same denominator, subtract them by subtracting their numerators.

$$sort(\frac{12}{35},\frac{\frac{19}{8}}{\frac{158-25}{40}})$$

Subtract 25 from 158 to get 133.

$$sort(\frac{12}{35},\frac{\frac{19}{8}}{\frac{133}{40}})$$

Divide \frac{19}{8} by \frac{133}{40} by multiplying \frac{19}{8} by the reciprocal of \frac{133}{40}.

$$sort(\frac{12}{35},\frac{19}{8}\times \frac{40}{133})$$

Multiply \frac{19}{8} times \frac{40}{133} by multiplying numerator times numerator and denominator times denominator.

$$sort(\frac{12}{35},\frac{19\times 40}{8\times 133})$$

Do the multiplications in the fraction \frac{19\times 40}{8\times 133}.

$$sort(\frac{12}{35},\frac{760}{1064})$$

Reduce the fraction \frac{760}{1064} to lowest terms by extracting and canceling out 152.

$$sort(\frac{12}{35},\frac{5}{7})$$

Least common denominator of the numbers in the list \frac{12}{35},\frac{5}{7} is 35. Convert numbers in the list to fractions with denominator 35.

$$\frac{12}{35},\frac{25}{35}$$

Evaluate

$$\frac{12}{35},\ \frac{5}{7}$$