Solve ((37/12-211/18+21/24)*1frac{5/31-frac{3}{52}(3frac{1}{2}+frac{5}{6}))*1frac{7}{13}}{frac{19}{84}:(5frac{13}{42}-2frac{13}{28}+frac{5}{24})+1frac{2}{27}-frac{1}{3}*frac{4}{9}}

$$\frac { ( ( 3 \frac { 7 } { 12 } - 2 \frac { 11 } { 18 } + 2 \frac { 1 } { 24 } ) \cdot 1 \frac { 5 } { 31 } - \frac { 3 } { 52 } ( 3 \frac { 1 } { 2 } + \frac { 5 } { 6 } ) ) \cdot 1 \frac { 7 } { 13 } } { \frac { 19 } { 84 } : ( 5 \frac { 13 } { 42 } - 2 \frac { 13 } { 28 } + \frac { 5 } { 24 } ) + 1 \frac { 2 } { 27 } - \frac { 1 } { 3 } \cdot \frac { 4 } { 9 } }$$

Evaluate

$$5$$

Solution Steps

Multiply 3 and 12 to get 36.

$$\frac{\left(\left(\frac{36+7}{12}-\frac{2\times 18+11}{18}+\frac{2\times 24+1}{24}\right)\times \frac{1\times 31+5}{31}-\frac{3}{52}\left(\frac{3\times 2+1}{2}+\frac{5}{6}\right)\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Add 36 and 7 to get 43.

$$\frac{\left(\left(\frac{43}{12}-\frac{2\times 18+11}{18}+\frac{2\times 24+1}{24}\right)\times \frac{1\times 31+5}{31}-\frac{3}{52}\left(\frac{3\times 2+1}{2}+\frac{5}{6}\right)\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Multiply 2 and 18 to get 36.

$$\frac{\left(\left(\frac{43}{12}-\frac{36+11}{18}+\frac{2\times 24+1}{24}\right)\times \frac{1\times 31+5}{31}-\frac{3}{52}\left(\frac{3\times 2+1}{2}+\frac{5}{6}\right)\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Add 36 and 11 to get 47.

$$\frac{\left(\left(\frac{43}{12}-\frac{47}{18}+\frac{2\times 24+1}{24}\right)\times \frac{1\times 31+5}{31}-\frac{3}{52}\left(\frac{3\times 2+1}{2}+\frac{5}{6}\right)\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Least common multiple of 12 and 18 is 36. Convert \frac{43}{12} and \frac{47}{18} to fractions with denominator 36.

$$\frac{\left(\left(\frac{129}{36}-\frac{94}{36}+\frac{2\times 24+1}{24}\right)\times \frac{1\times 31+5}{31}-\frac{3}{52}\left(\frac{3\times 2+1}{2}+\frac{5}{6}\right)\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Since \frac{129}{36} and \frac{94}{36} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(\left(\frac{129-94}{36}+\frac{2\times 24+1}{24}\right)\times \frac{1\times 31+5}{31}-\frac{3}{52}\left(\frac{3\times 2+1}{2}+\frac{5}{6}\right)\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Subtract 94 from 129 to get 35.

$$\frac{\left(\left(\frac{35}{36}+\frac{2\times 24+1}{24}\right)\times \frac{1\times 31+5}{31}-\frac{3}{52}\left(\frac{3\times 2+1}{2}+\frac{5}{6}\right)\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Multiply 2 and 24 to get 48.

$$\frac{\left(\left(\frac{35}{36}+\frac{48+1}{24}\right)\times \frac{1\times 31+5}{31}-\frac{3}{52}\left(\frac{3\times 2+1}{2}+\frac{5}{6}\right)\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Add 48 and 1 to get 49.

$$\frac{\left(\left(\frac{35}{36}+\frac{49}{24}\right)\times \frac{1\times 31+5}{31}-\frac{3}{52}\left(\frac{3\times 2+1}{2}+\frac{5}{6}\right)\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Least common multiple of 36 and 24 is 72. Convert \frac{35}{36} and \frac{49}{24} to fractions with denominator 72.

$$\frac{\left(\left(\frac{70}{72}+\frac{147}{72}\right)\times \frac{1\times 31+5}{31}-\frac{3}{52}\left(\frac{3\times 2+1}{2}+\frac{5}{6}\right)\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Since \frac{70}{72} and \frac{147}{72} have the same denominator, add them by adding their numerators.

$$\frac{\left(\frac{70+147}{72}\times \frac{1\times 31+5}{31}-\frac{3}{52}\left(\frac{3\times 2+1}{2}+\frac{5}{6}\right)\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Add 70 and 147 to get 217.

$$\frac{\left(\frac{217}{72}\times \frac{1\times 31+5}{31}-\frac{3}{52}\left(\frac{3\times 2+1}{2}+\frac{5}{6}\right)\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Multiply 1 and 31 to get 31.

$$\frac{\left(\frac{217}{72}\times \frac{31+5}{31}-\frac{3}{52}\left(\frac{3\times 2+1}{2}+\frac{5}{6}\right)\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Add 31 and 5 to get 36.

$$\frac{\left(\frac{217}{72}\times \frac{36}{31}-\frac{3}{52}\left(\frac{3\times 2+1}{2}+\frac{5}{6}\right)\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Multiply \frac{217}{72} times \frac{36}{31} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(\frac{217\times 36}{72\times 31}-\frac{3}{52}\left(\frac{3\times 2+1}{2}+\frac{5}{6}\right)\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Do the multiplications in the fraction \frac{217\times 36}{72\times 31}.

$$\frac{\left(\frac{7812}{2232}-\frac{3}{52}\left(\frac{3\times 2+1}{2}+\frac{5}{6}\right)\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Reduce the fraction \frac{7812}{2232} to lowest terms by extracting and canceling out 1116.

$$\frac{\left(\frac{7}{2}-\frac{3}{52}\left(\frac{3\times 2+1}{2}+\frac{5}{6}\right)\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Multiply 3 and 2 to get 6.

$$\frac{\left(\frac{7}{2}-\frac{3}{52}\left(\frac{6+1}{2}+\frac{5}{6}\right)\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Add 6 and 1 to get 7.

$$\frac{\left(\frac{7}{2}-\frac{3}{52}\left(\frac{7}{2}+\frac{5}{6}\right)\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Least common multiple of 2 and 6 is 6. Convert \frac{7}{2} and \frac{5}{6} to fractions with denominator 6.

$$\frac{\left(\frac{7}{2}-\frac{3}{52}\left(\frac{21}{6}+\frac{5}{6}\right)\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Since \frac{21}{6} and \frac{5}{6} have the same denominator, add them by adding their numerators.

$$\frac{\left(\frac{7}{2}-\frac{3}{52}\times \frac{21+5}{6}\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Add 21 and 5 to get 26.

$$\frac{\left(\frac{7}{2}-\frac{3}{52}\times \frac{26}{6}\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Reduce the fraction \frac{26}{6} to lowest terms by extracting and canceling out 2.

$$\frac{\left(\frac{7}{2}-\frac{3}{52}\times \frac{13}{3}\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Multiply \frac{3}{52} times \frac{13}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(\frac{7}{2}-\frac{3\times 13}{52\times 3}\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{\left(\frac{7}{2}-\frac{13}{52}\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Reduce the fraction \frac{13}{52} to lowest terms by extracting and canceling out 13.

$$\frac{\left(\frac{7}{2}-\frac{1}{4}\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Least common multiple of 2 and 4 is 4. Convert \frac{7}{2} and \frac{1}{4} to fractions with denominator 4.

$$\frac{\left(\frac{14}{4}-\frac{1}{4}\right)\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Since \frac{14}{4} and \frac{1}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{14-1}{4}\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Subtract 1 from 14 to get 13.

$$\frac{\frac{13}{4}\times \frac{1\times 13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Multiply 1 and 13 to get 13.

$$\frac{\frac{13}{4}\times \frac{13+7}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Add 13 and 7 to get 20.

$$\frac{\frac{13}{4}\times \frac{20}{13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Multiply \frac{13}{4} times \frac{20}{13} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{13\times 20}{4\times 13}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Cancel out 13 in both numerator and denominator.

$$\frac{\frac{20}{4}}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Divide 20 by 4 to get 5.

$$\frac{5}{\frac{\frac{19}{84}}{\frac{5\times 42+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Multiply 5 and 42 to get 210.

$$\frac{5}{\frac{\frac{19}{84}}{\frac{210+13}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Add 210 and 13 to get 223.

$$\frac{5}{\frac{\frac{19}{84}}{\frac{223}{42}-\frac{2\times 28+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Multiply 2 and 28 to get 56.

$$\frac{5}{\frac{\frac{19}{84}}{\frac{223}{42}-\frac{56+13}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Add 56 and 13 to get 69.

$$\frac{5}{\frac{\frac{19}{84}}{\frac{223}{42}-\frac{69}{28}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Least common multiple of 42 and 28 is 84. Convert \frac{223}{42} and \frac{69}{28} to fractions with denominator 84.

$$\frac{5}{\frac{\frac{19}{84}}{\frac{446}{84}-\frac{207}{84}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Since \frac{446}{84} and \frac{207}{84} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5}{\frac{\frac{19}{84}}{\frac{446-207}{84}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Subtract 207 from 446 to get 239.

$$\frac{5}{\frac{\frac{19}{84}}{\frac{239}{84}+\frac{5}{24}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Least common multiple of 84 and 24 is 168. Convert \frac{239}{84} and \frac{5}{24} to fractions with denominator 168.

$$\frac{5}{\frac{\frac{19}{84}}{\frac{478}{168}+\frac{35}{168}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Since \frac{478}{168} and \frac{35}{168} have the same denominator, add them by adding their numerators.

$$\frac{5}{\frac{\frac{19}{84}}{\frac{478+35}{168}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Add 478 and 35 to get 513.

$$\frac{5}{\frac{\frac{19}{84}}{\frac{513}{168}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Reduce the fraction \frac{513}{168} to lowest terms by extracting and canceling out 3.

$$\frac{5}{\frac{\frac{19}{84}}{\frac{171}{56}}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Divide \frac{19}{84} by \frac{171}{56} by multiplying \frac{19}{84} by the reciprocal of \frac{171}{56}.

$$\frac{5}{\frac{19}{84}\times \frac{56}{171}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Multiply \frac{19}{84} times \frac{56}{171} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5}{\frac{19\times 56}{84\times 171}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Do the multiplications in the fraction \frac{19\times 56}{84\times 171}.

$$\frac{5}{\frac{1064}{14364}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Reduce the fraction \frac{1064}{14364} to lowest terms by extracting and canceling out 532.

$$\frac{5}{\frac{2}{27}+\frac{1\times 27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Multiply 1 and 27 to get 27.

$$\frac{5}{\frac{2}{27}+\frac{27+2}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Add 27 and 2 to get 29.

$$\frac{5}{\frac{2}{27}+\frac{29}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Since \frac{2}{27} and \frac{29}{27} have the same denominator, add them by adding their numerators.

$$\frac{5}{\frac{2+29}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Add 2 and 29 to get 31.

$$\frac{5}{\frac{31}{27}-\frac{1}{3}\times \frac{4}{9}}$$

Multiply \frac{1}{3} times \frac{4}{9} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5}{\frac{31}{27}-\frac{1\times 4}{3\times 9}}$$

Do the multiplications in the fraction \frac{1\times 4}{3\times 9}.

$$\frac{5}{\frac{31}{27}-\frac{4}{27}}$$

Since \frac{31}{27} and \frac{4}{27} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5}{\frac{31-4}{27}}$$

Subtract 4 from 31 to get 27.

$$\frac{5}{\frac{27}{27}}$$

Divide 27 by 27 to get 1.

$$\frac{5}{1}$$

Anything divided by one gives itself.

$$5$$

Factor

$$5$$