Solve 5frac{2/3/1frac{1{4}}-left(21/2+11/3right)*left(3frac{1}{5}-2frac{4}{5}right)}{frac{5frac{2}{3}}{1frac{1}{2}}-2frac{1}{4}1frac{1}{6}-frac{11}{72}}

$$\frac{ \frac{ 5 \frac{ 2 }{ 3 } }{ 1 \frac{ 1 }{ 4 } } - \left( 2 \frac{ 1 }{ 2 } +1 \frac{ 1 }{ 3 } \right) \times \left( 3 \frac{ 1 }{ 5 } -2 \frac{ 4 }{ 5 } \right) }{ \frac{ 5 \frac{ 2 }{ 3 } }{ 1 \frac{ 1 }{ 2 } } -2 \frac{ 1 }{ 4 } 1 \frac{ 1 }{ 6 } - \frac{ 11 }{ 72 } }$$

Evaluate

$$3$$

Solution Steps

Divide \frac{5\times 3+2}{3} by \frac{1\times 4+1}{4} by multiplying \frac{5\times 3+2}{3} by the reciprocal of \frac{1\times 4+1}{4}.

$$\frac{\frac{\left(5\times 3+2\right)\times 4}{3\left(1\times 4+1\right)}-\left(\frac{2\times 2+1}{2}+\frac{1\times 3+1}{3}\right)\left(\frac{3\times 5+1}{5}-\frac{2\times 5+4}{5}\right)}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Multiply 5 and 3 to get 15.

$$\frac{\frac{\left(15+2\right)\times 4}{3\left(1\times 4+1\right)}-\left(\frac{2\times 2+1}{2}+\frac{1\times 3+1}{3}\right)\left(\frac{3\times 5+1}{5}-\frac{2\times 5+4}{5}\right)}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Add 15 and 2 to get 17.

$$\frac{\frac{17\times 4}{3\left(1\times 4+1\right)}-\left(\frac{2\times 2+1}{2}+\frac{1\times 3+1}{3}\right)\left(\frac{3\times 5+1}{5}-\frac{2\times 5+4}{5}\right)}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Multiply 17 and 4 to get 68.

$$\frac{\frac{68}{3\left(1\times 4+1\right)}-\left(\frac{2\times 2+1}{2}+\frac{1\times 3+1}{3}\right)\left(\frac{3\times 5+1}{5}-\frac{2\times 5+4}{5}\right)}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Multiply 1 and 4 to get 4.

$$\frac{\frac{68}{3\left(4+1\right)}-\left(\frac{2\times 2+1}{2}+\frac{1\times 3+1}{3}\right)\left(\frac{3\times 5+1}{5}-\frac{2\times 5+4}{5}\right)}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Add 4 and 1 to get 5.

$$\frac{\frac{68}{3\times 5}-\left(\frac{2\times 2+1}{2}+\frac{1\times 3+1}{3}\right)\left(\frac{3\times 5+1}{5}-\frac{2\times 5+4}{5}\right)}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Multiply 3 and 5 to get 15.

$$\frac{\frac{68}{15}-\left(\frac{2\times 2+1}{2}+\frac{1\times 3+1}{3}\right)\left(\frac{3\times 5+1}{5}-\frac{2\times 5+4}{5}\right)}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Multiply 2 and 2 to get 4.

$$\frac{\frac{68}{15}-\left(\frac{4+1}{2}+\frac{1\times 3+1}{3}\right)\left(\frac{3\times 5+1}{5}-\frac{2\times 5+4}{5}\right)}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Add 4 and 1 to get 5.

$$\frac{\frac{68}{15}-\left(\frac{5}{2}+\frac{1\times 3+1}{3}\right)\left(\frac{3\times 5+1}{5}-\frac{2\times 5+4}{5}\right)}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Multiply 1 and 3 to get 3.

$$\frac{\frac{68}{15}-\left(\frac{5}{2}+\frac{3+1}{3}\right)\left(\frac{3\times 5+1}{5}-\frac{2\times 5+4}{5}\right)}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Add 3 and 1 to get 4.

$$\frac{\frac{68}{15}-\left(\frac{5}{2}+\frac{4}{3}\right)\left(\frac{3\times 5+1}{5}-\frac{2\times 5+4}{5}\right)}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Least common multiple of 2 and 3 is 6. Convert \frac{5}{2} and \frac{4}{3} to fractions with denominator 6.

$$\frac{\frac{68}{15}-\left(\frac{15}{6}+\frac{8}{6}\right)\left(\frac{3\times 5+1}{5}-\frac{2\times 5+4}{5}\right)}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Since \frac{15}{6} and \frac{8}{6} have the same denominator, add them by adding their numerators.

$$\frac{\frac{68}{15}-\frac{15+8}{6}\left(\frac{3\times 5+1}{5}-\frac{2\times 5+4}{5}\right)}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Add 15 and 8 to get 23.

$$\frac{\frac{68}{15}-\frac{23}{6}\left(\frac{3\times 5+1}{5}-\frac{2\times 5+4}{5}\right)}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Multiply 3 and 5 to get 15.

$$\frac{\frac{68}{15}-\frac{23}{6}\left(\frac{15+1}{5}-\frac{2\times 5+4}{5}\right)}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Add 15 and 1 to get 16.

$$\frac{\frac{68}{15}-\frac{23}{6}\left(\frac{16}{5}-\frac{2\times 5+4}{5}\right)}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Multiply 2 and 5 to get 10.

$$\frac{\frac{68}{15}-\frac{23}{6}\left(\frac{16}{5}-\frac{10+4}{5}\right)}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Add 10 and 4 to get 14.

$$\frac{\frac{68}{15}-\frac{23}{6}\left(\frac{16}{5}-\frac{14}{5}\right)}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Since \frac{16}{5} and \frac{14}{5} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{68}{15}-\frac{23}{6}\times \frac{16-14}{5}}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Subtract 14 from 16 to get 2.

$$\frac{\frac{68}{15}-\frac{23}{6}\times \frac{2}{5}}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Multiply \frac{23}{6} times \frac{2}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{68}{15}-\frac{23\times 2}{6\times 5}}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Do the multiplications in the fraction \frac{23\times 2}{6\times 5}.

$$\frac{\frac{68}{15}-\frac{46}{30}}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Reduce the fraction \frac{46}{30} to lowest terms by extracting and canceling out 2.

$$\frac{\frac{68}{15}-\frac{23}{15}}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Since \frac{68}{15} and \frac{23}{15} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{68-23}{15}}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Subtract 23 from 68 to get 45.

$$\frac{\frac{45}{15}}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Divide 45 by 15 to get 3.

$$\frac{3}{\frac{\frac{5\times 3+2}{3}}{\frac{1\times 2+1}{2}}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Divide \frac{5\times 3+2}{3} by \frac{1\times 2+1}{2} by multiplying \frac{5\times 3+2}{3} by the reciprocal of \frac{1\times 2+1}{2}.

$$\frac{3}{\frac{\left(5\times 3+2\right)\times 2}{3\left(1\times 2+1\right)}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Multiply 5 and 3 to get 15.

$$\frac{3}{\frac{\left(15+2\right)\times 2}{3\left(1\times 2+1\right)}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Add 15 and 2 to get 17.

$$\frac{3}{\frac{17\times 2}{3\left(1\times 2+1\right)}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Multiply 17 and 2 to get 34.

$$\frac{3}{\frac{34}{3\left(1\times 2+1\right)}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Multiply 1 and 2 to get 2.

$$\frac{3}{\frac{34}{3\left(2+1\right)}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Add 2 and 1 to get 3.

$$\frac{3}{\frac{34}{3\times 3}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Multiply 3 and 3 to get 9.

$$\frac{3}{\frac{34}{9}-\frac{2\times 4+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Multiply 2 and 4 to get 8.

$$\frac{3}{\frac{34}{9}-\frac{8+1}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Add 8 and 1 to get 9.

$$\frac{3}{\frac{34}{9}-\frac{9}{4}\times \frac{1\times 6+1}{6}-\frac{11}{72}}$$

Multiply 1 and 6 to get 6.

$$\frac{3}{\frac{34}{9}-\frac{9}{4}\times \frac{6+1}{6}-\frac{11}{72}}$$

Add 6 and 1 to get 7.

$$\frac{3}{\frac{34}{9}-\frac{9}{4}\times \frac{7}{6}-\frac{11}{72}}$$

Multiply \frac{9}{4} times \frac{7}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3}{\frac{34}{9}-\frac{9\times 7}{4\times 6}-\frac{11}{72}}$$

Do the multiplications in the fraction \frac{9\times 7}{4\times 6}.

$$\frac{3}{\frac{34}{9}-\frac{63}{24}-\frac{11}{72}}$$

Reduce the fraction \frac{63}{24} to lowest terms by extracting and canceling out 3.

$$\frac{3}{\frac{34}{9}-\frac{21}{8}-\frac{11}{72}}$$

Least common multiple of 9 and 8 is 72. Convert \frac{34}{9} and \frac{21}{8} to fractions with denominator 72.

$$\frac{3}{\frac{272}{72}-\frac{189}{72}-\frac{11}{72}}$$

Since \frac{272}{72} and \frac{189}{72} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3}{\frac{272-189}{72}-\frac{11}{72}}$$

Subtract 189 from 272 to get 83.

$$\frac{3}{\frac{83}{72}-\frac{11}{72}}$$

Since \frac{83}{72} and \frac{11}{72} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3}{\frac{83-11}{72}}$$

Subtract 11 from 83 to get 72.

$$\frac{3}{\frac{72}{72}}$$

Divide 72 by 72 to get 1.

$$\frac{3}{1}$$

Anything divided by one gives itself.

$$3$$

Factor

$$3$$