Solve (4+1/5+1/6+1/7)*(1/5+frac{1}{6}+frac{1}{7}+frac{1}{8})-(4+frac{1}{5}+frac{1}{6}+frac{1}{7}+frac{1}{8})*(frac{1}{5}+frac{1}{6}+frac{1}{7})

$$( 4 + \frac { 1 } { 5 } + \frac { 1 } { 6 } + \frac { 1 } { 7 } ) \times ( \frac { 1 } { 5 } + \frac { 1 } { 6 } + \frac { 1 } { 7 } + \frac { 1 } { 8 } ) - ( 4 + \frac { 1 } { 5 } + \frac { 1 } { 6 } + \frac { 1 } { 7 } + \frac { 1 } { 8 } ) \times ( \frac { 1 } { 5 } + \frac { 1 } { 6 } + \frac { 1 } { 7 } )$$

Evaluate

$$\frac{1}{2}=0.5$$

Solution Steps

Convert 4 to fraction \frac{20}{5}.

$$\left(\frac{20}{5}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)-\left(4+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Since \frac{20}{5} and \frac{1}{5} have the same denominator, add them by adding their numerators.

$$\left(\frac{20+1}{5}+\frac{1}{6}+\frac{1}{7}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)-\left(4+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Add 20 and 1 to get 21.

$$\left(\frac{21}{5}+\frac{1}{6}+\frac{1}{7}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)-\left(4+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Least common multiple of 5 and 6 is 30. Convert \frac{21}{5} and \frac{1}{6} to fractions with denominator 30.

$$\left(\frac{126}{30}+\frac{5}{30}+\frac{1}{7}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)-\left(4+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Since \frac{126}{30} and \frac{5}{30} have the same denominator, add them by adding their numerators.

$$\left(\frac{126+5}{30}+\frac{1}{7}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)-\left(4+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Add 126 and 5 to get 131.

$$\left(\frac{131}{30}+\frac{1}{7}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)-\left(4+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Least common multiple of 30 and 7 is 210. Convert \frac{131}{30} and \frac{1}{7} to fractions with denominator 210.

$$\left(\frac{917}{210}+\frac{30}{210}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)-\left(4+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Since \frac{917}{210} and \frac{30}{210} have the same denominator, add them by adding their numerators.

$$\frac{917+30}{210}\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)-\left(4+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Add 917 and 30 to get 947.

$$\frac{947}{210}\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)-\left(4+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Least common multiple of 5 and 6 is 30. Convert \frac{1}{5} and \frac{1}{6} to fractions with denominator 30.

$$\frac{947}{210}\left(\frac{6}{30}+\frac{5}{30}+\frac{1}{7}+\frac{1}{8}\right)-\left(4+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Since \frac{6}{30} and \frac{5}{30} have the same denominator, add them by adding their numerators.

$$\frac{947}{210}\left(\frac{6+5}{30}+\frac{1}{7}+\frac{1}{8}\right)-\left(4+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Add 6 and 5 to get 11.

$$\frac{947}{210}\left(\frac{11}{30}+\frac{1}{7}+\frac{1}{8}\right)-\left(4+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Least common multiple of 30 and 7 is 210. Convert \frac{11}{30} and \frac{1}{7} to fractions with denominator 210.

$$\frac{947}{210}\left(\frac{77}{210}+\frac{30}{210}+\frac{1}{8}\right)-\left(4+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Since \frac{77}{210} and \frac{30}{210} have the same denominator, add them by adding their numerators.

$$\frac{947}{210}\left(\frac{77+30}{210}+\frac{1}{8}\right)-\left(4+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Add 77 and 30 to get 107.

$$\frac{947}{210}\left(\frac{107}{210}+\frac{1}{8}\right)-\left(4+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Least common multiple of 210 and 8 is 840. Convert \frac{107}{210} and \frac{1}{8} to fractions with denominator 840.

$$\frac{947}{210}\left(\frac{428}{840}+\frac{105}{840}\right)-\left(4+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Since \frac{428}{840} and \frac{105}{840} have the same denominator, add them by adding their numerators.

$$\frac{947}{210}\times \frac{428+105}{840}-\left(4+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Add 428 and 105 to get 533.

$$\frac{947}{210}\times \frac{533}{840}-\left(4+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Multiply \frac{947}{210} times \frac{533}{840} by multiplying numerator times numerator and denominator times denominator.

$$\frac{947\times 533}{210\times 840}-\left(4+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Do the multiplications in the fraction \frac{947\times 533}{210\times 840}.

$$\frac{504751}{176400}-\left(4+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Convert 4 to fraction \frac{20}{5}.

$$\frac{504751}{176400}-\left(\frac{20}{5}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Since \frac{20}{5} and \frac{1}{5} have the same denominator, add them by adding their numerators.

$$\frac{504751}{176400}-\left(\frac{20+1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Add 20 and 1 to get 21.

$$\frac{504751}{176400}-\left(\frac{21}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Least common multiple of 5 and 6 is 30. Convert \frac{21}{5} and \frac{1}{6} to fractions with denominator 30.

$$\frac{504751}{176400}-\left(\frac{126}{30}+\frac{5}{30}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Since \frac{126}{30} and \frac{5}{30} have the same denominator, add them by adding their numerators.

$$\frac{504751}{176400}-\left(\frac{126+5}{30}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Add 126 and 5 to get 131.

$$\frac{504751}{176400}-\left(\frac{131}{30}+\frac{1}{7}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Least common multiple of 30 and 7 is 210. Convert \frac{131}{30} and \frac{1}{7} to fractions with denominator 210.

$$\frac{504751}{176400}-\left(\frac{917}{210}+\frac{30}{210}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Since \frac{917}{210} and \frac{30}{210} have the same denominator, add them by adding their numerators.

$$\frac{504751}{176400}-\left(\frac{917+30}{210}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Add 917 and 30 to get 947.

$$\frac{504751}{176400}-\left(\frac{947}{210}+\frac{1}{8}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Least common multiple of 210 and 8 is 840. Convert \frac{947}{210} and \frac{1}{8} to fractions with denominator 840.

$$\frac{504751}{176400}-\left(\frac{3788}{840}+\frac{105}{840}\right)\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Since \frac{3788}{840} and \frac{105}{840} have the same denominator, add them by adding their numerators.

$$\frac{504751}{176400}-\frac{3788+105}{840}\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Add 3788 and 105 to get 3893.

$$\frac{504751}{176400}-\frac{3893}{840}\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)$$

Least common multiple of 5 and 6 is 30. Convert \frac{1}{5} and \frac{1}{6} to fractions with denominator 30.

$$\frac{504751}{176400}-\frac{3893}{840}\left(\frac{6}{30}+\frac{5}{30}+\frac{1}{7}\right)$$

Since \frac{6}{30} and \frac{5}{30} have the same denominator, add them by adding their numerators.

$$\frac{504751}{176400}-\frac{3893}{840}\left(\frac{6+5}{30}+\frac{1}{7}\right)$$

Add 6 and 5 to get 11.

$$\frac{504751}{176400}-\frac{3893}{840}\left(\frac{11}{30}+\frac{1}{7}\right)$$

Least common multiple of 30 and 7 is 210. Convert \frac{11}{30} and \frac{1}{7} to fractions with denominator 210.

$$\frac{504751}{176400}-\frac{3893}{840}\left(\frac{77}{210}+\frac{30}{210}\right)$$

Since \frac{77}{210} and \frac{30}{210} have the same denominator, add them by adding their numerators.

$$\frac{504751}{176400}-\frac{3893}{840}\times \frac{77+30}{210}$$

Add 77 and 30 to get 107.

$$\frac{504751}{176400}-\frac{3893}{840}\times \frac{107}{210}$$

Multiply \frac{3893}{840} times \frac{107}{210} by multiplying numerator times numerator and denominator times denominator.

$$\frac{504751}{176400}-\frac{3893\times 107}{840\times 210}$$

Do the multiplications in the fraction \frac{3893\times 107}{840\times 210}.

$$\frac{504751}{176400}-\frac{416551}{176400}$$

Since \frac{504751}{176400} and \frac{416551}{176400} have the same denominator, subtract them by subtracting their numerators.

$$\frac{504751-416551}{176400}$$

Subtract 416551 from 504751 to get 88200.

$$\frac{88200}{176400}$$

Reduce the fraction \frac{88200}{176400} to lowest terms by extracting and canceling out 88200.

$$\frac{1}{2}$$

Factor

$$\frac{1}{2} = 0.5$$