Solve (1+1/2+1/3+1/4)*(1/2+frac{1}{3}+frac{1}{4}+frac{1}{5})-(1+frac{1}{2}+frac{1}{3}+frac{1}{4}+frac{1}{5}*(frac{1}{2}+frac{1}{3}+frac{1}{4})

$$( 1 + \frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 4 } ) \times ( \frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 4 } + \frac { 1 } { 5 } ) - ( 1 + \frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 4 } + \frac { 1 } { 5 } \times ( \frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 4 } )$$

Evaluate

$$\frac{269}{720}\approx 0.373611111$$

Solution Steps

Convert 1 to fraction \frac{2}{2}.

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

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

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

Add 2 and 1 to get 3.

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

Least common multiple of 2 and 3 is 6. Convert \frac{3}{2} and \frac{1}{3} to fractions with denominator 6.

$$\left(\frac{9}{6}+\frac{2}{6}+\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Since \frac{9}{6} and \frac{2}{6} have the same denominator, add them by adding their numerators.

$$\left(\frac{9+2}{6}+\frac{1}{4}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Add 9 and 2 to get 11.

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

Least common multiple of 6 and 4 is 12. Convert \frac{11}{6} and \frac{1}{4} to fractions with denominator 12.

$$\left(\frac{22}{12}+\frac{3}{12}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Since \frac{22}{12} and \frac{3}{12} have the same denominator, add them by adding their numerators.

$$\frac{22+3}{12}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Add 22 and 3 to get 25.

$$\frac{25}{12}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Least common multiple of 2 and 3 is 6. Convert \frac{1}{2} and \frac{1}{3} to fractions with denominator 6.

$$\frac{25}{12}\left(\frac{3}{6}+\frac{2}{6}+\frac{1}{4}+\frac{1}{5}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Since \frac{3}{6} and \frac{2}{6} have the same denominator, add them by adding their numerators.

$$\frac{25}{12}\left(\frac{3+2}{6}+\frac{1}{4}+\frac{1}{5}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Add 3 and 2 to get 5.

$$\frac{25}{12}\left(\frac{5}{6}+\frac{1}{4}+\frac{1}{5}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Least common multiple of 6 and 4 is 12. Convert \frac{5}{6} and \frac{1}{4} to fractions with denominator 12.

$$\frac{25}{12}\left(\frac{10}{12}+\frac{3}{12}+\frac{1}{5}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Since \frac{10}{12} and \frac{3}{12} have the same denominator, add them by adding their numerators.

$$\frac{25}{12}\left(\frac{10+3}{12}+\frac{1}{5}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Add 10 and 3 to get 13.

$$\frac{25}{12}\left(\frac{13}{12}+\frac{1}{5}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Least common multiple of 12 and 5 is 60. Convert \frac{13}{12} and \frac{1}{5} to fractions with denominator 60.

$$\frac{25}{12}\left(\frac{65}{60}+\frac{12}{60}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Since \frac{65}{60} and \frac{12}{60} have the same denominator, add them by adding their numerators.

$$\frac{25}{12}\times \frac{65+12}{60}-\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Add 65 and 12 to get 77.

$$\frac{25}{12}\times \frac{77}{60}-\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Multiply \frac{25}{12} times \frac{77}{60} by multiplying numerator times numerator and denominator times denominator.

$$\frac{25\times 77}{12\times 60}-\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Do the multiplications in the fraction \frac{25\times 77}{12\times 60}.

$$\frac{1925}{720}-\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Reduce the fraction \frac{1925}{720} to lowest terms by extracting and canceling out 5.

$$\frac{385}{144}-\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Convert 1 to fraction \frac{2}{2}.

$$\frac{385}{144}-\left(\frac{2}{2}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

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

$$\frac{385}{144}-\left(\frac{2+1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Add 2 and 1 to get 3.

$$\frac{385}{144}-\left(\frac{3}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Least common multiple of 2 and 3 is 6. Convert \frac{3}{2} and \frac{1}{3} to fractions with denominator 6.

$$\frac{385}{144}-\left(\frac{9}{6}+\frac{2}{6}+\frac{1}{4}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Since \frac{9}{6} and \frac{2}{6} have the same denominator, add them by adding their numerators.

$$\frac{385}{144}-\left(\frac{9+2}{6}+\frac{1}{4}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Add 9 and 2 to get 11.

$$\frac{385}{144}-\left(\frac{11}{6}+\frac{1}{4}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Least common multiple of 6 and 4 is 12. Convert \frac{11}{6} and \frac{1}{4} to fractions with denominator 12.

$$\frac{385}{144}-\left(\frac{22}{12}+\frac{3}{12}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Since \frac{22}{12} and \frac{3}{12} have the same denominator, add them by adding their numerators.

$$\frac{385}{144}-\left(\frac{22+3}{12}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Add 22 and 3 to get 25.

$$\frac{385}{144}-\left(\frac{25}{12}+\frac{1}{5}\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)\right)$$

Least common multiple of 2 and 3 is 6. Convert \frac{1}{2} and \frac{1}{3} to fractions with denominator 6.

$$\frac{385}{144}-\left(\frac{25}{12}+\frac{1}{5}\left(\frac{3}{6}+\frac{2}{6}+\frac{1}{4}\right)\right)$$

Since \frac{3}{6} and \frac{2}{6} have the same denominator, add them by adding their numerators.

$$\frac{385}{144}-\left(\frac{25}{12}+\frac{1}{5}\left(\frac{3+2}{6}+\frac{1}{4}\right)\right)$$

Add 3 and 2 to get 5.

$$\frac{385}{144}-\left(\frac{25}{12}+\frac{1}{5}\left(\frac{5}{6}+\frac{1}{4}\right)\right)$$

Least common multiple of 6 and 4 is 12. Convert \frac{5}{6} and \frac{1}{4} to fractions with denominator 12.

$$\frac{385}{144}-\left(\frac{25}{12}+\frac{1}{5}\left(\frac{10}{12}+\frac{3}{12}\right)\right)$$

Since \frac{10}{12} and \frac{3}{12} have the same denominator, add them by adding their numerators.

$$\frac{385}{144}-\left(\frac{25}{12}+\frac{1}{5}\times \frac{10+3}{12}\right)$$

Add 10 and 3 to get 13.

$$\frac{385}{144}-\left(\frac{25}{12}+\frac{1}{5}\times \frac{13}{12}\right)$$

Multiply \frac{1}{5} times \frac{13}{12} by multiplying numerator times numerator and denominator times denominator.

$$\frac{385}{144}-\left(\frac{25}{12}+\frac{1\times 13}{5\times 12}\right)$$

Do the multiplications in the fraction \frac{1\times 13}{5\times 12}.

$$\frac{385}{144}-\left(\frac{25}{12}+\frac{13}{60}\right)$$

Least common multiple of 12 and 60 is 60. Convert \frac{25}{12} and \frac{13}{60} to fractions with denominator 60.

$$\frac{385}{144}-\left(\frac{125}{60}+\frac{13}{60}\right)$$

Since \frac{125}{60} and \frac{13}{60} have the same denominator, add them by adding their numerators.

$$\frac{385}{144}-\frac{125+13}{60}$$

Add 125 and 13 to get 138.

$$\frac{385}{144}-\frac{138}{60}$$

Reduce the fraction \frac{138}{60} to lowest terms by extracting and canceling out 6.

$$\frac{385}{144}-\frac{23}{10}$$

Least common multiple of 144 and 10 is 720. Convert \frac{385}{144} and \frac{23}{10} to fractions with denominator 720.

$$\frac{1925}{720}-\frac{1656}{720}$$

Since \frac{1925}{720} and \frac{1656}{720} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1925-1656}{720}$$

Subtract 1656 from 1925 to get 269.

$$\frac{269}{720}$$

Factor

$$\frac{269}{2 ^ {4} \cdot 3 ^ {2} \cdot 5} = 0.3736111111111111$$