Solve {l}{1+1/2+1/3+1/4+1/5+frac{1}{6}}{+frac{1}{7}+frac{1}{8}+frac{1}{9}+frac{1}{10}+frac{1}{11}}{+frac{1}{12}+frac{1}{13}+frac{1}{14}+frac{11}{15}+frac{1}{16}}

$$\left. \begin{array} { l } { 1 + \frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 4 } + \frac { 1 } { 5 } + \frac { 1 } { 6 } } \\ { + \frac { 1 } { 7 } + \frac { 1 } { 8 } + \frac { 1 } { 9 } + \frac { 1 } { 10 } + \frac { 1 } { 11 } } \\ { + \frac { 1 } { 12 } + \frac { 1 } { 13 } + \frac { 1 } { 14 } + \frac { 11 } { 15 } + \frac { 1 } { 16 } } \end{array} \right.$$

Evaluate

$$\frac{2917039}{720720}\approx 4.04739566$$

Solution Steps

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

$$\frac{2}{2}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

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

$$\frac{2+1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Add 2 and 1 to get 3.

$$\frac{3}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

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

$$\frac{9}{6}+\frac{2}{6}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

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

$$\frac{9+2}{6}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Add 9 and 2 to get 11.

$$\frac{11}{6}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

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

$$\frac{22}{12}+\frac{3}{12}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

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

$$\frac{22+3}{12}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Add 22 and 3 to get 25.

$$\frac{25}{12}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Least common multiple of 12 and 5 is 60. Convert \frac{25}{12} and \frac{1}{5} to fractions with denominator 60.

$$\frac{125}{60}+\frac{12}{60}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Since \frac{125}{60} and \frac{12}{60} have the same denominator, add them by adding their numerators.

$$\frac{125+12}{60}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Add 125 and 12 to get 137.

$$\frac{137}{60}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Least common multiple of 60 and 6 is 60. Convert \frac{137}{60} and \frac{1}{6} to fractions with denominator 60.

$$\frac{137}{60}+\frac{10}{60}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Since \frac{137}{60} and \frac{10}{60} have the same denominator, add them by adding their numerators.

$$\frac{137+10}{60}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Add 137 and 10 to get 147.

$$\frac{147}{60}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Reduce the fraction \frac{147}{60} to lowest terms by extracting and canceling out 3.

$$\frac{49}{20}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Least common multiple of 20 and 7 is 140. Convert \frac{49}{20} and \frac{1}{7} to fractions with denominator 140.

$$\frac{343}{140}+\frac{20}{140}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Since \frac{343}{140} and \frac{20}{140} have the same denominator, add them by adding their numerators.

$$\frac{343+20}{140}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Add 343 and 20 to get 363.

$$\frac{363}{140}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Least common multiple of 140 and 8 is 280. Convert \frac{363}{140} and \frac{1}{8} to fractions with denominator 280.

$$\frac{726}{280}+\frac{35}{280}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Since \frac{726}{280} and \frac{35}{280} have the same denominator, add them by adding their numerators.

$$\frac{726+35}{280}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Add 726 and 35 to get 761.

$$\frac{761}{280}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Least common multiple of 280 and 9 is 2520. Convert \frac{761}{280} and \frac{1}{9} to fractions with denominator 2520.

$$\frac{6849}{2520}+\frac{280}{2520}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Since \frac{6849}{2520} and \frac{280}{2520} have the same denominator, add them by adding their numerators.

$$\frac{6849+280}{2520}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Add 6849 and 280 to get 7129.

$$\frac{7129}{2520}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Least common multiple of 2520 and 10 is 2520. Convert \frac{7129}{2520} and \frac{1}{10} to fractions with denominator 2520.

$$\frac{7129}{2520}+\frac{252}{2520}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Since \frac{7129}{2520} and \frac{252}{2520} have the same denominator, add them by adding their numerators.

$$\frac{7129+252}{2520}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Add 7129 and 252 to get 7381.

$$\frac{7381}{2520}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Least common multiple of 2520 and 11 is 27720. Convert \frac{7381}{2520} and \frac{1}{11} to fractions with denominator 27720.

$$\frac{81191}{27720}+\frac{2520}{27720}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Since \frac{81191}{27720} and \frac{2520}{27720} have the same denominator, add them by adding their numerators.

$$\frac{81191+2520}{27720}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Add 81191 and 2520 to get 83711.

$$\frac{83711}{27720}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Least common multiple of 27720 and 12 is 27720. Convert \frac{83711}{27720} and \frac{1}{12} to fractions with denominator 27720.

$$\frac{83711}{27720}+\frac{2310}{27720}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Since \frac{83711}{27720} and \frac{2310}{27720} have the same denominator, add them by adding their numerators.

$$\frac{83711+2310}{27720}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Add 83711 and 2310 to get 86021.

$$\frac{86021}{27720}+\frac{1}{13}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Least common multiple of 27720 and 13 is 360360. Convert \frac{86021}{27720} and \frac{1}{13} to fractions with denominator 360360.

$$\frac{1118273}{360360}+\frac{27720}{360360}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Since \frac{1118273}{360360} and \frac{27720}{360360} have the same denominator, add them by adding their numerators.

$$\frac{1118273+27720}{360360}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Add 1118273 and 27720 to get 1145993.

$$\frac{1145993}{360360}+\frac{1}{14}+\frac{11}{15}+\frac{1}{16}$$

Least common multiple of 360360 and 14 is 360360. Convert \frac{1145993}{360360} and \frac{1}{14} to fractions with denominator 360360.

$$\frac{1145993}{360360}+\frac{25740}{360360}+\frac{11}{15}+\frac{1}{16}$$

Since \frac{1145993}{360360} and \frac{25740}{360360} have the same denominator, add them by adding their numerators.

$$\frac{1145993+25740}{360360}+\frac{11}{15}+\frac{1}{16}$$

Add 1145993 and 25740 to get 1171733.

$$\frac{1171733}{360360}+\frac{11}{15}+\frac{1}{16}$$

Least common multiple of 360360 and 15 is 360360. Convert \frac{1171733}{360360} and \frac{11}{15} to fractions with denominator 360360.

$$\frac{1171733}{360360}+\frac{264264}{360360}+\frac{1}{16}$$

Since \frac{1171733}{360360} and \frac{264264}{360360} have the same denominator, add them by adding their numerators.

$$\frac{1171733+264264}{360360}+\frac{1}{16}$$

Add 1171733 and 264264 to get 1435997.

$$\frac{1435997}{360360}+\frac{1}{16}$$

Least common multiple of 360360 and 16 is 720720. Convert \frac{1435997}{360360} and \frac{1}{16} to fractions with denominator 720720.

$$\frac{2871994}{720720}+\frac{45045}{720720}$$

Since \frac{2871994}{720720} and \frac{45045}{720720} have the same denominator, add them by adding their numerators.

$$\frac{2871994+45045}{720720}$$

Add 2871994 and 45045 to get 2917039.

$$\frac{2917039}{720720}$$

Factor

$$\frac{1693 \cdot 1723}{2 ^ {4} \cdot 3 ^ {2} \cdot 5 \cdot 7 \cdot 11 \cdot 13} = 4\frac{34159}{720720} = 4.04739565989566$$