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}+2}

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

Evaluate

$$\frac{130373}{27720}\approx 4.703210678$$

Solution Steps

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

$$\frac{2}{2}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

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

$$\frac{2+1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

Add 2 and 1 to get 3.

$$\frac{3}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

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

$$\frac{9}{6}+\frac{2}{6}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

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

$$\frac{9+2}{6}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

Add 9 and 2 to get 11.

$$\frac{11}{6}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

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

$$\frac{22}{12}+\frac{3}{12}-\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

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

$$\frac{22+3}{12}-\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

Add 22 and 3 to get 25.

$$\frac{25}{12}-\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

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

$$\frac{125}{60}-\frac{12}{60}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

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

$$\frac{125-12}{60}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

Subtract 12 from 125 to get 113.

$$\frac{113}{60}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

Least common multiple of 60 and 6 is 60. Convert \frac{113}{60} and \frac{1}{6} to fractions with denominator 60.

$$\frac{113}{60}+\frac{10}{60}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

Since \frac{113}{60} and \frac{10}{60} have the same denominator, add them by adding their numerators.

$$\frac{113+10}{60}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

Add 113 and 10 to get 123.

$$\frac{123}{60}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

Reduce the fraction \frac{123}{60} to lowest terms by extracting and canceling out 3.

$$\frac{41}{20}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

Least common multiple of 20 and 7 is 140. Convert \frac{41}{20} and \frac{1}{7} to fractions with denominator 140.

$$\frac{287}{140}+\frac{20}{140}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

Since \frac{287}{140} and \frac{20}{140} have the same denominator, add them by adding their numerators.

$$\frac{287+20}{140}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

Add 287 and 20 to get 307.

$$\frac{307}{140}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

Least common multiple of 140 and 8 is 280. Convert \frac{307}{140} and \frac{1}{8} to fractions with denominator 280.

$$\frac{614}{280}+\frac{35}{280}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

Since \frac{614}{280} and \frac{35}{280} have the same denominator, add them by adding their numerators.

$$\frac{614+35}{280}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

Add 614 and 35 to get 649.

$$\frac{649}{280}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

Least common multiple of 280 and 9 is 2520. Convert \frac{649}{280} and \frac{1}{9} to fractions with denominator 2520.

$$\frac{5841}{2520}+\frac{280}{2520}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

Since \frac{5841}{2520} and \frac{280}{2520} have the same denominator, add them by adding their numerators.

$$\frac{5841+280}{2520}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

Add 5841 and 280 to get 6121.

$$\frac{6121}{2520}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+2$$

Least common multiple of 2520 and 10 is 2520. Convert \frac{6121}{2520} and \frac{1}{10} to fractions with denominator 2520.

$$\frac{6121}{2520}+\frac{252}{2520}+\frac{1}{11}+\frac{1}{12}+2$$

Since \frac{6121}{2520} and \frac{252}{2520} have the same denominator, add them by adding their numerators.

$$\frac{6121+252}{2520}+\frac{1}{11}+\frac{1}{12}+2$$

Add 6121 and 252 to get 6373.

$$\frac{6373}{2520}+\frac{1}{11}+\frac{1}{12}+2$$

Least common multiple of 2520 and 11 is 27720. Convert \frac{6373}{2520} and \frac{1}{11} to fractions with denominator 27720.

$$\frac{70103}{27720}+\frac{2520}{27720}+\frac{1}{12}+2$$

Since \frac{70103}{27720} and \frac{2520}{27720} have the same denominator, add them by adding their numerators.

$$\frac{70103+2520}{27720}+\frac{1}{12}+2$$

Add 70103 and 2520 to get 72623.

$$\frac{72623}{27720}+\frac{1}{12}+2$$

Least common multiple of 27720 and 12 is 27720. Convert \frac{72623}{27720} and \frac{1}{12} to fractions with denominator 27720.

$$\frac{72623}{27720}+\frac{2310}{27720}+2$$

Since \frac{72623}{27720} and \frac{2310}{27720} have the same denominator, add them by adding their numerators.

$$\frac{72623+2310}{27720}+2$$

Add 72623 and 2310 to get 74933.

$$\frac{74933}{27720}+2$$

Convert 2 to fraction \frac{55440}{27720}.

$$\frac{74933}{27720}+\frac{55440}{27720}$$

Since \frac{74933}{27720} and \frac{55440}{27720} have the same denominator, add them by adding their numerators.

$$\frac{74933+55440}{27720}$$

Add 74933 and 55440 to get 130373.

$$\frac{130373}{27720}$$

Factor

$$\frac{17 \cdot 7669}{2 ^ {3} \cdot 3 ^ {2} \cdot 5 \cdot 7 \cdot 11} = 4\frac{19493}{27720} = 4.703210678210678$$