Solve 1/2+5/6+11/12+19/20+frac{29}{30}+frac{41}{42}+frac{55}{56}+frac{71}{72}+frac{89}{90}+frac{109}{110}=

$$\frac { 1 } { 2 } + \frac { 5 } { 6 } + \frac { 11 } { 12 } + \frac { 19 } { 20 } + \frac { 29 } { 30 } + \frac { 41 } { 42 } + \frac { 55 } { 56 } + \frac { 71 } { 72 } + \frac { 89 } { 90 } + \frac { 109 } { 110 } =$$

Evaluate

$$\frac{100}{11}\approx 9.090909091$$

Solution Steps

Least common multiple of 2 and 6 is 6. Convert \frac{1}{2} and \frac{5}{6} to fractions with denominator 6.

$$\frac{3}{6}+\frac{5}{6}+\frac{11}{12}+\frac{19}{20}+\frac{29}{30}+\frac{41}{42}+\frac{55}{56}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Since \frac{3}{6} and \frac{5}{6} have the same denominator, add them by adding their numerators.

$$\frac{3+5}{6}+\frac{11}{12}+\frac{19}{20}+\frac{29}{30}+\frac{41}{42}+\frac{55}{56}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Add 3 and 5 to get 8.

$$\frac{8}{6}+\frac{11}{12}+\frac{19}{20}+\frac{29}{30}+\frac{41}{42}+\frac{55}{56}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Reduce the fraction \frac{8}{6} to lowest terms by extracting and canceling out 2.

$$\frac{4}{3}+\frac{11}{12}+\frac{19}{20}+\frac{29}{30}+\frac{41}{42}+\frac{55}{56}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Least common multiple of 3 and 12 is 12. Convert \frac{4}{3} and \frac{11}{12} to fractions with denominator 12.

$$\frac{16}{12}+\frac{11}{12}+\frac{19}{20}+\frac{29}{30}+\frac{41}{42}+\frac{55}{56}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Since \frac{16}{12} and \frac{11}{12} have the same denominator, add them by adding their numerators.

$$\frac{16+11}{12}+\frac{19}{20}+\frac{29}{30}+\frac{41}{42}+\frac{55}{56}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Add 16 and 11 to get 27.

$$\frac{27}{12}+\frac{19}{20}+\frac{29}{30}+\frac{41}{42}+\frac{55}{56}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Reduce the fraction \frac{27}{12} to lowest terms by extracting and canceling out 3.

$$\frac{9}{4}+\frac{19}{20}+\frac{29}{30}+\frac{41}{42}+\frac{55}{56}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Least common multiple of 4 and 20 is 20. Convert \frac{9}{4} and \frac{19}{20} to fractions with denominator 20.

$$\frac{45}{20}+\frac{19}{20}+\frac{29}{30}+\frac{41}{42}+\frac{55}{56}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Since \frac{45}{20} and \frac{19}{20} have the same denominator, add them by adding their numerators.

$$\frac{45+19}{20}+\frac{29}{30}+\frac{41}{42}+\frac{55}{56}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Add 45 and 19 to get 64.

$$\frac{64}{20}+\frac{29}{30}+\frac{41}{42}+\frac{55}{56}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Reduce the fraction \frac{64}{20} to lowest terms by extracting and canceling out 4.

$$\frac{16}{5}+\frac{29}{30}+\frac{41}{42}+\frac{55}{56}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Least common multiple of 5 and 30 is 30. Convert \frac{16}{5} and \frac{29}{30} to fractions with denominator 30.

$$\frac{96}{30}+\frac{29}{30}+\frac{41}{42}+\frac{55}{56}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Since \frac{96}{30} and \frac{29}{30} have the same denominator, add them by adding their numerators.

$$\frac{96+29}{30}+\frac{41}{42}+\frac{55}{56}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Add 96 and 29 to get 125.

$$\frac{125}{30}+\frac{41}{42}+\frac{55}{56}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Reduce the fraction \frac{125}{30} to lowest terms by extracting and canceling out 5.

$$\frac{25}{6}+\frac{41}{42}+\frac{55}{56}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Least common multiple of 6 and 42 is 42. Convert \frac{25}{6} and \frac{41}{42} to fractions with denominator 42.

$$\frac{175}{42}+\frac{41}{42}+\frac{55}{56}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Since \frac{175}{42} and \frac{41}{42} have the same denominator, add them by adding their numerators.

$$\frac{175+41}{42}+\frac{55}{56}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Add 175 and 41 to get 216.

$$\frac{216}{42}+\frac{55}{56}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Reduce the fraction \frac{216}{42} to lowest terms by extracting and canceling out 6.

$$\frac{36}{7}+\frac{55}{56}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Least common multiple of 7 and 56 is 56. Convert \frac{36}{7} and \frac{55}{56} to fractions with denominator 56.

$$\frac{288}{56}+\frac{55}{56}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Since \frac{288}{56} and \frac{55}{56} have the same denominator, add them by adding their numerators.

$$\frac{288+55}{56}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Add 288 and 55 to get 343.

$$\frac{343}{56}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Reduce the fraction \frac{343}{56} to lowest terms by extracting and canceling out 7.

$$\frac{49}{8}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Least common multiple of 8 and 72 is 72. Convert \frac{49}{8} and \frac{71}{72} to fractions with denominator 72.

$$\frac{441}{72}+\frac{71}{72}+\frac{89}{90}+\frac{109}{110}$$

Since \frac{441}{72} and \frac{71}{72} have the same denominator, add them by adding their numerators.

$$\frac{441+71}{72}+\frac{89}{90}+\frac{109}{110}$$

Add 441 and 71 to get 512.

$$\frac{512}{72}+\frac{89}{90}+\frac{109}{110}$$

Reduce the fraction \frac{512}{72} to lowest terms by extracting and canceling out 8.

$$\frac{64}{9}+\frac{89}{90}+\frac{109}{110}$$

Least common multiple of 9 and 90 is 90. Convert \frac{64}{9} and \frac{89}{90} to fractions with denominator 90.

$$\frac{640}{90}+\frac{89}{90}+\frac{109}{110}$$

Since \frac{640}{90} and \frac{89}{90} have the same denominator, add them by adding their numerators.

$$\frac{640+89}{90}+\frac{109}{110}$$

Add 640 and 89 to get 729.

$$\frac{729}{90}+\frac{109}{110}$$

Reduce the fraction \frac{729}{90} to lowest terms by extracting and canceling out 9.

$$\frac{81}{10}+\frac{109}{110}$$

Least common multiple of 10 and 110 is 110. Convert \frac{81}{10} and \frac{109}{110} to fractions with denominator 110.

$$\frac{891}{110}+\frac{109}{110}$$

Since \frac{891}{110} and \frac{109}{110} have the same denominator, add them by adding their numerators.

$$\frac{891+109}{110}$$

Add 891 and 109 to get 1000.

$$\frac{1000}{110}$$

Reduce the fraction \frac{1000}{110} to lowest terms by extracting and canceling out 10.

$$\frac{100}{11}$$

Factor

$$\frac{2 ^ {2} \cdot 5 ^ {2}}{11} = 9\frac{1}{11} = 9.090909090909092$$