Solve 1/2+3/4+5/6+7/6+frac{9}{10}+frac{11}{12}....

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

Evaluate

$$\frac{76}{15}\approx 5.066666667$$

Solution Steps

Least common multiple of 2 and 4 is 4. Convert \frac{1}{2} and \frac{3}{4} to fractions with denominator 4.

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

Since \frac{2}{4} and \frac{3}{4} have the same denominator, add them by adding their numerators.

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

Add 2 and 3 to get 5.

$$\frac{5}{4}+\frac{5}{6}+\frac{7}{6}+\frac{9}{10}+\frac{11}{12}$$

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

$$\frac{15}{12}+\frac{10}{12}+\frac{7}{6}+\frac{9}{10}+\frac{11}{12}$$

Since \frac{15}{12} and \frac{10}{12} have the same denominator, add them by adding their numerators.

$$\frac{15+10}{12}+\frac{7}{6}+\frac{9}{10}+\frac{11}{12}$$

Add 15 and 10 to get 25.

$$\frac{25}{12}+\frac{7}{6}+\frac{9}{10}+\frac{11}{12}$$

Least common multiple of 12 and 6 is 12. Convert \frac{25}{12} and \frac{7}{6} to fractions with denominator 12.

$$\frac{25}{12}+\frac{14}{12}+\frac{9}{10}+\frac{11}{12}$$

Since \frac{25}{12} and \frac{14}{12} have the same denominator, add them by adding their numerators.

$$\frac{25+14}{12}+\frac{9}{10}+\frac{11}{12}$$

Add 25 and 14 to get 39.

$$\frac{39}{12}+\frac{9}{10}+\frac{11}{12}$$

Reduce the fraction \frac{39}{12} to lowest terms by extracting and canceling out 3.

$$\frac{13}{4}+\frac{9}{10}+\frac{11}{12}$$

Least common multiple of 4 and 10 is 20. Convert \frac{13}{4} and \frac{9}{10} to fractions with denominator 20.

$$\frac{65}{20}+\frac{18}{20}+\frac{11}{12}$$

Since \frac{65}{20} and \frac{18}{20} have the same denominator, add them by adding their numerators.

$$\frac{65+18}{20}+\frac{11}{12}$$

Add 65 and 18 to get 83.

$$\frac{83}{20}+\frac{11}{12}$$

Least common multiple of 20 and 12 is 60. Convert \frac{83}{20} and \frac{11}{12} to fractions with denominator 60.

$$\frac{249}{60}+\frac{55}{60}$$

Since \frac{249}{60} and \frac{55}{60} have the same denominator, add them by adding their numerators.

$$\frac{249+55}{60}$$

Add 249 and 55 to get 304.

$$\frac{304}{60}$$

Reduce the fraction \frac{304}{60} to lowest terms by extracting and canceling out 4.

$$\frac{76}{15}$$

Factor

$$\frac{2 ^ {2} \cdot 19}{3 \cdot 5} = 5\frac{1}{15} = 5.066666666666666$$