Solve 1/2+3/4+5/6+7/8+frac{9}{10}+frac{11}{12}....

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

Evaluate

$$\frac{191}{40}=4.775$$

Solution Steps

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

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

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

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

Add 2 and 3 to get 5.

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

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

$$\frac{15}{12}+\frac{10}{12}+\frac{7}{8}+\frac{9}{10}+\frac{11}{12}$$

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

$$\frac{15+10}{12}+\frac{7}{8}+\frac{9}{10}+\frac{11}{12}$$

Add 15 and 10 to get 25.

$$\frac{25}{12}+\frac{7}{8}+\frac{9}{10}+\frac{11}{12}$$

Least common multiple of 12 and 8 is 24. Convert \frac{25}{12} and \frac{7}{8} to fractions with denominator 24.

$$\frac{50}{24}+\frac{21}{24}+\frac{9}{10}+\frac{11}{12}$$

Since \frac{50}{24} and \frac{21}{24} have the same denominator, add them by adding their numerators.

$$\frac{50+21}{24}+\frac{9}{10}+\frac{11}{12}$$

Add 50 and 21 to get 71.

$$\frac{71}{24}+\frac{9}{10}+\frac{11}{12}$$

Least common multiple of 24 and 10 is 120. Convert \frac{71}{24} and \frac{9}{10} to fractions with denominator 120.

$$\frac{355}{120}+\frac{108}{120}+\frac{11}{12}$$

Since \frac{355}{120} and \frac{108}{120} have the same denominator, add them by adding their numerators.

$$\frac{355+108}{120}+\frac{11}{12}$$

Add 355 and 108 to get 463.

$$\frac{463}{120}+\frac{11}{12}$$

Least common multiple of 120 and 12 is 120. Convert \frac{463}{120} and \frac{11}{12} to fractions with denominator 120.

$$\frac{463}{120}+\frac{110}{120}$$

Since \frac{463}{120} and \frac{110}{120} have the same denominator, add them by adding their numerators.

$$\frac{463+110}{120}$$

Add 463 and 110 to get 573.

$$\frac{573}{120}$$

Reduce the fraction \frac{573}{120} to lowest terms by extracting and canceling out 3.

$$\frac{191}{40}$$

Factor

$$\frac{191}{2 ^ {3} \cdot 5} = 4\frac{31}{40} = 4.775$$