Solve |1/12-1/11|+|1/13-1/12|+|frac{1}{14}-frac{1}{13}

$$| \frac { 1 } { 12 } - \frac { 1 } { 11 } | + | \frac { 1 } { 13 } - \frac { 1 } { 12 } | + | \frac { 1 } { 14 } - \frac { 1 } { 13 }$$

Evaluate

$$\frac{3}{154}\approx 0.019480519$$

Solution Steps

Least common multiple of 12 and 11 is 132. Convert \frac{1}{12} and \frac{1}{11} to fractions with denominator 132.

$$|\frac{11}{132}-\frac{12}{132}|+|\frac{1}{13}-\frac{1}{12}|+|\frac{1}{14}-\frac{1}{13}|$$

Since \frac{11}{132} and \frac{12}{132} have the same denominator, subtract them by subtracting their numerators.

$$|\frac{11-12}{132}|+|\frac{1}{13}-\frac{1}{12}|+|\frac{1}{14}-\frac{1}{13}|$$

Subtract 12 from 11 to get -1.

$$|-\frac{1}{132}|+|\frac{1}{13}-\frac{1}{12}|+|\frac{1}{14}-\frac{1}{13}|$$

The absolute value of a real number a is a when a\geq 0, or -a when a<0. The absolute value of -\frac{1}{132} is \frac{1}{132}.

$$\frac{1}{132}+|\frac{1}{13}-\frac{1}{12}|+|\frac{1}{14}-\frac{1}{13}|$$

Least common multiple of 13 and 12 is 156. Convert \frac{1}{13} and \frac{1}{12} to fractions with denominator 156.

$$\frac{1}{132}+|\frac{12}{156}-\frac{13}{156}|+|\frac{1}{14}-\frac{1}{13}|$$

Since \frac{12}{156} and \frac{13}{156} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{132}+|\frac{12-13}{156}|+|\frac{1}{14}-\frac{1}{13}|$$

Subtract 13 from 12 to get -1.

$$\frac{1}{132}+|-\frac{1}{156}|+|\frac{1}{14}-\frac{1}{13}|$$

The absolute value of a real number a is a when a\geq 0, or -a when a<0. The absolute value of -\frac{1}{156} is \frac{1}{156}.

$$\frac{1}{132}+\frac{1}{156}+|\frac{1}{14}-\frac{1}{13}|$$

Least common multiple of 132 and 156 is 1716. Convert \frac{1}{132} and \frac{1}{156} to fractions with denominator 1716.

$$\frac{13}{1716}+\frac{11}{1716}+|\frac{1}{14}-\frac{1}{13}|$$

Since \frac{13}{1716} and \frac{11}{1716} have the same denominator, add them by adding their numerators.

$$\frac{13+11}{1716}+|\frac{1}{14}-\frac{1}{13}|$$

Add 13 and 11 to get 24.

$$\frac{24}{1716}+|\frac{1}{14}-\frac{1}{13}|$$

Reduce the fraction \frac{24}{1716} to lowest terms by extracting and canceling out 12.

$$\frac{2}{143}+|\frac{1}{14}-\frac{1}{13}|$$

Least common multiple of 14 and 13 is 182. Convert \frac{1}{14} and \frac{1}{13} to fractions with denominator 182.

$$\frac{2}{143}+|\frac{13}{182}-\frac{14}{182}|$$

Since \frac{13}{182} and \frac{14}{182} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2}{143}+|\frac{13-14}{182}|$$

Subtract 14 from 13 to get -1.

$$\frac{2}{143}+|-\frac{1}{182}|$$

The absolute value of a real number a is a when a\geq 0, or -a when a<0. The absolute value of -\frac{1}{182} is \frac{1}{182}.

$$\frac{2}{143}+\frac{1}{182}$$

Least common multiple of 143 and 182 is 2002. Convert \frac{2}{143} and \frac{1}{182} to fractions with denominator 2002.

$$\frac{28}{2002}+\frac{11}{2002}$$

Since \frac{28}{2002} and \frac{11}{2002} have the same denominator, add them by adding their numerators.

$$\frac{28+11}{2002}$$

Add 28 and 11 to get 39.

$$\frac{39}{2002}$$

Reduce the fraction \frac{39}{2002} to lowest terms by extracting and canceling out 13.

$$\frac{3}{154}$$

Factor

$$\frac{3}{2 \cdot 7 \cdot 11} = 0.01948051948051948$$