Solve left|1/2011-1/2010right|+left|1/2012-1/2011right|-left|frac{1}{2012}-frac{1}{2010}right|

$$\left| \frac{ 1 }{ 2011 } - \frac{ 1 }{ 2010 } \right| + \left| \frac{ 1 }{ 2012 } - \frac{ 1 }{ 2011 } \right| - \left| \frac{ 1 }{ 2012 } - \frac{ 1 }{ 2010 } \right|$$

Evaluate

$$0$$

Solution Steps

Least common multiple of 2011 and 2010 is 4042110. Convert \frac{1}{2011} and \frac{1}{2010} to fractions with denominator 4042110.

$$|\frac{2010}{4042110}-\frac{2011}{4042110}|+|\frac{1}{2012}-\frac{1}{2011}|-|\frac{1}{2012}-\frac{1}{2010}|$$

Since \frac{2010}{4042110} and \frac{2011}{4042110} have the same denominator, subtract them by subtracting their numerators.

$$|\frac{2010-2011}{4042110}|+|\frac{1}{2012}-\frac{1}{2011}|-|\frac{1}{2012}-\frac{1}{2010}|$$

Subtract 2011 from 2010 to get -1.

$$|-\frac{1}{4042110}|+|\frac{1}{2012}-\frac{1}{2011}|-|\frac{1}{2012}-\frac{1}{2010}|$$

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}{4042110} is \frac{1}{4042110}.

$$\frac{1}{4042110}+|\frac{1}{2012}-\frac{1}{2011}|-|\frac{1}{2012}-\frac{1}{2010}|$$

Least common multiple of 2012 and 2011 is 4046132. Convert \frac{1}{2012} and \frac{1}{2011} to fractions with denominator 4046132.

$$\frac{1}{4042110}+|\frac{2011}{4046132}-\frac{2012}{4046132}|-|\frac{1}{2012}-\frac{1}{2010}|$$

Since \frac{2011}{4046132} and \frac{2012}{4046132} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{4042110}+|\frac{2011-2012}{4046132}|-|\frac{1}{2012}-\frac{1}{2010}|$$

Subtract 2012 from 2011 to get -1.

$$\frac{1}{4042110}+|-\frac{1}{4046132}|-|\frac{1}{2012}-\frac{1}{2010}|$$

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}{4046132} is \frac{1}{4046132}.

$$\frac{1}{4042110}+\frac{1}{4046132}-|\frac{1}{2012}-\frac{1}{2010}|$$

Least common multiple of 4042110 and 4046132 is 4066362660. Convert \frac{1}{4042110} and \frac{1}{4046132} to fractions with denominator 4066362660.

$$\frac{1006}{4066362660}+\frac{1005}{4066362660}-|\frac{1}{2012}-\frac{1}{2010}|$$

Since \frac{1006}{4066362660} and \frac{1005}{4066362660} have the same denominator, add them by adding their numerators.

$$\frac{1006+1005}{4066362660}-|\frac{1}{2012}-\frac{1}{2010}|$$

Add 1006 and 1005 to get 2011.

$$\frac{2011}{4066362660}-|\frac{1}{2012}-\frac{1}{2010}|$$

Reduce the fraction \frac{2011}{4066362660} to lowest terms by extracting and canceling out 2011.

$$\frac{1}{2022060}-|\frac{1}{2012}-\frac{1}{2010}|$$

Least common multiple of 2012 and 2010 is 2022060. Convert \frac{1}{2012} and \frac{1}{2010} to fractions with denominator 2022060.

$$\frac{1}{2022060}-|\frac{1005}{2022060}-\frac{1006}{2022060}|$$

Since \frac{1005}{2022060} and \frac{1006}{2022060} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{2022060}-|\frac{1005-1006}{2022060}|$$

Subtract 1006 from 1005 to get -1.

$$\frac{1}{2022060}-|-\frac{1}{2022060}|$$

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}{2022060} is \frac{1}{2022060}.

$$\frac{1}{2022060}-\frac{1}{2022060}$$

Subtract \frac{1}{2022060} from \frac{1}{2022060} to get 0.

$$0$$

Factor

$$0$$