Solve 1/5-1/6|+|1/7-1/6|-|frac{1}{5}-frac{1}{7}|

$$\frac { 1 } { 5 } - \frac { 1 } { 6 } | + | \frac { 1 } { 7 } - \frac { 1 } { 6 } | - | \frac { 1 } { 5 } - \frac { 1 } { 7 } |$$

Evaluate

$$\frac{7}{36}\approx 0.194444444$$

Solution Steps

Least common multiple of 7 and 6 is 42. Convert \frac{1}{7} and \frac{1}{6} to fractions with denominator 42.

$$\frac{1}{5}-\frac{1}{6}||\frac{6}{42}-\frac{7}{42}|-|\frac{1}{5}-\frac{1}{7}||$$

Since \frac{6}{42} and \frac{7}{42} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{5}-\frac{1}{6}||\frac{6-7}{42}|-|\frac{1}{5}-\frac{1}{7}||$$

Subtract 7 from 6 to get -1.

$$\frac{1}{5}-\frac{1}{6}||-\frac{1}{42}|-|\frac{1}{5}-\frac{1}{7}||$$

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}{42} is \frac{1}{42}.

$$\frac{1}{5}-\frac{1}{6}|\frac{1}{42}-|\frac{1}{5}-\frac{1}{7}||$$

Least common multiple of 5 and 7 is 35. Convert \frac{1}{5} and \frac{1}{7} to fractions with denominator 35.

$$\frac{1}{5}-\frac{1}{6}|\frac{1}{42}-|\frac{7}{35}-\frac{5}{35}||$$

Since \frac{7}{35} and \frac{5}{35} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{5}-\frac{1}{6}|\frac{1}{42}-|\frac{7-5}{35}||$$

Subtract 5 from 7 to get 2.

$$\frac{1}{5}-\frac{1}{6}|\frac{1}{42}-|\frac{2}{35}||$$

The absolute value of a real number a is a when a\geq 0, or -a when a<0. The absolute value of \frac{2}{35} is \frac{2}{35}.

$$\frac{1}{5}-\frac{1}{6}|\frac{1}{42}-\frac{2}{35}|$$

Least common multiple of 42 and 35 is 210. Convert \frac{1}{42} and \frac{2}{35} to fractions with denominator 210.

$$\frac{1}{5}-\frac{1}{6}|\frac{5}{210}-\frac{12}{210}|$$

Since \frac{5}{210} and \frac{12}{210} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{5}-\frac{1}{6}|\frac{5-12}{210}|$$

Subtract 12 from 5 to get -7.

$$\frac{1}{5}-\frac{1}{6}|\frac{-7}{210}|$$

Reduce the fraction \frac{-7}{210} to lowest terms by extracting and canceling out 7.

$$\frac{1}{5}-\frac{1}{6}|-\frac{1}{30}|$$

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}{30} is \frac{1}{30}.

$$\frac{1}{5}-\frac{1}{6}\times \frac{1}{30}$$

Multiply \frac{1}{6} times \frac{1}{30} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{5}-\frac{1\times 1}{6\times 30}$$

Do the multiplications in the fraction \frac{1\times 1}{6\times 30}.

$$\frac{1}{5}-\frac{1}{180}$$

Least common multiple of 5 and 180 is 180. Convert \frac{1}{5} and \frac{1}{180} to fractions with denominator 180.

$$\frac{36}{180}-\frac{1}{180}$$

Since \frac{36}{180} and \frac{1}{180} have the same denominator, subtract them by subtracting their numerators.

$$\frac{36-1}{180}$$

Subtract 1 from 36 to get 35.

$$\frac{35}{180}$$

Reduce the fraction \frac{35}{180} to lowest terms by extracting and canceling out 5.

$$\frac{7}{36}$$

Factor

$$\frac{7}{2 ^ {2} \cdot 3 ^ {2}} = 0.19444444444444445$$