Solve 11/5-1/3-{2/5-5/6-[frac{3}{4}-frac{1}{2}-(frac{7}{30}-frac{4}{5}-1)-frac{1}{4}]}-1

$$\frac { 11 } { 5 } - \frac { 1 } { 3 } - \{ \frac { 2 } { 5 } - \frac { 5 } { 6 } - [ \frac { 3 } { 4 } - \frac { 1 } { 2 } - ( \frac { 7 } { 30 } - \frac { 4 } { 5 } - 1 ) - \frac { 1 } { 4 } ] \} - 1$$

Evaluate

$$\frac{43}{15}\approx 2.866666667$$

Solution Steps

Least common multiple of 5 and 3 is 15. Convert \frac{11}{5} and \frac{1}{3} to fractions with denominator 15.

$$\frac{33}{15}-\frac{5}{15}-\left(\frac{2}{5}-\frac{5}{6}-\left(\frac{3}{4}-\frac{1}{2}-\left(\frac{7}{30}-\frac{4}{5}-1\right)-\frac{1}{4}\right)\right)-1$$

Since \frac{33}{15} and \frac{5}{15} have the same denominator, subtract them by subtracting their numerators.

$$\frac{33-5}{15}-\left(\frac{2}{5}-\frac{5}{6}-\left(\frac{3}{4}-\frac{1}{2}-\left(\frac{7}{30}-\frac{4}{5}-1\right)-\frac{1}{4}\right)\right)-1$$

Subtract 5 from 33 to get 28.

$$\frac{28}{15}-\left(\frac{2}{5}-\frac{5}{6}-\left(\frac{3}{4}-\frac{1}{2}-\left(\frac{7}{30}-\frac{4}{5}-1\right)-\frac{1}{4}\right)\right)-1$$

Least common multiple of 5 and 6 is 30. Convert \frac{2}{5} and \frac{5}{6} to fractions with denominator 30.

$$\frac{28}{15}-\left(\frac{12}{30}-\frac{25}{30}-\left(\frac{3}{4}-\frac{1}{2}-\left(\frac{7}{30}-\frac{4}{5}-1\right)-\frac{1}{4}\right)\right)-1$$

Since \frac{12}{30} and \frac{25}{30} have the same denominator, subtract them by subtracting their numerators.

$$\frac{28}{15}-\left(\frac{12-25}{30}-\left(\frac{3}{4}-\frac{1}{2}-\left(\frac{7}{30}-\frac{4}{5}-1\right)-\frac{1}{4}\right)\right)-1$$

Subtract 25 from 12 to get -13.

$$\frac{28}{15}-\left(-\frac{13}{30}-\left(\frac{3}{4}-\frac{1}{2}-\left(\frac{7}{30}-\frac{4}{5}-1\right)-\frac{1}{4}\right)\right)-1$$

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

$$\frac{28}{15}-\left(-\frac{13}{30}-\left(\frac{3}{4}-\frac{2}{4}-\left(\frac{7}{30}-\frac{4}{5}-1\right)-\frac{1}{4}\right)\right)-1$$

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

$$\frac{28}{15}-\left(-\frac{13}{30}-\left(\frac{3-2}{4}-\left(\frac{7}{30}-\frac{4}{5}-1\right)-\frac{1}{4}\right)\right)-1$$

Subtract 2 from 3 to get 1.

$$\frac{28}{15}-\left(-\frac{13}{30}-\left(\frac{1}{4}-\left(\frac{7}{30}-\frac{4}{5}-1\right)-\frac{1}{4}\right)\right)-1$$

Least common multiple of 30 and 5 is 30. Convert \frac{7}{30} and \frac{4}{5} to fractions with denominator 30.

$$\frac{28}{15}-\left(-\frac{13}{30}-\left(\frac{1}{4}-\left(\frac{7}{30}-\frac{24}{30}-1\right)-\frac{1}{4}\right)\right)-1$$

Since \frac{7}{30} and \frac{24}{30} have the same denominator, subtract them by subtracting their numerators.

$$\frac{28}{15}-\left(-\frac{13}{30}-\left(\frac{1}{4}-\left(\frac{7-24}{30}-1\right)-\frac{1}{4}\right)\right)-1$$

Subtract 24 from 7 to get -17.

$$\frac{28}{15}-\left(-\frac{13}{30}-\left(\frac{1}{4}-\left(-\frac{17}{30}-1\right)-\frac{1}{4}\right)\right)-1$$

Convert 1 to fraction \frac{30}{30}.

$$\frac{28}{15}-\left(-\frac{13}{30}-\left(\frac{1}{4}-\left(-\frac{17}{30}-\frac{30}{30}\right)-\frac{1}{4}\right)\right)-1$$

Since -\frac{17}{30} and \frac{30}{30} have the same denominator, subtract them by subtracting their numerators.

$$\frac{28}{15}-\left(-\frac{13}{30}-\left(\frac{1}{4}-\frac{-17-30}{30}-\frac{1}{4}\right)\right)-1$$

Subtract 30 from -17 to get -47.

$$\frac{28}{15}-\left(-\frac{13}{30}-\left(\frac{1}{4}-\left(-\frac{47}{30}\right)-\frac{1}{4}\right)\right)-1$$

The opposite of -\frac{47}{30} is \frac{47}{30}.

$$\frac{28}{15}-\left(-\frac{13}{30}-\left(\frac{1}{4}+\frac{47}{30}-\frac{1}{4}\right)\right)-1$$

Least common multiple of 4 and 30 is 60. Convert \frac{1}{4} and \frac{47}{30} to fractions with denominator 60.

$$\frac{28}{15}-\left(-\frac{13}{30}-\left(\frac{15}{60}+\frac{94}{60}-\frac{1}{4}\right)\right)-1$$

Since \frac{15}{60} and \frac{94}{60} have the same denominator, add them by adding their numerators.

$$\frac{28}{15}-\left(-\frac{13}{30}-\left(\frac{15+94}{60}-\frac{1}{4}\right)\right)-1$$

Add 15 and 94 to get 109.

$$\frac{28}{15}-\left(-\frac{13}{30}-\left(\frac{109}{60}-\frac{1}{4}\right)\right)-1$$

Least common multiple of 60 and 4 is 60. Convert \frac{109}{60} and \frac{1}{4} to fractions with denominator 60.

$$\frac{28}{15}-\left(-\frac{13}{30}-\left(\frac{109}{60}-\frac{15}{60}\right)\right)-1$$

Since \frac{109}{60} and \frac{15}{60} have the same denominator, subtract them by subtracting their numerators.

$$\frac{28}{15}-\left(-\frac{13}{30}-\frac{109-15}{60}\right)-1$$

Subtract 15 from 109 to get 94.

$$\frac{28}{15}-\left(-\frac{13}{30}-\frac{94}{60}\right)-1$$

Reduce the fraction \frac{94}{60} to lowest terms by extracting and canceling out 2.

$$\frac{28}{15}-\left(-\frac{13}{30}-\frac{47}{30}\right)-1$$

Since -\frac{13}{30} and \frac{47}{30} have the same denominator, subtract them by subtracting their numerators.

$$\frac{28}{15}-\frac{-13-47}{30}-1$$

Subtract 47 from -13 to get -60.

$$\frac{28}{15}-\frac{-60}{30}-1$$

Divide -60 by 30 to get -2.

$$\frac{28}{15}-\left(-2\right)-1$$

The opposite of -2 is 2.

$$\frac{28}{15}+2-1$$

Convert 2 to fraction \frac{30}{15}.

$$\frac{28}{15}+\frac{30}{15}-1$$

Since \frac{28}{15} and \frac{30}{15} have the same denominator, add them by adding their numerators.

$$\frac{28+30}{15}-1$$

Add 28 and 30 to get 58.

$$\frac{58}{15}-1$$

Convert 1 to fraction \frac{15}{15}.

$$\frac{58}{15}-\frac{15}{15}$$

Since \frac{58}{15} and \frac{15}{15} have the same denominator, subtract them by subtracting their numerators.

$$\frac{58-15}{15}$$

Subtract 15 from 58 to get 43.

$$\frac{43}{15}$$

Factor

$$\frac{43}{3 \cdot 5} = 2\frac{13}{15} = 2.8666666666666667$$