Solve 1+frac{1/1+frac{1{2}}}{1-1/1-frac{1{2}}}-2*3/4-2frac{3/4}{1+frac{1}{1-frac{3}{4}}}

$$\frac { 1 + \frac { 1 } { 1 + \frac { 1 } { 2 } } } { 1 - \frac { 1 } { 1 - \frac { 1 } { 2 } } } - \frac { 2 \cdot \frac { 3 } { 4 } - 2 \frac { 3 } { 4 } } { 1 + \frac { 1 } { 1 - \frac { 3 } { 4 } } }$$

Evaluate

$$-\frac{17}{12}\approx -1.416666667$$

Solution Steps

Convert 1 to fraction \frac{2}{2}.

$$\frac{1+\frac{1}{\frac{2}{2}+\frac{1}{2}}}{1-\frac{1}{1-\frac{1}{2}}}-\frac{2\times \frac{3}{4}-\frac{2\times 4+3}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Since \frac{2}{2} and \frac{1}{2} have the same denominator, add them by adding their numerators.

$$\frac{1+\frac{1}{\frac{2+1}{2}}}{1-\frac{1}{1-\frac{1}{2}}}-\frac{2\times \frac{3}{4}-\frac{2\times 4+3}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Add 2 and 1 to get 3.

$$\frac{1+\frac{1}{\frac{3}{2}}}{1-\frac{1}{1-\frac{1}{2}}}-\frac{2\times \frac{3}{4}-\frac{2\times 4+3}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Divide 1 by \frac{3}{2} by multiplying 1 by the reciprocal of \frac{3}{2}.

$$\frac{1+1\times \frac{2}{3}}{1-\frac{1}{1-\frac{1}{2}}}-\frac{2\times \frac{3}{4}-\frac{2\times 4+3}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Multiply 1 and \frac{2}{3} to get \frac{2}{3}.

$$\frac{1+\frac{2}{3}}{1-\frac{1}{1-\frac{1}{2}}}-\frac{2\times \frac{3}{4}-\frac{2\times 4+3}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Convert 1 to fraction \frac{3}{3}.

$$\frac{\frac{3}{3}+\frac{2}{3}}{1-\frac{1}{1-\frac{1}{2}}}-\frac{2\times \frac{3}{4}-\frac{2\times 4+3}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

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

$$\frac{\frac{3+2}{3}}{1-\frac{1}{1-\frac{1}{2}}}-\frac{2\times \frac{3}{4}-\frac{2\times 4+3}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Add 3 and 2 to get 5.

$$\frac{\frac{5}{3}}{1-\frac{1}{1-\frac{1}{2}}}-\frac{2\times \frac{3}{4}-\frac{2\times 4+3}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Convert 1 to fraction \frac{2}{2}.

$$\frac{\frac{5}{3}}{1-\frac{1}{\frac{2}{2}-\frac{1}{2}}}-\frac{2\times \frac{3}{4}-\frac{2\times 4+3}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Since \frac{2}{2} and \frac{1}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{5}{3}}{1-\frac{1}{\frac{2-1}{2}}}-\frac{2\times \frac{3}{4}-\frac{2\times 4+3}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Subtract 1 from 2 to get 1.

$$\frac{\frac{5}{3}}{1-\frac{1}{\frac{1}{2}}}-\frac{2\times \frac{3}{4}-\frac{2\times 4+3}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Divide 1 by \frac{1}{2} by multiplying 1 by the reciprocal of \frac{1}{2}.

$$\frac{\frac{5}{3}}{1-1\times 2}-\frac{2\times \frac{3}{4}-\frac{2\times 4+3}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Multiply 1 and 2 to get 2.

$$\frac{\frac{5}{3}}{1-2}-\frac{2\times \frac{3}{4}-\frac{2\times 4+3}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Subtract 2 from 1 to get -1.

$$\frac{\frac{5}{3}}{-1}-\frac{2\times \frac{3}{4}-\frac{2\times 4+3}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Express \frac{\frac{5}{3}}{-1} as a single fraction.

$$\frac{5}{3\left(-1\right)}-\frac{2\times \frac{3}{4}-\frac{2\times 4+3}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Multiply 3 and -1 to get -3.

$$\frac{5}{-3}-\frac{2\times \frac{3}{4}-\frac{2\times 4+3}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Fraction \frac{5}{-3} can be rewritten as -\frac{5}{3} by extracting the negative sign.

$$-\frac{5}{3}-\frac{2\times \frac{3}{4}-\frac{2\times 4+3}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Express 2\times \frac{3}{4} as a single fraction.

$$-\frac{5}{3}-\frac{\frac{2\times 3}{4}-\frac{2\times 4+3}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Multiply 2 and 3 to get 6.

$$-\frac{5}{3}-\frac{\frac{6}{4}-\frac{2\times 4+3}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Reduce the fraction \frac{6}{4} to lowest terms by extracting and canceling out 2.

$$-\frac{5}{3}-\frac{\frac{3}{2}-\frac{2\times 4+3}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Multiply 2 and 4 to get 8.

$$-\frac{5}{3}-\frac{\frac{3}{2}-\frac{8+3}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Add 8 and 3 to get 11.

$$-\frac{5}{3}-\frac{\frac{3}{2}-\frac{11}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Least common multiple of 2 and 4 is 4. Convert \frac{3}{2} and \frac{11}{4} to fractions with denominator 4.

$$-\frac{5}{3}-\frac{\frac{6}{4}-\frac{11}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Since \frac{6}{4} and \frac{11}{4} have the same denominator, subtract them by subtracting their numerators.

$$-\frac{5}{3}-\frac{\frac{6-11}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Subtract 11 from 6 to get -5.

$$-\frac{5}{3}-\frac{-\frac{5}{4}}{1+\frac{1}{1-\frac{3}{4}}}$$

Convert 1 to fraction \frac{4}{4}.

$$-\frac{5}{3}-\frac{-\frac{5}{4}}{1+\frac{1}{\frac{4}{4}-\frac{3}{4}}}$$

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

$$-\frac{5}{3}-\frac{-\frac{5}{4}}{1+\frac{1}{\frac{4-3}{4}}}$$

Subtract 3 from 4 to get 1.

$$-\frac{5}{3}-\frac{-\frac{5}{4}}{1+\frac{1}{\frac{1}{4}}}$$

Divide 1 by \frac{1}{4} by multiplying 1 by the reciprocal of \frac{1}{4}.

$$-\frac{5}{3}-\frac{-\frac{5}{4}}{1+1\times 4}$$

Multiply 1 and 4 to get 4.

$$-\frac{5}{3}-\frac{-\frac{5}{4}}{1+4}$$

Add 1 and 4 to get 5.

$$-\frac{5}{3}-\frac{-\frac{5}{4}}{5}$$

Express \frac{-\frac{5}{4}}{5} as a single fraction.

$$-\frac{5}{3}-\frac{-5}{4\times 5}$$

Multiply 4 and 5 to get 20.

$$-\frac{5}{3}-\frac{-5}{20}$$

Reduce the fraction \frac{-5}{20} to lowest terms by extracting and canceling out 5.

$$-\frac{5}{3}-\left(-\frac{1}{4}\right)$$

The opposite of -\frac{1}{4} is \frac{1}{4}.

$$-\frac{5}{3}+\frac{1}{4}$$

Least common multiple of 3 and 4 is 12. Convert -\frac{5}{3} and \frac{1}{4} to fractions with denominator 12.

$$-\frac{20}{12}+\frac{3}{12}$$

Since -\frac{20}{12} and \frac{3}{12} have the same denominator, add them by adding their numerators.

$$\frac{-20+3}{12}$$

Add -20 and 3 to get -17.

$$-\frac{17}{12}$$

Factor

$$-\frac{17}{12} = -1\frac{5}{12} = -1.4166666666666667$$