Solve 1+2/3+5{1-frac{1/3}}quad2-1/1-1{1-frac{1/2}}

$$1 + \frac { 2 } { 3 + \frac { 5 } { 1 - \frac { 1 } { 3 } } } \quad 2 - \frac { 1 } { 1 - \frac { 1 } { 1 - \frac { 1 } { 2 } } }$$

Evaluate

$$\frac{50}{21}\approx 2.380952381$$

Solution Steps

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

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

Since \frac{3}{3} and \frac{1}{3} have the same denominator, subtract them by subtracting their numerators.

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

Subtract 1 from 3 to get 2.

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

Divide 5 by \frac{2}{3} by multiplying 5 by the reciprocal of \frac{2}{3}.

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

Express 5\times \frac{3}{2} as a single fraction.

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

Multiply 5 and 3 to get 15.

$$1+\frac{2}{3+\frac{15}{2}}\times 2-\frac{1}{1-\frac{1}{1-\frac{1}{2}}}$$

Convert 3 to fraction \frac{6}{2}.

$$1+\frac{2}{\frac{6}{2}+\frac{15}{2}}\times 2-\frac{1}{1-\frac{1}{1-\frac{1}{2}}}$$

Since \frac{6}{2} and \frac{15}{2} have the same denominator, add them by adding their numerators.

$$1+\frac{2}{\frac{6+15}{2}}\times 2-\frac{1}{1-\frac{1}{1-\frac{1}{2}}}$$

Add 6 and 15 to get 21.

$$1+\frac{2}{\frac{21}{2}}\times 2-\frac{1}{1-\frac{1}{1-\frac{1}{2}}}$$

Divide 2 by \frac{21}{2} by multiplying 2 by the reciprocal of \frac{21}{2}.

$$1+2\times \frac{2}{21}\times 2-\frac{1}{1-\frac{1}{1-\frac{1}{2}}}$$

Express 2\times \frac{2}{21} as a single fraction.

$$1+\frac{2\times 2}{21}\times 2-\frac{1}{1-\frac{1}{1-\frac{1}{2}}}$$

Multiply 2 and 2 to get 4.

$$1+\frac{4}{21}\times 2-\frac{1}{1-\frac{1}{1-\frac{1}{2}}}$$

Express \frac{4}{21}\times 2 as a single fraction.

$$1+\frac{4\times 2}{21}-\frac{1}{1-\frac{1}{1-\frac{1}{2}}}$$

Multiply 4 and 2 to get 8.

$$1+\frac{8}{21}-\frac{1}{1-\frac{1}{1-\frac{1}{2}}}$$

Convert 1 to fraction \frac{21}{21}.

$$\frac{21}{21}+\frac{8}{21}-\frac{1}{1-\frac{1}{1-\frac{1}{2}}}$$

Since \frac{21}{21} and \frac{8}{21} have the same denominator, add them by adding their numerators.

$$\frac{21+8}{21}-\frac{1}{1-\frac{1}{1-\frac{1}{2}}}$$

Add 21 and 8 to get 29.

$$\frac{29}{21}-\frac{1}{1-\frac{1}{1-\frac{1}{2}}}$$

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

$$\frac{29}{21}-\frac{1}{1-\frac{1}{\frac{2}{2}-\frac{1}{2}}}$$

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

$$\frac{29}{21}-\frac{1}{1-\frac{1}{\frac{2-1}{2}}}$$

Subtract 1 from 2 to get 1.

$$\frac{29}{21}-\frac{1}{1-\frac{1}{\frac{1}{2}}}$$

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

$$\frac{29}{21}-\frac{1}{1-1\times 2}$$

Multiply 1 and 2 to get 2.

$$\frac{29}{21}-\frac{1}{1-2}$$

Subtract 2 from 1 to get -1.

$$\frac{29}{21}-\frac{1}{-1}$$

Divide 1 by -1 to get -1.

$$\frac{29}{21}-\left(-1\right)$$

The opposite of -1 is 1.

$$\frac{29}{21}+1$$

Convert 1 to fraction \frac{21}{21}.

$$\frac{29}{21}+\frac{21}{21}$$

Since \frac{29}{21} and \frac{21}{21} have the same denominator, add them by adding their numerators.

$$\frac{29+21}{21}$$

Add 29 and 21 to get 50.

$$\frac{50}{21}$$

Factor

$$\frac{2 \cdot 5 ^ {2}}{3 \cdot 7} = 2\frac{8}{21} = 2.380952380952381$$