Solve 1+1/1+1{1+frac{1/1}}+1/1{frac{1/1+1}+1}+1

$$1 + \frac { 1 } { 1 + \frac { 1 } { 1 + \frac { 1 } { 1 } } } + \frac { 1 } { \frac { 1 } { \frac { 1 } { 1 } + 1 } + 1 } + 1$$

Evaluate

$$\frac{10}{3}\approx 3.333333333$$

Solution Steps

Divide 1 by 1 to get 1.

$$1+\frac{1}{1+\frac{1}{1+1}}+\frac{1}{\frac{1}{\frac{1}{1}+1}+1}+1$$

Divide 1 by 1 to get 1.

$$1+\frac{1}{1+\frac{1}{1+1}}+\frac{1}{\frac{1}{1+1}+1}+1$$

Add 1 and 1 to get 2.

$$1+\frac{1}{1+\frac{1}{2}}+\frac{1}{\frac{1}{1+1}+1}+1$$

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

$$1+\frac{1}{\frac{2}{2}+\frac{1}{2}}+\frac{1}{\frac{1}{1+1}+1}+1$$

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

$$1+\frac{1}{\frac{2+1}{2}}+\frac{1}{\frac{1}{1+1}+1}+1$$

Add 2 and 1 to get 3.

$$1+\frac{1}{\frac{3}{2}}+\frac{1}{\frac{1}{1+1}+1}+1$$

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

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

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

$$1+\frac{2}{3}+\frac{1}{\frac{1}{1+1}+1}+1$$

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

$$\frac{3}{3}+\frac{2}{3}+\frac{1}{\frac{1}{1+1}+1}+1$$

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

$$\frac{3+2}{3}+\frac{1}{\frac{1}{1+1}+1}+1$$

Add 3 and 2 to get 5.

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

Add 1 and 1 to get 2.

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

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

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

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

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

Add 1 and 2 to get 3.

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

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

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

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

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

Since \frac{5}{3} and \frac{2}{3} have the same denominator, add them by adding their numerators.

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

Add 5 and 2 to get 7.

$$\frac{7}{3}+1$$

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

$$\frac{7}{3}+\frac{3}{3}$$

Since \frac{7}{3} and \frac{3}{3} have the same denominator, add them by adding their numerators.

$$\frac{7+3}{3}$$

Add 7 and 3 to get 10.

$$\frac{10}{3}$$

Factor

$$\frac{2 \cdot 5}{3} = 3\frac{1}{3} = 3.3333333333333335$$