Solve 2/3+2{3+frac{2/3+frac{2{3}}}}

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

Evaluate

$$\frac{78}{139}\approx 0.561151079$$

Solution Steps

Convert 3 to fraction \frac{9}{3}.

$$\frac{2}{3+\frac{2}{3+\frac{2}{\frac{9}{3}+\frac{2}{3}}}}$$

Since \frac{9}{3} and \frac{2}{3} have the same denominator, add them by adding their numerators.

$$\frac{2}{3+\frac{2}{3+\frac{2}{\frac{9+2}{3}}}}$$

Add 9 and 2 to get 11.

$$\frac{2}{3+\frac{2}{3+\frac{2}{\frac{11}{3}}}}$$

Divide 2 by \frac{11}{3} by multiplying 2 by the reciprocal of \frac{11}{3}.

$$\frac{2}{3+\frac{2}{3+2\times \frac{3}{11}}}$$

Express 2\times \frac{3}{11} as a single fraction.

$$\frac{2}{3+\frac{2}{3+\frac{2\times 3}{11}}}$$

Multiply 2 and 3 to get 6.

$$\frac{2}{3+\frac{2}{3+\frac{6}{11}}}$$

Convert 3 to fraction \frac{33}{11}.

$$\frac{2}{3+\frac{2}{\frac{33}{11}+\frac{6}{11}}}$$

Since \frac{33}{11} and \frac{6}{11} have the same denominator, add them by adding their numerators.

$$\frac{2}{3+\frac{2}{\frac{33+6}{11}}}$$

Add 33 and 6 to get 39.

$$\frac{2}{3+\frac{2}{\frac{39}{11}}}$$

Divide 2 by \frac{39}{11} by multiplying 2 by the reciprocal of \frac{39}{11}.

$$\frac{2}{3+2\times \frac{11}{39}}$$

Express 2\times \frac{11}{39} as a single fraction.

$$\frac{2}{3+\frac{2\times 11}{39}}$$

Multiply 2 and 11 to get 22.

$$\frac{2}{3+\frac{22}{39}}$$

Convert 3 to fraction \frac{117}{39}.

$$\frac{2}{\frac{117}{39}+\frac{22}{39}}$$

Since \frac{117}{39} and \frac{22}{39} have the same denominator, add them by adding their numerators.

$$\frac{2}{\frac{117+22}{39}}$$

Add 117 and 22 to get 139.

$$\frac{2}{\frac{139}{39}}$$

Divide 2 by \frac{139}{39} by multiplying 2 by the reciprocal of \frac{139}{39}.

$$2\times \frac{39}{139}$$

Express 2\times \frac{39}{139} as a single fraction.

$$\frac{2\times 39}{139}$$

Multiply 2 and 39 to get 78.

$$\frac{78}{139}$$

Factor

$$\frac{2 \cdot 3 \cdot 13}{139} = 0.5611510791366906$$