Solve (1+1/3)*(1/2+1/3)/(1+frac{1{3})+(frac{1}{2}+frac{1}{3})}+frac{1}{3}

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

Evaluate

$$\frac{11}{13}\approx 0.846153846$$

Solution Steps

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

$$\frac{\left(\frac{3}{3}+\frac{1}{3}\right)\left(\frac{1}{2}+\frac{1}{3}\right)}{1+\frac{1}{3}+\frac{1}{2}+\frac{1}{3}}+\frac{1}{3}$$

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

$$\frac{\frac{3+1}{3}\left(\frac{1}{2}+\frac{1}{3}\right)}{1+\frac{1}{3}+\frac{1}{2}+\frac{1}{3}}+\frac{1}{3}$$

Add 3 and 1 to get 4.

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

Least common multiple of 2 and 3 is 6. Convert \frac{1}{2} and \frac{1}{3} to fractions with denominator 6.

$$\frac{\frac{4}{3}\left(\frac{3}{6}+\frac{2}{6}\right)}{1+\frac{1}{3}+\frac{1}{2}+\frac{1}{3}}+\frac{1}{3}$$

Since \frac{3}{6} and \frac{2}{6} have the same denominator, add them by adding their numerators.

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

Add 3 and 2 to get 5.

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

Multiply \frac{4}{3} times \frac{5}{6} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{4\times 5}{3\times 6}.

$$\frac{\frac{20}{18}}{1+\frac{1}{3}+\frac{1}{2}+\frac{1}{3}}+\frac{1}{3}$$

Reduce the fraction \frac{20}{18} to lowest terms by extracting and canceling out 2.

$$\frac{\frac{10}{9}}{1+\frac{1}{3}+\frac{1}{2}+\frac{1}{3}}+\frac{1}{3}$$

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

$$\frac{\frac{10}{9}}{\frac{3}{3}+\frac{1}{3}+\frac{1}{2}+\frac{1}{3}}+\frac{1}{3}$$

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

$$\frac{\frac{10}{9}}{\frac{3+1}{3}+\frac{1}{2}+\frac{1}{3}}+\frac{1}{3}$$

Add 3 and 1 to get 4.

$$\frac{\frac{10}{9}}{\frac{4}{3}+\frac{1}{2}+\frac{1}{3}}+\frac{1}{3}$$

Least common multiple of 3 and 2 is 6. Convert \frac{4}{3} and \frac{1}{2} to fractions with denominator 6.

$$\frac{\frac{10}{9}}{\frac{8}{6}+\frac{3}{6}+\frac{1}{3}}+\frac{1}{3}$$

Since \frac{8}{6} and \frac{3}{6} have the same denominator, add them by adding their numerators.

$$\frac{\frac{10}{9}}{\frac{8+3}{6}+\frac{1}{3}}+\frac{1}{3}$$

Add 8 and 3 to get 11.

$$\frac{\frac{10}{9}}{\frac{11}{6}+\frac{1}{3}}+\frac{1}{3}$$

Least common multiple of 6 and 3 is 6. Convert \frac{11}{6} and \frac{1}{3} to fractions with denominator 6.

$$\frac{\frac{10}{9}}{\frac{11}{6}+\frac{2}{6}}+\frac{1}{3}$$

Since \frac{11}{6} and \frac{2}{6} have the same denominator, add them by adding their numerators.

$$\frac{\frac{10}{9}}{\frac{11+2}{6}}+\frac{1}{3}$$

Add 11 and 2 to get 13.

$$\frac{\frac{10}{9}}{\frac{13}{6}}+\frac{1}{3}$$

Divide \frac{10}{9} by \frac{13}{6} by multiplying \frac{10}{9} by the reciprocal of \frac{13}{6}.

$$\frac{10}{9}\times \frac{6}{13}+\frac{1}{3}$$

Multiply \frac{10}{9} times \frac{6}{13} by multiplying numerator times numerator and denominator times denominator.

$$\frac{10\times 6}{9\times 13}+\frac{1}{3}$$

Do the multiplications in the fraction \frac{10\times 6}{9\times 13}.

$$\frac{60}{117}+\frac{1}{3}$$

Reduce the fraction \frac{60}{117} to lowest terms by extracting and canceling out 3.

$$\frac{20}{39}+\frac{1}{3}$$

Least common multiple of 39 and 3 is 39. Convert \frac{20}{39} and \frac{1}{3} to fractions with denominator 39.

$$\frac{20}{39}+\frac{13}{39}$$

Since \frac{20}{39} and \frac{13}{39} have the same denominator, add them by adding their numerators.

$$\frac{20+13}{39}$$

Add 20 and 13 to get 33.

$$\frac{33}{39}$$

Reduce the fraction \frac{33}{39} to lowest terms by extracting and canceling out 3.

$$\frac{11}{13}$$

Factor

$$\frac{11}{13} = 0.8461538461538461$$