Solve 3/11(2/19div3/38*frac{2/3+frac{3}{38}divfrac{2}{19}text{of}frac{2}{3})}{(frac{1}{3}+frac{3}{11})div3frac{1}{2}}text{of}frac{31}{29}

$$\frac { \frac { 3 } { 11 } ( \frac { 2 } { 19 } \div \frac { 3 } { 38 } \times \frac { 2 } { 3 } + \frac { 3 } { 38 } \div \frac { 2 } { 19 } \text { of } \frac { 2 } { 3 } ) } { ( \frac { 1 } { 3 } + \frac { 3 } { 11 } ) \div 3 \frac { 1 } { 2 } } \text { of } \frac { 31 } { 29 }$$

Evaluate

$$\frac{1085}{464}\approx 2.338362069$$

Solution Steps

Divide \frac{2}{19} by \frac{3}{38} by multiplying \frac{2}{19} by the reciprocal of \frac{3}{38}.

$$\frac{\frac{3}{11}\left(\frac{2}{19}\times \frac{38}{3}\times \frac{2}{3}+\frac{\frac{3}{38}}{\frac{2}{19}}\times \frac{2}{3}\right)}{\frac{\frac{1}{3}+\frac{3}{11}}{\frac{3\times 2+1}{2}}}\times \frac{31}{29}$$

Multiply \frac{2}{19} times \frac{38}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{3}{11}\left(\frac{2\times 38}{19\times 3}\times \frac{2}{3}+\frac{\frac{3}{38}}{\frac{2}{19}}\times \frac{2}{3}\right)}{\frac{\frac{1}{3}+\frac{3}{11}}{\frac{3\times 2+1}{2}}}\times \frac{31}{29}$$

Do the multiplications in the fraction \frac{2\times 38}{19\times 3}.

$$\frac{\frac{3}{11}\left(\frac{76}{57}\times \frac{2}{3}+\frac{\frac{3}{38}}{\frac{2}{19}}\times \frac{2}{3}\right)}{\frac{\frac{1}{3}+\frac{3}{11}}{\frac{3\times 2+1}{2}}}\times \frac{31}{29}$$

Reduce the fraction \frac{76}{57} to lowest terms by extracting and canceling out 19.

$$\frac{\frac{3}{11}\left(\frac{4}{3}\times \frac{2}{3}+\frac{\frac{3}{38}}{\frac{2}{19}}\times \frac{2}{3}\right)}{\frac{\frac{1}{3}+\frac{3}{11}}{\frac{3\times 2+1}{2}}}\times \frac{31}{29}$$

Multiply \frac{4}{3} times \frac{2}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{3}{11}\left(\frac{4\times 2}{3\times 3}+\frac{\frac{3}{38}}{\frac{2}{19}}\times \frac{2}{3}\right)}{\frac{\frac{1}{3}+\frac{3}{11}}{\frac{3\times 2+1}{2}}}\times \frac{31}{29}$$

Do the multiplications in the fraction \frac{4\times 2}{3\times 3}.

$$\frac{\frac{3}{11}\left(\frac{8}{9}+\frac{\frac{3}{38}}{\frac{2}{19}}\times \frac{2}{3}\right)}{\frac{\frac{1}{3}+\frac{3}{11}}{\frac{3\times 2+1}{2}}}\times \frac{31}{29}$$

Divide \frac{3}{38} by \frac{2}{19} by multiplying \frac{3}{38} by the reciprocal of \frac{2}{19}.

$$\frac{\frac{3}{11}\left(\frac{8}{9}+\frac{3}{38}\times \frac{19}{2}\times \frac{2}{3}\right)}{\frac{\frac{1}{3}+\frac{3}{11}}{\frac{3\times 2+1}{2}}}\times \frac{31}{29}$$

Multiply \frac{3}{38} times \frac{19}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{3}{11}\left(\frac{8}{9}+\frac{3\times 19}{38\times 2}\times \frac{2}{3}\right)}{\frac{\frac{1}{3}+\frac{3}{11}}{\frac{3\times 2+1}{2}}}\times \frac{31}{29}$$

Do the multiplications in the fraction \frac{3\times 19}{38\times 2}.

$$\frac{\frac{3}{11}\left(\frac{8}{9}+\frac{57}{76}\times \frac{2}{3}\right)}{\frac{\frac{1}{3}+\frac{3}{11}}{\frac{3\times 2+1}{2}}}\times \frac{31}{29}$$

Reduce the fraction \frac{57}{76} to lowest terms by extracting and canceling out 19.

$$\frac{\frac{3}{11}\left(\frac{8}{9}+\frac{3}{4}\times \frac{2}{3}\right)}{\frac{\frac{1}{3}+\frac{3}{11}}{\frac{3\times 2+1}{2}}}\times \frac{31}{29}$$

Multiply \frac{3}{4} times \frac{2}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{3}{11}\left(\frac{8}{9}+\frac{3\times 2}{4\times 3}\right)}{\frac{\frac{1}{3}+\frac{3}{11}}{\frac{3\times 2+1}{2}}}\times \frac{31}{29}$$

Cancel out 3 in both numerator and denominator.

$$\frac{\frac{3}{11}\left(\frac{8}{9}+\frac{2}{4}\right)}{\frac{\frac{1}{3}+\frac{3}{11}}{\frac{3\times 2+1}{2}}}\times \frac{31}{29}$$

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

$$\frac{\frac{3}{11}\left(\frac{8}{9}+\frac{1}{2}\right)}{\frac{\frac{1}{3}+\frac{3}{11}}{\frac{3\times 2+1}{2}}}\times \frac{31}{29}$$

Least common multiple of 9 and 2 is 18. Convert \frac{8}{9} and \frac{1}{2} to fractions with denominator 18.

$$\frac{\frac{3}{11}\left(\frac{16}{18}+\frac{9}{18}\right)}{\frac{\frac{1}{3}+\frac{3}{11}}{\frac{3\times 2+1}{2}}}\times \frac{31}{29}$$

Since \frac{16}{18} and \frac{9}{18} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3}{11}\times \frac{16+9}{18}}{\frac{\frac{1}{3}+\frac{3}{11}}{\frac{3\times 2+1}{2}}}\times \frac{31}{29}$$

Add 16 and 9 to get 25.

$$\frac{\frac{3}{11}\times \frac{25}{18}}{\frac{\frac{1}{3}+\frac{3}{11}}{\frac{3\times 2+1}{2}}}\times \frac{31}{29}$$

Multiply \frac{3}{11} times \frac{25}{18} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{3\times 25}{11\times 18}}{\frac{\frac{1}{3}+\frac{3}{11}}{\frac{3\times 2+1}{2}}}\times \frac{31}{29}$$

Do the multiplications in the fraction \frac{3\times 25}{11\times 18}.

$$\frac{\frac{75}{198}}{\frac{\frac{1}{3}+\frac{3}{11}}{\frac{3\times 2+1}{2}}}\times \frac{31}{29}$$

Reduce the fraction \frac{75}{198} to lowest terms by extracting and canceling out 3.

$$\frac{\frac{25}{66}}{\frac{\frac{1}{3}+\frac{3}{11}}{\frac{3\times 2+1}{2}}}\times \frac{31}{29}$$

Least common multiple of 3 and 11 is 33. Convert \frac{1}{3} and \frac{3}{11} to fractions with denominator 33.

$$\frac{\frac{25}{66}}{\frac{\frac{11}{33}+\frac{9}{33}}{\frac{3\times 2+1}{2}}}\times \frac{31}{29}$$

Since \frac{11}{33} and \frac{9}{33} have the same denominator, add them by adding their numerators.

$$\frac{\frac{25}{66}}{\frac{\frac{11+9}{33}}{\frac{3\times 2+1}{2}}}\times \frac{31}{29}$$

Add 11 and 9 to get 20.

$$\frac{\frac{25}{66}}{\frac{\frac{20}{33}}{\frac{3\times 2+1}{2}}}\times \frac{31}{29}$$

Multiply 3 and 2 to get 6.

$$\frac{\frac{25}{66}}{\frac{\frac{20}{33}}{\frac{6+1}{2}}}\times \frac{31}{29}$$

Add 6 and 1 to get 7.

$$\frac{\frac{25}{66}}{\frac{\frac{20}{33}}{\frac{7}{2}}}\times \frac{31}{29}$$

Divide \frac{20}{33} by \frac{7}{2} by multiplying \frac{20}{33} by the reciprocal of \frac{7}{2}.

$$\frac{\frac{25}{66}}{\frac{20}{33}\times \frac{2}{7}}\times \frac{31}{29}$$

Multiply \frac{20}{33} times \frac{2}{7} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{25}{66}}{\frac{20\times 2}{33\times 7}}\times \frac{31}{29}$$

Do the multiplications in the fraction \frac{20\times 2}{33\times 7}.

$$\frac{\frac{25}{66}}{\frac{40}{231}}\times \frac{31}{29}$$

Divide \frac{25}{66} by \frac{40}{231} by multiplying \frac{25}{66} by the reciprocal of \frac{40}{231}.

$$\frac{25}{66}\times \frac{231}{40}\times \frac{31}{29}$$

Multiply \frac{25}{66} times \frac{231}{40} by multiplying numerator times numerator and denominator times denominator.

$$\frac{25\times 231}{66\times 40}\times \frac{31}{29}$$

Do the multiplications in the fraction \frac{25\times 231}{66\times 40}.

$$\frac{5775}{2640}\times \frac{31}{29}$$

Reduce the fraction \frac{5775}{2640} to lowest terms by extracting and canceling out 165.

$$\frac{35}{16}\times \frac{31}{29}$$

Multiply \frac{35}{16} times \frac{31}{29} by multiplying numerator times numerator and denominator times denominator.

$$\frac{35\times 31}{16\times 29}$$

Do the multiplications in the fraction \frac{35\times 31}{16\times 29}.

$$\frac{1085}{464}$$

Factor

$$\frac{5 \cdot 7 \cdot 31}{2 ^ {4} \cdot 29} = 2\frac{157}{464} = 2.3383620689655173$$