Solve 2+1/3/7+1-frac{1/4/3}{frac{frac{1}{2}}{frac{1}{4}}-frac{frac{1}{4}}{frac{3}{5}}}left(frac{2}{7}+frac{4}{19}right)

$$\frac{ \frac{ 2+ \frac{ 1 }{ 3 } }{ 7 } + \frac{ 1- \frac{ 1 }{ 4 } }{ 3 } }{ \frac{ \frac{ 1 }{ 2 } }{ \frac{ 1 }{ 4 } } - \frac{ \frac{ 1 }{ 4 } }{ \frac{ 3 }{ 5 } } } \left( \frac{ 2 }{ 7 } + \frac{ 4 }{ 19 } \right)$$

Evaluate

$$\frac{66}{361}\approx 0.182825485$$

Solution Steps

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

$$\frac{\frac{\frac{6}{3}+\frac{1}{3}}{7}+\frac{1-\frac{1}{4}}{3}}{\frac{\frac{1}{2}}{\frac{1}{4}}-\frac{\frac{1}{4}}{\frac{3}{5}}}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Since \frac{6}{3} and \frac{1}{3} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\frac{6+1}{3}}{7}+\frac{1-\frac{1}{4}}{3}}{\frac{\frac{1}{2}}{\frac{1}{4}}-\frac{\frac{1}{4}}{\frac{3}{5}}}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Add 6 and 1 to get 7.

$$\frac{\frac{\frac{7}{3}}{7}+\frac{1-\frac{1}{4}}{3}}{\frac{\frac{1}{2}}{\frac{1}{4}}-\frac{\frac{1}{4}}{\frac{3}{5}}}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Express \frac{\frac{7}{3}}{7} as a single fraction.

$$\frac{\frac{7}{3\times 7}+\frac{1-\frac{1}{4}}{3}}{\frac{\frac{1}{2}}{\frac{1}{4}}-\frac{\frac{1}{4}}{\frac{3}{5}}}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Cancel out 7 in both numerator and denominator.

$$\frac{\frac{1}{3}+\frac{1-\frac{1}{4}}{3}}{\frac{\frac{1}{2}}{\frac{1}{4}}-\frac{\frac{1}{4}}{\frac{3}{5}}}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Convert 1 to fraction \frac{4}{4}.

$$\frac{\frac{1}{3}+\frac{\frac{4}{4}-\frac{1}{4}}{3}}{\frac{\frac{1}{2}}{\frac{1}{4}}-\frac{\frac{1}{4}}{\frac{3}{5}}}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Since \frac{4}{4} and \frac{1}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{1}{3}+\frac{\frac{4-1}{4}}{3}}{\frac{\frac{1}{2}}{\frac{1}{4}}-\frac{\frac{1}{4}}{\frac{3}{5}}}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Subtract 1 from 4 to get 3.

$$\frac{\frac{1}{3}+\frac{\frac{3}{4}}{3}}{\frac{\frac{1}{2}}{\frac{1}{4}}-\frac{\frac{1}{4}}{\frac{3}{5}}}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Express \frac{\frac{3}{4}}{3} as a single fraction.

$$\frac{\frac{1}{3}+\frac{3}{4\times 3}}{\frac{\frac{1}{2}}{\frac{1}{4}}-\frac{\frac{1}{4}}{\frac{3}{5}}}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Cancel out 3 in both numerator and denominator.

$$\frac{\frac{1}{3}+\frac{1}{4}}{\frac{\frac{1}{2}}{\frac{1}{4}}-\frac{\frac{1}{4}}{\frac{3}{5}}}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Least common multiple of 3 and 4 is 12. Convert \frac{1}{3} and \frac{1}{4} to fractions with denominator 12.

$$\frac{\frac{4}{12}+\frac{3}{12}}{\frac{\frac{1}{2}}{\frac{1}{4}}-\frac{\frac{1}{4}}{\frac{3}{5}}}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Since \frac{4}{12} and \frac{3}{12} have the same denominator, add them by adding their numerators.

$$\frac{\frac{4+3}{12}}{\frac{\frac{1}{2}}{\frac{1}{4}}-\frac{\frac{1}{4}}{\frac{3}{5}}}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Add 4 and 3 to get 7.

$$\frac{\frac{7}{12}}{\frac{\frac{1}{2}}{\frac{1}{4}}-\frac{\frac{1}{4}}{\frac{3}{5}}}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Divide \frac{1}{2} by \frac{1}{4} by multiplying \frac{1}{2} by the reciprocal of \frac{1}{4}.

$$\frac{\frac{7}{12}}{\frac{1}{2}\times 4-\frac{\frac{1}{4}}{\frac{3}{5}}}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Multiply \frac{1}{2} and 4 to get \frac{4}{2}.

$$\frac{\frac{7}{12}}{\frac{4}{2}-\frac{\frac{1}{4}}{\frac{3}{5}}}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Divide 4 by 2 to get 2.

$$\frac{\frac{7}{12}}{2-\frac{\frac{1}{4}}{\frac{3}{5}}}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Divide \frac{1}{4} by \frac{3}{5} by multiplying \frac{1}{4} by the reciprocal of \frac{3}{5}.

$$\frac{\frac{7}{12}}{2-\frac{1}{4}\times \frac{5}{3}}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Multiply \frac{1}{4} times \frac{5}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{7}{12}}{2-\frac{1\times 5}{4\times 3}}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Do the multiplications in the fraction \frac{1\times 5}{4\times 3}.

$$\frac{\frac{7}{12}}{2-\frac{5}{12}}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Convert 2 to fraction \frac{24}{12}.

$$\frac{\frac{7}{12}}{\frac{24}{12}-\frac{5}{12}}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Since \frac{24}{12} and \frac{5}{12} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{7}{12}}{\frac{24-5}{12}}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Subtract 5 from 24 to get 19.

$$\frac{\frac{7}{12}}{\frac{19}{12}}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Divide \frac{7}{12} by \frac{19}{12} by multiplying \frac{7}{12} by the reciprocal of \frac{19}{12}.

$$\frac{7}{12}\times \frac{12}{19}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Multiply \frac{7}{12} times \frac{12}{19} by multiplying numerator times numerator and denominator times denominator.

$$\frac{7\times 12}{12\times 19}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Cancel out 12 in both numerator and denominator.

$$\frac{7}{19}\left(\frac{2}{7}+\frac{4}{19}\right)$$

Least common multiple of 7 and 19 is 133. Convert \frac{2}{7} and \frac{4}{19} to fractions with denominator 133.

$$\frac{7}{19}\left(\frac{38}{133}+\frac{28}{133}\right)$$

Since \frac{38}{133} and \frac{28}{133} have the same denominator, add them by adding their numerators.

$$\frac{7}{19}\times \frac{38+28}{133}$$

Add 38 and 28 to get 66.

$$\frac{7}{19}\times \frac{66}{133}$$

Multiply \frac{7}{19} times \frac{66}{133} by multiplying numerator times numerator and denominator times denominator.

$$\frac{7\times 66}{19\times 133}$$

Do the multiplications in the fraction \frac{7\times 66}{19\times 133}.

$$\frac{462}{2527}$$

Reduce the fraction \frac{462}{2527} to lowest terms by extracting and canceling out 7.

$$\frac{66}{361}$$

Factor

$$\frac{2 \cdot 3 \cdot 11}{19 ^ {2}} = 0.18282548476454294$$