Solve [1/2+frac{1/3/frac{2{3}-1/2}-frac{4+frac{1}{5}}{frac{14}{5^2}}]}{sqrt{frac{16}{25}}(frac{1}{2})^2}

$$\frac { [ \frac { \frac { 1 } { 2 } + \frac { 1 } { 3 } } { \frac { 2 } { 3 } - \frac { 1 } { 2 } } - \frac { 4 + \frac { 1 } { 5 } } { \frac { 14 } { 5 ^ { 2 } } } ] } { \sqrt { \frac { 16 } { 25 } } ( \frac { 1 } { 2 } ) ^ { 2 } }$$

Evaluate

$$-\frac{25}{2}=-12.5$$

Solution Steps

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

$$\frac{\frac{\frac{3}{6}+\frac{2}{6}}{\frac{2}{3}-\frac{1}{2}}-\frac{4+\frac{1}{5}}{\frac{14}{5^{2}}}}{\sqrt{\frac{16}{25}}\times \left(\frac{1}{2}\right)^{2}}$$

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

$$\frac{\frac{\frac{3+2}{6}}{\frac{2}{3}-\frac{1}{2}}-\frac{4+\frac{1}{5}}{\frac{14}{5^{2}}}}{\sqrt{\frac{16}{25}}\times \left(\frac{1}{2}\right)^{2}}$$

Add 3 and 2 to get 5.

$$\frac{\frac{\frac{5}{6}}{\frac{2}{3}-\frac{1}{2}}-\frac{4+\frac{1}{5}}{\frac{14}{5^{2}}}}{\sqrt{\frac{16}{25}}\times \left(\frac{1}{2}\right)^{2}}$$

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

$$\frac{\frac{\frac{5}{6}}{\frac{4}{6}-\frac{3}{6}}-\frac{4+\frac{1}{5}}{\frac{14}{5^{2}}}}{\sqrt{\frac{16}{25}}\times \left(\frac{1}{2}\right)^{2}}$$

Since \frac{4}{6} and \frac{3}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\frac{5}{6}}{\frac{4-3}{6}}-\frac{4+\frac{1}{5}}{\frac{14}{5^{2}}}}{\sqrt{\frac{16}{25}}\times \left(\frac{1}{2}\right)^{2}}$$

Subtract 3 from 4 to get 1.

$$\frac{\frac{\frac{5}{6}}{\frac{1}{6}}-\frac{4+\frac{1}{5}}{\frac{14}{5^{2}}}}{\sqrt{\frac{16}{25}}\times \left(\frac{1}{2}\right)^{2}}$$

Divide \frac{5}{6} by \frac{1}{6} by multiplying \frac{5}{6} by the reciprocal of \frac{1}{6}.

$$\frac{\frac{5}{6}\times 6-\frac{4+\frac{1}{5}}{\frac{14}{5^{2}}}}{\sqrt{\frac{16}{25}}\times \left(\frac{1}{2}\right)^{2}}$$

Cancel out 6 and 6.

$$\frac{5-\frac{4+\frac{1}{5}}{\frac{14}{5^{2}}}}{\sqrt{\frac{16}{25}}\times \left(\frac{1}{2}\right)^{2}}$$

Convert 4 to fraction \frac{20}{5}.

$$\frac{5-\frac{\frac{20}{5}+\frac{1}{5}}{\frac{14}{5^{2}}}}{\sqrt{\frac{16}{25}}\times \left(\frac{1}{2}\right)^{2}}$$

Since \frac{20}{5} and \frac{1}{5} have the same denominator, add them by adding their numerators.

$$\frac{5-\frac{\frac{20+1}{5}}{\frac{14}{5^{2}}}}{\sqrt{\frac{16}{25}}\times \left(\frac{1}{2}\right)^{2}}$$

Add 20 and 1 to get 21.

$$\frac{5-\frac{\frac{21}{5}}{\frac{14}{5^{2}}}}{\sqrt{\frac{16}{25}}\times \left(\frac{1}{2}\right)^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{5-\frac{\frac{21}{5}}{\frac{14}{25}}}{\sqrt{\frac{16}{25}}\times \left(\frac{1}{2}\right)^{2}}$$

Divide \frac{21}{5} by \frac{14}{25} by multiplying \frac{21}{5} by the reciprocal of \frac{14}{25}.

$$\frac{5-\frac{21}{5}\times \frac{25}{14}}{\sqrt{\frac{16}{25}}\times \left(\frac{1}{2}\right)^{2}}$$

Multiply \frac{21}{5} times \frac{25}{14} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5-\frac{21\times 25}{5\times 14}}{\sqrt{\frac{16}{25}}\times \left(\frac{1}{2}\right)^{2}}$$

Do the multiplications in the fraction \frac{21\times 25}{5\times 14}.

$$\frac{5-\frac{525}{70}}{\sqrt{\frac{16}{25}}\times \left(\frac{1}{2}\right)^{2}}$$

Reduce the fraction \frac{525}{70} to lowest terms by extracting and canceling out 35.

$$\frac{5-\frac{15}{2}}{\sqrt{\frac{16}{25}}\times \left(\frac{1}{2}\right)^{2}}$$

Convert 5 to fraction \frac{10}{2}.

$$\frac{\frac{10}{2}-\frac{15}{2}}{\sqrt{\frac{16}{25}}\times \left(\frac{1}{2}\right)^{2}}$$

Since \frac{10}{2} and \frac{15}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{10-15}{2}}{\sqrt{\frac{16}{25}}\times \left(\frac{1}{2}\right)^{2}}$$

Subtract 15 from 10 to get -5.

$$\frac{-\frac{5}{2}}{\sqrt{\frac{16}{25}}\times \left(\frac{1}{2}\right)^{2}}$$

Rewrite the square root of the division \frac{16}{25} as the division of square roots \frac{\sqrt{16}}{\sqrt{25}}. Take the square root of both numerator and denominator.

$$\frac{-\frac{5}{2}}{\frac{4}{5}\times \left(\frac{1}{2}\right)^{2}}$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{-\frac{5}{2}}{\frac{4}{5}\times \frac{1}{4}}$$

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

$$\frac{-\frac{5}{2}}{\frac{4\times 1}{5\times 4}}$$

Cancel out 4 in both numerator and denominator.

$$\frac{-\frac{5}{2}}{\frac{1}{5}}$$

Divide -\frac{5}{2} by \frac{1}{5} by multiplying -\frac{5}{2} by the reciprocal of \frac{1}{5}.

$$-\frac{5}{2}\times 5$$

Express -\frac{5}{2}\times 5 as a single fraction.

$$\frac{-5\times 5}{2}$$

Multiply -5 and 5 to get -25.

$$\frac{-25}{2}$$

Fraction \frac{-25}{2} can be rewritten as -\frac{25}{2} by extracting the negative sign.

$$-\frac{25}{2}$$

Factor

$$-\frac{25}{2} = -12\frac{1}{2} = -12.5$$