Solve {3+3/5/1+frac{1{5}}(3/2-1/3)+sqrt{2+frac{1}{4}}}

$$\{ \frac { 3 + \frac { 3 } { 5 } } { 1 + \frac { 1 } { 5 } } ( \frac { 3 } { 2 } - \frac { 1 } { 3 } ) + \sqrt { 2 + \frac { 1 } { 4 } } \}$$

Evaluate

$$5$$

Solution Steps

Convert 3 to fraction \frac{15}{5}.

$$\frac{\frac{15}{5}+\frac{3}{5}}{1+\frac{1}{5}}\left(\frac{3}{2}-\frac{1}{3}\right)+\sqrt{2+\frac{1}{4}}$$

Since \frac{15}{5} and \frac{3}{5} have the same denominator, add them by adding their numerators.

$$\frac{\frac{15+3}{5}}{1+\frac{1}{5}}\left(\frac{3}{2}-\frac{1}{3}\right)+\sqrt{2+\frac{1}{4}}$$

Add 15 and 3 to get 18.

$$\frac{\frac{18}{5}}{1+\frac{1}{5}}\left(\frac{3}{2}-\frac{1}{3}\right)+\sqrt{2+\frac{1}{4}}$$

Convert 1 to fraction \frac{5}{5}.

$$\frac{\frac{18}{5}}{\frac{5}{5}+\frac{1}{5}}\left(\frac{3}{2}-\frac{1}{3}\right)+\sqrt{2+\frac{1}{4}}$$

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

$$\frac{\frac{18}{5}}{\frac{5+1}{5}}\left(\frac{3}{2}-\frac{1}{3}\right)+\sqrt{2+\frac{1}{4}}$$

Add 5 and 1 to get 6.

$$\frac{\frac{18}{5}}{\frac{6}{5}}\left(\frac{3}{2}-\frac{1}{3}\right)+\sqrt{2+\frac{1}{4}}$$

Divide \frac{18}{5} by \frac{6}{5} by multiplying \frac{18}{5} by the reciprocal of \frac{6}{5}.

$$\frac{18}{5}\times \frac{5}{6}\left(\frac{3}{2}-\frac{1}{3}\right)+\sqrt{2+\frac{1}{4}}$$

Multiply \frac{18}{5} times \frac{5}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{18\times 5}{5\times 6}\left(\frac{3}{2}-\frac{1}{3}\right)+\sqrt{2+\frac{1}{4}}$$

Cancel out 5 in both numerator and denominator.

$$\frac{18}{6}\left(\frac{3}{2}-\frac{1}{3}\right)+\sqrt{2+\frac{1}{4}}$$

Divide 18 by 6 to get 3.

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

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

$$3\left(\frac{9}{6}-\frac{2}{6}\right)+\sqrt{2+\frac{1}{4}}$$

Since \frac{9}{6} and \frac{2}{6} have the same denominator, subtract them by subtracting their numerators.

$$3\times \frac{9-2}{6}+\sqrt{2+\frac{1}{4}}$$

Subtract 2 from 9 to get 7.

$$3\times \frac{7}{6}+\sqrt{2+\frac{1}{4}}$$

Express 3\times \frac{7}{6} as a single fraction.

$$\frac{3\times 7}{6}+\sqrt{2+\frac{1}{4}}$$

Multiply 3 and 7 to get 21.

$$\frac{21}{6}+\sqrt{2+\frac{1}{4}}$$

Reduce the fraction \frac{21}{6} to lowest terms by extracting and canceling out 3.

$$\frac{7}{2}+\sqrt{2+\frac{1}{4}}$$

Convert 2 to fraction \frac{8}{4}.

$$\frac{7}{2}+\sqrt{\frac{8}{4}+\frac{1}{4}}$$

Since \frac{8}{4} and \frac{1}{4} have the same denominator, add them by adding their numerators.

$$\frac{7}{2}+\sqrt{\frac{8+1}{4}}$$

Add 8 and 1 to get 9.

$$\frac{7}{2}+\sqrt{\frac{9}{4}}$$

Rewrite the square root of the division \frac{9}{4} as the division of square roots \frac{\sqrt{9}}{\sqrt{4}}. Take the square root of both numerator and denominator.

$$\frac{7}{2}+\frac{3}{2}$$

Since \frac{7}{2} and \frac{3}{2} have the same denominator, add them by adding their numerators.

$$\frac{7+3}{2}$$

Add 7 and 3 to get 10.

$$\frac{10}{2}$$

Divide 10 by 2 to get 5.

$$5$$

Factor

$$5$$