Solve 1/1{1/30+frac{1/170}+1/frac{1{x}+frac{1}{85}}}+frac{1}{34}

$$\frac { 1 } { \frac { 1 } { \frac { 1 } { 30 } + \frac { 1 } { 170 } } + \frac { 1 } { \frac { 1 } { x } + \frac { 1 } { 85 } } } + \frac { 1 } { 34 }$$

Evaluate

$$\frac{x+35}{2\left(13x+255\right)}$$

Short Solution Steps

Least common multiple of 30 and 170 is 510. Convert \frac{1}{30} and \frac{1}{170} to fractions with denominator 510.

$$\frac{1}{\frac{1}{\frac{17}{510}+\frac{3}{510}}+\frac{1}{\frac{1}{x}+\frac{1}{85}}}+\frac{1}{34}$$

Since \frac{17}{510} and \frac{3}{510} have the same denominator, add them by adding their numerators.

$$\frac{1}{\frac{1}{\frac{17+3}{510}}+\frac{1}{\frac{1}{x}+\frac{1}{85}}}+\frac{1}{34}$$

Add 17 and 3 to get 20.

$$\frac{1}{\frac{1}{\frac{20}{510}}+\frac{1}{\frac{1}{x}+\frac{1}{85}}}+\frac{1}{34}$$

Reduce the fraction \frac{20}{510} to lowest terms by extracting and canceling out 10.

$$\frac{1}{\frac{1}{\frac{2}{51}}+\frac{1}{\frac{1}{x}+\frac{1}{85}}}+\frac{1}{34}$$

Divide 1 by \frac{2}{51} by multiplying 1 by the reciprocal of \frac{2}{51}.

$$\frac{1}{1\times \frac{51}{2}+\frac{1}{\frac{1}{x}+\frac{1}{85}}}+\frac{1}{34}$$

Multiply 1 and \frac{51}{2} to get \frac{51}{2}.

$$\frac{1}{\frac{51}{2}+\frac{1}{\frac{1}{x}+\frac{1}{85}}}+\frac{1}{34}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and 85 is 85x. Multiply \frac{1}{x} times \frac{85}{85}. Multiply \frac{1}{85} times \frac{x}{x}.

$$\frac{1}{\frac{51}{2}+\frac{1}{\frac{85}{85x}+\frac{x}{85x}}}+\frac{1}{34}$$

Since \frac{85}{85x} and \frac{x}{85x} have the same denominator, add them by adding their numerators.

$$\frac{1}{\frac{51}{2}+\frac{1}{\frac{85+x}{85x}}}+\frac{1}{34}$$

Divide 1 by \frac{85+x}{85x} by multiplying 1 by the reciprocal of \frac{85+x}{85x}.

$$\frac{1}{\frac{51}{2}+\frac{85x}{85+x}}+\frac{1}{34}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 85+x is 2\left(x+85\right). Multiply \frac{51}{2} times \frac{x+85}{x+85}. Multiply \frac{85x}{85+x} times \frac{2}{2}.

$$\frac{1}{\frac{51\left(x+85\right)}{2\left(x+85\right)}+\frac{2\times 85x}{2\left(x+85\right)}}+\frac{1}{34}$$

Since \frac{51\left(x+85\right)}{2\left(x+85\right)} and \frac{2\times 85x}{2\left(x+85\right)} have the same denominator, add them by adding their numerators.

$$\frac{1}{\frac{51\left(x+85\right)+2\times 85x}{2\left(x+85\right)}}+\frac{1}{34}$$

Do the multiplications in 51\left(x+85\right)+2\times 85x.

$$\frac{1}{\frac{51x+4335+170x}{2\left(x+85\right)}}+\frac{1}{34}$$

Combine like terms in 51x+4335+170x.

$$\frac{1}{\frac{221x+4335}{2\left(x+85\right)}}+\frac{1}{34}$$

Divide 1 by \frac{221x+4335}{2\left(x+85\right)} by multiplying 1 by the reciprocal of \frac{221x+4335}{2\left(x+85\right)}.

$$\frac{2\left(x+85\right)}{221x+4335}+\frac{1}{34}$$

Factor 221x+4335.

$$\frac{2\left(x+85\right)}{17\left(13x+255\right)}+\frac{1}{34}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 17\left(13x+255\right) and 34 is 34\left(13x+255\right). Multiply \frac{2\left(x+85\right)}{17\left(13x+255\right)} times \frac{2}{2}. Multiply \frac{1}{34} times \frac{13x+255}{13x+255}.

$$\frac{2\times 2\left(x+85\right)}{34\left(13x+255\right)}+\frac{13x+255}{34\left(13x+255\right)}$$

Since \frac{2\times 2\left(x+85\right)}{34\left(13x+255\right)} and \frac{13x+255}{34\left(13x+255\right)} have the same denominator, add them by adding their numerators.

$$\frac{2\times 2\left(x+85\right)+13x+255}{34\left(13x+255\right)}$$

Do the multiplications in 2\times 2\left(x+85\right)+13x+255.

$$\frac{4x+340+13x+255}{34\left(13x+255\right)}$$

Combine like terms in 4x+340+13x+255.

$$\frac{17x+595}{34\left(13x+255\right)}$$

Factor the expressions that are not already factored in \frac{17x+595}{34\left(13x+255\right)}.

$$\frac{17\left(x+35\right)}{34\left(13x+255\right)}$$

Cancel out 17 in both numerator and denominator.

$$\frac{x+35}{2\left(13x+255\right)}$$

Expand 2\left(13x+255\right).

$$\frac{x+35}{26x+510}$$

Factor

$$\frac{x+35}{2\left(13x+255\right)}$$