Solve frac{r_{1}}{100}+frac{v_{1}-v_{2}}{250}+frac{v_{3}}{200}+frac{v_{3}-v_{2}}{400}+frac{v_{3}-(v_{2}+128)}{500}

$$\frac { r _ { 1 } } { 100 } + \frac { v _ { 1 } - v _ { 2 } } { 250 } + \frac { v _ { 3 } } { 200 } + \frac { v _ { 3 } - v _ { 2 } } { 400 } + \frac { v _ { 3 } - ( v _ { 2 } + 128 ) } { 500 }$$

Evaluate

$$\frac{r_{1}}{100}+\frac{v_{1}}{250}+\frac{19v_{3}}{2000}-\frac{17v_{2}}{2000}-\frac{32}{125}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 100 and 250 is 500. Multiply \frac{r_{1}}{100} times \frac{5}{5}. Multiply \frac{v_{1}-v_{2}}{250} times \frac{2}{2}.

$$\frac{5r_{1}}{500}+\frac{2\left(v_{1}-v_{2}\right)}{500}+\frac{v_{3}}{200}+\frac{v_{3}-v_{2}}{400}+\frac{v_{3}-\left(v_{2}+128\right)}{500}$$

Since \frac{5r_{1}}{500} and \frac{2\left(v_{1}-v_{2}\right)}{500} have the same denominator, add them by adding their numerators.

$$\frac{5r_{1}+2\left(v_{1}-v_{2}\right)}{500}+\frac{v_{3}}{200}+\frac{v_{3}-v_{2}}{400}+\frac{v_{3}-\left(v_{2}+128\right)}{500}$$

Do the multiplications in 5r_{1}+2\left(v_{1}-v_{2}\right).

$$\frac{5r_{1}+2v_{1}-2v_{2}}{500}+\frac{v_{3}}{200}+\frac{v_{3}-v_{2}}{400}+\frac{v_{3}-\left(v_{2}+128\right)}{500}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 500 and 200 is 1000. Multiply \frac{5r_{1}+2v_{1}-2v_{2}}{500} times \frac{2}{2}. Multiply \frac{v_{3}}{200} times \frac{5}{5}.

$$\frac{2\left(5r_{1}+2v_{1}-2v_{2}\right)}{1000}+\frac{5v_{3}}{1000}+\frac{v_{3}-v_{2}}{400}+\frac{v_{3}-\left(v_{2}+128\right)}{500}$$

Since \frac{2\left(5r_{1}+2v_{1}-2v_{2}\right)}{1000} and \frac{5v_{3}}{1000} have the same denominator, add them by adding their numerators.

$$\frac{2\left(5r_{1}+2v_{1}-2v_{2}\right)+5v_{3}}{1000}+\frac{v_{3}-v_{2}}{400}+\frac{v_{3}-\left(v_{2}+128\right)}{500}$$

Do the multiplications in 2\left(5r_{1}+2v_{1}-2v_{2}\right)+5v_{3}.

$$\frac{10r_{1}+4v_{1}-4v_{2}+5v_{3}}{1000}+\frac{v_{3}-v_{2}}{400}+\frac{v_{3}-\left(v_{2}+128\right)}{500}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 1000 and 400 is 2000. Multiply \frac{10r_{1}+4v_{1}-4v_{2}+5v_{3}}{1000} times \frac{2}{2}. Multiply \frac{v_{3}-v_{2}}{400} times \frac{5}{5}.

$$\frac{2\left(10r_{1}+4v_{1}-4v_{2}+5v_{3}\right)}{2000}+\frac{5\left(v_{3}-v_{2}\right)}{2000}+\frac{v_{3}-\left(v_{2}+128\right)}{500}$$

Since \frac{2\left(10r_{1}+4v_{1}-4v_{2}+5v_{3}\right)}{2000} and \frac{5\left(v_{3}-v_{2}\right)}{2000} have the same denominator, add them by adding their numerators.

$$\frac{2\left(10r_{1}+4v_{1}-4v_{2}+5v_{3}\right)+5\left(v_{3}-v_{2}\right)}{2000}+\frac{v_{3}-\left(v_{2}+128\right)}{500}$$

Do the multiplications in 2\left(10r_{1}+4v_{1}-4v_{2}+5v_{3}\right)+5\left(v_{3}-v_{2}\right).

$$\frac{20r_{1}+8v_{1}-8v_{2}+10v_{3}+5v_{3}-5v_{2}}{2000}+\frac{v_{3}-\left(v_{2}+128\right)}{500}$$

Combine like terms in 20r_{1}+8v_{1}-8v_{2}+10v_{3}+5v_{3}-5v_{2}.

$$\frac{20r_{1}+8v_{1}-13v_{2}+15v_{3}}{2000}+\frac{v_{3}-\left(v_{2}+128\right)}{500}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2000 and 500 is 2000. Multiply \frac{v_{3}-\left(v_{2}+128\right)}{500} times \frac{4}{4}.

$$\frac{20r_{1}+8v_{1}-13v_{2}+15v_{3}}{2000}+\frac{4\left(v_{3}-\left(v_{2}+128\right)\right)}{2000}$$

Since \frac{20r_{1}+8v_{1}-13v_{2}+15v_{3}}{2000} and \frac{4\left(v_{3}-\left(v_{2}+128\right)\right)}{2000} have the same denominator, add them by adding their numerators.

$$\frac{20r_{1}+8v_{1}-13v_{2}+15v_{3}+4\left(v_{3}-\left(v_{2}+128\right)\right)}{2000}$$

Do the multiplications in 20r_{1}+8v_{1}-13v_{2}+15v_{3}+4\left(v_{3}-\left(v_{2}+128\right)\right).

$$\frac{20r_{1}+8v_{1}-13v_{2}+15v_{3}+4v_{3}-4v_{2}-512}{2000}$$

Combine like terms in 20r_{1}+8v_{1}-13v_{2}+15v_{3}+4v_{3}-4v_{2}-512.

$$\frac{20r_{1}+8v_{1}-17v_{2}+19v_{3}-512}{2000}$$

Expand

$$\frac{r_{1}}{100}+\frac{v_{1}}{250}+\frac{19v_{3}}{2000}-\frac{17v_{2}}{2000}-\frac{32}{125}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 100 and 250 is 500. Multiply \frac{r_{1}}{100} times \frac{5}{5}. Multiply \frac{v_{1}-v_{2}}{250} times \frac{2}{2}.

$$\frac{5r_{1}}{500}+\frac{2\left(v_{1}-v_{2}\right)}{500}+\frac{v_{3}}{200}+\frac{v_{3}-v_{2}}{400}+\frac{v_{3}-\left(v_{2}+128\right)}{500}$$

Since \frac{5r_{1}}{500} and \frac{2\left(v_{1}-v_{2}\right)}{500} have the same denominator, add them by adding their numerators.

$$\frac{5r_{1}+2\left(v_{1}-v_{2}\right)}{500}+\frac{v_{3}}{200}+\frac{v_{3}-v_{2}}{400}+\frac{v_{3}-\left(v_{2}+128\right)}{500}$$

Do the multiplications in 5r_{1}+2\left(v_{1}-v_{2}\right).

$$\frac{5r_{1}+2v_{1}-2v_{2}}{500}+\frac{v_{3}}{200}+\frac{v_{3}-v_{2}}{400}+\frac{v_{3}-\left(v_{2}+128\right)}{500}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 500 and 200 is 1000. Multiply \frac{5r_{1}+2v_{1}-2v_{2}}{500} times \frac{2}{2}. Multiply \frac{v_{3}}{200} times \frac{5}{5}.

$$\frac{2\left(5r_{1}+2v_{1}-2v_{2}\right)}{1000}+\frac{5v_{3}}{1000}+\frac{v_{3}-v_{2}}{400}+\frac{v_{3}-\left(v_{2}+128\right)}{500}$$

Since \frac{2\left(5r_{1}+2v_{1}-2v_{2}\right)}{1000} and \frac{5v_{3}}{1000} have the same denominator, add them by adding their numerators.

$$\frac{2\left(5r_{1}+2v_{1}-2v_{2}\right)+5v_{3}}{1000}+\frac{v_{3}-v_{2}}{400}+\frac{v_{3}-\left(v_{2}+128\right)}{500}$$

Do the multiplications in 2\left(5r_{1}+2v_{1}-2v_{2}\right)+5v_{3}.

$$\frac{10r_{1}+4v_{1}-4v_{2}+5v_{3}}{1000}+\frac{v_{3}-v_{2}}{400}+\frac{v_{3}-\left(v_{2}+128\right)}{500}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 1000 and 400 is 2000. Multiply \frac{10r_{1}+4v_{1}-4v_{2}+5v_{3}}{1000} times \frac{2}{2}. Multiply \frac{v_{3}-v_{2}}{400} times \frac{5}{5}.

$$\frac{2\left(10r_{1}+4v_{1}-4v_{2}+5v_{3}\right)}{2000}+\frac{5\left(v_{3}-v_{2}\right)}{2000}+\frac{v_{3}-\left(v_{2}+128\right)}{500}$$

Since \frac{2\left(10r_{1}+4v_{1}-4v_{2}+5v_{3}\right)}{2000} and \frac{5\left(v_{3}-v_{2}\right)}{2000} have the same denominator, add them by adding their numerators.

$$\frac{2\left(10r_{1}+4v_{1}-4v_{2}+5v_{3}\right)+5\left(v_{3}-v_{2}\right)}{2000}+\frac{v_{3}-\left(v_{2}+128\right)}{500}$$

Do the multiplications in 2\left(10r_{1}+4v_{1}-4v_{2}+5v_{3}\right)+5\left(v_{3}-v_{2}\right).

$$\frac{20r_{1}+8v_{1}-8v_{2}+10v_{3}+5v_{3}-5v_{2}}{2000}+\frac{v_{3}-\left(v_{2}+128\right)}{500}$$

Combine like terms in 20r_{1}+8v_{1}-8v_{2}+10v_{3}+5v_{3}-5v_{2}.

$$\frac{20r_{1}+8v_{1}-13v_{2}+15v_{3}}{2000}+\frac{v_{3}-\left(v_{2}+128\right)}{500}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2000 and 500 is 2000. Multiply \frac{v_{3}-\left(v_{2}+128\right)}{500} times \frac{4}{4}.

$$\frac{20r_{1}+8v_{1}-13v_{2}+15v_{3}}{2000}+\frac{4\left(v_{3}-\left(v_{2}+128\right)\right)}{2000}$$

Since \frac{20r_{1}+8v_{1}-13v_{2}+15v_{3}}{2000} and \frac{4\left(v_{3}-\left(v_{2}+128\right)\right)}{2000} have the same denominator, add them by adding their numerators.

$$\frac{20r_{1}+8v_{1}-13v_{2}+15v_{3}+4\left(v_{3}-\left(v_{2}+128\right)\right)}{2000}$$

Do the multiplications in 20r_{1}+8v_{1}-13v_{2}+15v_{3}+4\left(v_{3}-\left(v_{2}+128\right)\right).

$$\frac{20r_{1}+8v_{1}-13v_{2}+15v_{3}+4v_{3}-4v_{2}-512}{2000}$$

Combine like terms in 20r_{1}+8v_{1}-13v_{2}+15v_{3}+4v_{3}-4v_{2}-512.

$$\frac{20r_{1}+8v_{1}-17v_{2}+19v_{3}-512}{2000}$$