Solve for r 1/r=1/2+1/3

To write ${\displaystyle \frac{1}{2}}$ as a fraction with a common denominator, multiply by ${\displaystyle \frac{3}{3}}$.

To write ${\displaystyle \frac{1}{3}}$ as a fraction with a common denominator, multiply by ${\displaystyle \frac{2}{2}}$.

Write each expression with a common denominator of ${\displaystyle 6}$, by multiplying each by an appropriate factor of ${\displaystyle 1}$.

Combine the numerators over the common denominator.

Add ${\displaystyle 3}$ and ${\displaystyle 2}$.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

Since ${\displaystyle r,6}$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part ${\displaystyle 1,6}$ then find LCM for the variable part ${\displaystyle r^1}$.

The LCM is the smallest positive number that all of the numbers divide into evenly.

The number ${\displaystyle 1}$ is not a prime number because it only has one positive factor, which is itself.

${\displaystyle 6}$ has factors of ${\displaystyle 2}$ and ${\displaystyle 3}$.

Multiply ${\displaystyle 2}$ by ${\displaystyle 3}$.

The factor for ${\displaystyle r^1}$ is ${\displaystyle r}$ itself.

The LCM of ${\displaystyle r^1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

The LCM for ${\displaystyle r,6}$ is the numeric part ${\displaystyle 6}$ multiplied by the variable part.

Multiply each term in ${\displaystyle \frac{1}{r}=\frac{5}{6}}$ by ${\displaystyle 6r}$.

Simplify the left side.

Simplify the right side.

Rewrite the equation as ${\displaystyle 5r=6}$.

Divide each term in ${\displaystyle 5r=6}$ by ${\displaystyle 5}$ and simplify.