$\displaystyle \frac{2}{15x} + \frac{7}{18}$
Can someone please go through the details on how to solve this? The correct answer is$\displaystyle \frac{12+35x}{90x}$
Edit: Title should read variable Fractions obviously my mistake.
I think that clarifies that question a few more on the same subject though.
1.in this example for canceling variables we have
$\displaystyle \frac{2+x}{3} = \frac{2}{x} + \frac{x}{x} = \frac{2}{x} +1$
Note that in this example the x in the denominator is applied to the 2 and x in the numerator. Shown here $\displaystyle \frac{2+x}{x}$, the x in the denominator goes to both numerators thus $\displaystyle \frac{2}{x} + \frac{x}{x}$
2.However in this example
$\displaystyle \frac{(2 + 3x)(x-1)}{2 + 3x}$ For the solution the author does not apply the whole denominator to both of the numerators and this is confusing me greatly. Working this problem out we have $\displaystyle \frac{(2 + 3x)(x-1)}{2 + 3x}=\frac{2+3x}{2+3x} * \frac{x-1}{1}= x -1$
why is the 2 + 3x not applied to both parts of this numerator resulting in $\displaystyle \frac{2+3x}{2+3x} * \frac{x-1}{2+3x}$?
thanks in advance
The first expression is made up of addends (terms that are added together). You must have a common denominator to combine them.
Your second expression is a product. You do not need a common denominator. You simply multiply numerators and denominators and factor out any common elements in the product or along the way.
$\displaystyle \frac{(2 + 3x)(x-1)}{2 + 3x}=\frac{2+3x}{2+3x} * \frac{x-1}{1}=\not[\frac{2+3x}{2+3x}] * \frac{x-1}{1}= \frac{1}{1} * \frac{x-1}{1}=x -1$
$\displaystyle \frac{2+3x}{2+3x}$ reduces to $\displaystyle \frac{1}{1}$
Look at a numerical example:
$\displaystyle \frac{(2)(3)}{6} \neq \frac{2}{6} * \frac{3}{6} \neq \frac{6}{36} \neq \frac{1}{6}$
Instead, it's this:
$\displaystyle \frac{(2)(3)}{6}=\frac{2}{6} * \frac{3}{1} = \frac{6}{6} = 1$
Generally,
$\displaystyle \frac{ab}{c}=\frac{a}{c}*\frac{b}{1}$
If we did it as you suggest, we would have
$\displaystyle \frac{ab}{c}=\frac{a}{c} * \frac{b}{c} = \frac{ab}{c^2} \neq \frac{ab}{c}$