Simplify $\frac{x}{c} - \frac{x}{c-d}$

Question

There's a long time that I don't solve questions like this one. I'm having problems to simplify this one:

$$\frac{x}{c} - \frac{x}{c-d}$$

Answer 1

First recall that \[ \frac{a}{b}-\frac{c}{d}= \frac{ad-bc}{bd} \] Then \[ \frac{x}{c}-\frac{x}{c-d}= x\left(\frac{1}{c}-\frac{1}{c-d}\right)= x\left(\frac{c-d-c}{c(c-d)}\right)= \frac{-xd}{c^2-cd} \] I attempted to be as clear as possible.

Answer 2

Multiply the first term by $\frac{c-d}{c-d}$ and the second term by $\frac{c}{c}$ Then you will have $\frac{x(c-d)-x(c)}{c(c-d)} = \frac{-xd}{c(c-d)}$

Answer 3

If you can be happy with this

\[ \frac{x}{c}-\frac{x}{c-d}= \frac{x}{c}-\frac{c x}{c(c-d)}= \frac{x}{c} (1-\frac{c}{c-d})= \frac{x}{c} (1-\frac{1}{1-\frac{d}{c}}) \]

Answer 4

$\frac{x}{c}-\frac{x}{c-d}= {x}(\frac{1}{c}-\frac{1}{c-d})={x} (\frac{c-d-(c)}{c(c-d)})= \frac{-xd}{c(c-d)}$