Fraction confusion

Question

I read in a set of memoranda that if $ \frac{b-x}{x}=\frac{b}{a}$, then $$x = \frac{ab}{a+b}$$ How is this true? I tried working it out but I could not understand. Please help.

Answer 1

You have

$$\frac{b-x}{x} = \frac{b}{a} \Leftrightarrow b-x = x \frac{b}{a}$$

That implies

$$b = x\left(1+\frac{b}{a}\right)$$

And finally, you get

$$x= \frac{b}{1+\frac{b}{a}}$$

And it's the same as

$$x= \frac{ab}{a+b}$$

Answer 2

$$\frac{b-x}{x} = \frac{b}{x} - 1 $$ I think this is the step you're looking for : $$\frac{b-x}{x} = \frac{b}{a}$$

$$\frac{b}{x} = \frac{b}{a}+1 = \frac{a+b}{a}$$

I think you're now able to do it...