algebra precalculus - Equation solving, why does this simplification work?

$$\dfrac{154}{3.2-x} = \dfrac{66}{1.6-x}$$

How can that equation be simplified as such? :

$$154(1.6-x) = 66(3.2-x)$$

Can someone explain?

Answer

Apart from the fact that, as long as denominators are not zero, the formal definition of fractions is that the two equations are equivalent, we have:

$$\begin{align} \frac{154}{3.2-x} &= \frac{66}{1.6-x}\\ \frac{154}{3.2-x}\cdot (1.6 - x)& = \frac{66}{1.6-x}\cdot (1.6 - x)\\ \frac{154}{3.2-x}\cdot (1.6 - x) &= \frac{66}{\color{red}{(1.6-x)}}\color{red}{\cdot (1.6 - x)}\\ \frac{154}{3.2-x}\cdot (1.6 - x) &= 66\\ \frac{154}{3.2-x}\cdot (1.6 - x)\cdot(3.2-x) &= 66\cdot(3.2 - x)\\ \frac{154}{\color{red}{(3.2-x)}}\cdot (1.6 - x)\cdot\color{red}{(3.2-x)} &= 66\cdot(3.2 - x)\\ 154\cdot (1.6 - x) &= 66\cdot(3.2 - x) \end{align}$$
where all the red parts cancel eachother.