Multiplying by $10$'s place voodoo: Why is $30\times 50= 15\times 100$? This is a very trivial question, but I can't seem to reason out why 
$$30\times 50= 15\times 100$$ 
As a kid, I never really thought about why it works, but now I can't figure it out and the idea is really troubling me. I understand that we can break up the problem like this: 
$$3\times 10\times 5\times 10$$ 
but at this point I feel like I've lost the intuitive aspect of the problem. Can someone plz help and provide some intuition?

If you're looking for an intuitive explanation, maybe consider it like a change of units. $30\cdot50$ is, say, the area of a rectangle 30 millimeters by 50 millimeters in square millimeters, but you can switch back and forth between units. So Instead you think of it as a rectangle 3 centimeters by 5 centimeters, and so the area is just 15 square centimeters, which, converting back to millimeters, is $15 \text{cm}^2 \cdot \left(\frac{10\text{mm}}{1 \text{cm}}\right)^2=1500\text{mm}$
