Q:

Intuition behind multiplication of decimals that are greater than 0.1

How can I visualize the multiplication of 2 decimals that both are greater than 0.1? For example, 0.2×0.25. 
I understand that 0.1×0.1 is to:
step 1. divide a size into 10 parts
2. divide 1 of those parts into another 10 parts
3. retrieve 1 part from the product of step 2
Overall, it's equivalent to dividing the initial size into 100 parts, then get 1 part from it.

A:

You can do the same with your example by writing $0.2=\frac 15, 0.25=\frac 14, \frac 15 \cdot \frac 14=\frac 1{20}=0.05$
The problem comes when your decimals do not convert to fractions.  I think it is better to think of doing multiplication, then finding where the decimal point goes.  Counting places past the decimal point works for me.  If you multiply $0.1416 \cdot 0.71828$, ignoring the decimal points you get $101708448$.  Now the number of decmals in the product is the sum of the numbers in the things you multiplied, so there are $9$ decimals and the result is $0.101708448$.  You can also approximate $0.1416 \approx \frac 17, 0.71828 \approx \frac 7{10},$ so the product is about $\frac 1{10}$

