How to divide decimals by decimals - simple explanation and why it works
To divide by a decimal, first change the ENTIRE division problem to another with a whole-number divisor, yet with the same answer. How? Simply multiply both numbers in the division problem (the dividend and the divisor) by 10 repeatedly, until the divisor is a whole number. (The dividend doesn't matter; it can be a decimal).
The shortcut for multiplying by 10 is of course that you move the decimal point, and most school books only give you the shortcut, without explaining this principle for division of decimals that in reality, we are MULTIPLYING both the dividend & divisor by 10, 100, 1000 (powers of ten).
We can do this process, because in it, the answer (the quotient) does not change. Think about it: 0.2 goes into 0.7 as many times as 2 goes into 7, or 20 goes into 70. Similarly, if you have a division problem with a decimal divisor, such as 4.392 / 0.7, you're solving how many times 0.7 goes into 4.392. But that is the same as finding how many times 7 goes into 43.92.
This process works equally well whether we are using mental math to divide or long division.
See also
Math Mammoth Decimals 2 — downloadable worktext
Divide decimals with mental math, part 1: sharing divisions (whole-number divisor) — video lesson