##10##

First, note that ##0.1## just has a ##1## in the ##10^”th”s## position. As a fraction, then, we can represent it as ##0.1 = 1/10##

Next, remember that the multiplicative inverse of any nonzero number is the number whose product with the original number is ##1##. So, for any nonzero number ##x##, the multiplicative inverse of ##x## is ##1/x##. This is because ##x xx 1/x = x/x = 1##

So, if we want the multiplicative inverse of ##0.1##, we look at ##1/0.1## and get

##1/0.1 = 1/(1/10)##

##=1/(1/10) xx 10/10##

(multiplying the numerator and denominator by the same number doesn’t change our value)

##=10/((1/10)xx10)##

##=10/1##

##=10##

So the multiplicative inverse of ##0.1## is ##10##.

As a side note, it’s a useful trick to remember that ##1/(1/x) = x## for any nonzero ##x##. Multiplying by ##x/x = 1## shows it to be true, as done above.