Why the fraction equivalent of 0.1 is 1/10?

To try to explain, I'm going to give you a new way to read fractions. For a/b, say "b of them make a". For example, 1/3 is "3 of them make 1", because 1/3+1/3+1/3=1. 2/7 is "7 of them make 2", because 2/7+2/7+2/7+2/7+2/7+2/7+2/7=2. This works for whole numbers a and b.

You can change the numbers like so: "3 of them make 1" is the same as "6 of them make 2", and so 1/3=2/6. Similarly "7 of them make 2" is the same as "14 of them make 4" so 2/7=4/14. "2 of them make 12" is the same as "1 of them makes 6", so 12/2=6/1=6.

Now 1/0.1 is "0.1 of them make 1". Whatever "they" are, you only need 0.1 of them to make 1, so "they" must be bigger than 1. How much bigger? Following the pattern, "0.1 of them make 1" is the same as "0.2 of them make 2", "0.5 of them make 5", and "1 of them makes 10". So 1/0.1=10

It is equivalent because of the way you solved it.