0.1 x 100 = 10
0.1 times 100 is 10. A tenth of one hundred is ten. The same move converts the decimal 0.1 into 10 percent, and it is where the most common percentage mistake begins.
0.1 × 100 = 10
The working
0.1 is one tenth, so this is just 100 divided by 10, which is 10. By decimal point: shift it two places right and 0.1 becomes 1. and then 10.
0.1 as a percentage
Multiplying by 100 is the decimal-to-percent conversion, so 0.1 = 10%. This is the pair most people are really checking when they run the calculation.
The mistake that is a hundred times out
0.1% of 100 is 0.1, not 10. A percent sign means the number has already been divided by a hundred, so a value written as 0.1% is one thousandth, not one tenth. The two answers sit a factor of 100 apart, and because both are small and plausible the error survives a sanity check. Decide which form you are holding before you multiply.
In the same family, 0.1 hours is 6 minutes, since decimal hours convert with 60 rather than 100.
A note for anyone doing this in code
0.1 has no exact representation in binary, so a computer stores it as roughly 0.1000000000000000055. That is why 0.1 + 0.2 prints 0.30000000000000004 in most languages, and why 0.1 x 3 does the same. This particular product is safe: 0.1 x 100 rounds back to exactly 10 in both Python and JavaScript. The result you get here is the exact one either way.