I've struggled with this concept and have generally interpreted it one way for all of my life, which leads me to believe people are incorrect when they state the other form. And honestly I'm not really sure this is even the correct SE site but I figured there wasn't a better choice.
Here it is:
If I have a basket of 10 apples and I remove 5 apples, do I have 50% fewer apples? Or 100% fewer apples? I tend to consider the correct phrasing by performing:
(original amount as a fraction / (new amount as a fraction) - 1) * 100%
So therefore 5 apples is now 0.5 of what I previously had, giving
((1 / .5) - 1) * 100%, giving 100% less. I think this phrasing I choose is motivated by the fact that when someone says:
Basket A has 10 apples and basket B has 20 apples, therefore basket B has 100% more apples. So it leads me to believe twice as much is 100% more, and then half as much would be 100% less. It would not make sense to me if twice as many apples were 100% more, yet half as many apples were 50% less.
Where this gets confusing
If my interpretation above is correct, then I commonly see people making the mistake when dealing with different amounts that don't make this obvious. For instance about a year ago, a professor of mine said:
Processor A runs at 9/10 the speed of processor B, therefore it is 10% slower. I raised my hand and explained it should be
11.1% slower since
(1 / 0.9) - 1 * 100% gives 11.1%, arguing that if processor B ran at half the speed, we wouldn't claim it ran
He agreed, but I'm still not sure I use this phrase correctly.
Going along the same lines, I generally struggle with the differences between
% of an amount vs. % more than an amount, for amounts greater than 100.
It seems logical than
50% of is plain multiplcation. Therefore
Qty A is 50% of Qty B means
Qty A = Qty B * 0.5. And also
50% more than would be
100% + 0.5 * amount, so
Qty A is 50% more than Qty B means
Qty A = 1.00 + Qty B * 0.5.
So if that is true, then
of means pure multiplication, while
more than means
multiplication by 2nd amount and added to the original.
But this seems to be very confusing for amount larger than 100%.
Qty A is 200% of Qty B So
Qty A = Qty B * 2? And
Qty A is 200% more than Qty B is
Qty A = 1.00 + 2 * Qty B? That means the phrasing
200% more than really means 3x as much!