Is half of an amount stated as 50% less or 100% less? [closed]

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 50% slower

He agreed, but I'm still not sure I use this phrase correctly.

Additionally

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%.

For example: 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!

The correct expression is 50% fewer generally, as 50% of 10 is of course 5, while 100% less would be exactly zero.

Notice that 100% more would be 20 and that 50% more would be 15.

Then again, 100% of the original would be the same (10), while 300% would be 30.

I was a high school math teacher for a little while, and explaining the connection between English phrases like this and the corresponding math is a huge part of teaching story problems. Most of the math teachers I spoke with at the time agreed that these concepts should be taught in both English and math classes, and the English teachers were beginning to have story problem segments in their classes.

The more than/less than phrases are based on addition and subtraction, and the of phrasing is based on multiplication.

x% more than / x% less than

The phrase more than indicates addition, and the phrase less than indicates subtraction. When we say 50% more we mean to add 50% of the original amount. Similarly, when we say 50% less, we mean to subtract 50% of the original value.

x% more than y = y + y*(x/100)

So 50% more than 10 is 10 + 5 = 15, and 250% more than 10 is 10 + 25 = 35.

Note that the difference between the 250% more value and the 50% more value is 35 - 15 = 20, which is 200% of the original value. This is as it should be, as 250% - 50% = 200%.

x% less than y = y - y*(x/100)

So 50% less than 10 is 10 - 5 = 5. We don't often say less than with a percentage greater than 100%, but 250% less than 10 is 10 - 25 = -15.

Common Confusion

I do sometimes see this phrasing used incorrectly when speakers attempt to reverse statements. This will lead to one speaker saying something along the lines of "we had 20% more revenue this year than last year", and then another speaker will say something like "...last year we had 20% less revenue", but this second statement is not correct.

This is because if A is x% less than B, it is not the case that B is x% more than A. Example:

• 8 is 20% less than 10. 10 is 25% more than 8.
• 9 is 10% less than 10. 10 is 11.1% more than 9.
• 5 is 50% less than 10. 10 is 100% more than 5.
• 5 is 75% less than 20. 20 is 300% more than 5.

I expect you probably noticed this, and concluded that you were getting one of the numbers wrong because the values did not match. It's okay, they're not supposed to match.

Note: The only place I can think of where I've seen percentages handled in the way you describe is in the math of some video games. The phrasing of the effects in question is usually not 50% less but instead something like 50% reduced _____. Specifically, in the Borderlands series of games, having a 100% reduction in something results in 50% as much of the thing. For instance, 100% damage reduction results in taking 50% damage, 200% damage reduction results in taking 33% damage, 300% damage reduction results in taking 25% damage, and so on.