Percentage as Standard for Comparison

Date: 05/12/2003 at 13:42:05
From: Samantha
Subject: More than 100%?

How can you have more than 100% of something?

100% indicates you have a complete item, so how can it be more than 
complete?

Date: 05/12/2003 at 17:05:10
From: Doctor Peterson
Subject: Re: More than 100%?

Hi, Samantha.

Good question!

In some situations, a percentage refers to a part of all there is: I 
can't do more than 100% of a job, or eat more than 100% of the one 
pizza I ordered. More than 100% of the sky can't be covered by clouds, 
and more than 100% of the population can't be poor.

But at other times, a percentage refers to something that is not an 
absolute limit, but just a standard for comparison. For example, the 
orange juice you drink in the morning may contain 100% of your daily 
requirement of vitamin C; if you eat any more foods containing 
vitamin C during the day, you will have eaten more than 100% of the 
RDA. For some vitamins, if you ate too much, it might be bad for you; 
however 130% in this case just means that you didn't need to have as 
much as you did, but it's okay to go beyond that. You've just taken in 
more than the recommended amount.

As another example, you might plan to work 10 hours to put together a 
project. If you keep track of the time you've used, when you have 
spent 6 hours on it you will have used 60% of the planned time. But 
you might find that you aren't quite finished when the planned time 
is used up. If you spend 3 extra hours to make everything just right, 
making a total of 13 hours of work, you will have used 130% of the 
planned time.

Now, if there had been only 10 hours left before the project was due, 
you couldn't have taken 130% of that time. But you had just PLANNED 
to use 10 hours, and it was possible (though possibly bad for your 
health, or your other activities) to use extra time for the project. 
Just like the vitamin C requirement, 100% didn't mean "all there is," 
but just "all we expect." The 130% means that you are "30% over 
budget."

Do you see the difference in these two kinds of situations? The same 
thing happens with fractions. I can eat 1/2 a pizza, or 2/3 of a 
pizza; but I can also eat 2 3/4 pizzas, because the one pizza isn't 
all there is. I can fill 1/2 of my stomach, or 2/3 of my stomach, but 
not 2 3/4 stomachs, because I only have one. Fractions can be greater 
than 1, and percentages can be greater than 100. It's only when the 
fraction or percentage refers to a part of a whole that we can't go 
beyond the whole.

Back to the nutrition label: a bag of potato chips can't be more than 
100% fat, because the fat is PART of the chip, and the part can't be 
bigger than the whole. But it might conceivably have more than 100% 
of your daily requirement of fat, because the fat in the chips is 
only being COMPARED to the amount you're supposed to eat. Nothing 
prevents you from eating more. And finally, the number of calories 
you get from fat today can't be more than 100% of the total calories 
you consume, because they are part of that total. That gives us three 
different ways to express the fat in the chips as a percentage, and 
each percentage has a different meaning.

Here's yet another way that a percentage can be used: A copier may 
have a setting for enlarging or reducing what you copy. The normal 
setting would be 100%, meaning that what comes out is the same size 
as what went in. You can set it to 50%, meaning the copy will be half 
the size of the original. Or, you can set it to 200%, and the copy 
will be twice as large. There's nothing wrong with that, because the 
copy is not part of the original; we are only comparing it to the 
original. And that's what makes the difference.

Here is an old answer to a question like yours:

   More Than 100 Percent
   http://mathforum.org/library/drmath/view/58166.html 

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/