A recent discussion with a student I was tutoring face to face, about an ambiguously worded problem, led me to gather a few answers we’ve given related to the words we use associated with percentages.

## Ambiguous percentages

Here is a question from 2003:

Clarifying Percentages vs. Percentage Points

What is the difference between measuring using percentages versus measuring using percentage points?  What is meant by a percentage point?

I answered, starting with why we need the terms:

The term "percentage point" is used to get around an ambiguity in English when we are comparing two different percentages.  The problem is that "percent" implicitly refers to a relative change (some fraction of an original amount, like a salary increase of 10%) rather than an absolute change (some specified amount, like a salary increase of $1000). What do we say when we want to treat a percentage as an absolute amount? If, for example, the current tax rate were 10% and we increased it to 12%, we might say that we increased it by 2 percent. But that would be taken to mean that we increased it by 2% _of the original 10%_ (that is, by 2/100 of 10%, or 0.2%), to 10.2%. The question is, are we using "percent" to mean one of the units called percent, or a percentage of that percentage? To avoid this problem, we say instead that we are increasing the tax rate by "two percentage points". This unambiguously refers to the number 2% itself as a unit, rather than to 2% of something else. So the percent increase from 10% to 12% is the difference, 2%, divided by the original amount, 10%, which is 0.20, or 20%. But the percentage point increase is just the 2%. If we replaced the percentages given with a unit, say dollars, we would say that the percent increase from$10 to $12 is the difference,$2, divided by the original amount, $10, which is 0.20, or 20%; the dollar increase is$2. This is what I mean by saying we treat percentage points as a unit.

On the other hand, if we actually wanted to say that the tax increased to 10.2%, it would be a good idea to clarify that as well, perhaps by saying explicitly that it increased by 2% of its old rate, or by stating the old and new values.  Technically, however, it is correct to say that it increased by 2%.

In summary, I wouldn't say that we "measure" using one or the other; rather, we use the one term to clarify our meaning where the other would be ambiguous, because we are switching perspective from thinking of a percentage as a fraction of something else, to treating it as a number that stands on its own.  A percentage change is a difference divided by some base number, while a percentage _point_ change is a simple addition or subtraction.

The issue with my student was in this area. She was working on the following problem:

In 1950, Americans spent 22% of their budget on food. This has decreased at an average rate of approximately 0.25% per year since then.

Find a linear function in slope-intercept form that models this description. It should model the percentage of total spending, p(x), by Americans x years after 1950.

She knew that a decrease of 0.25% means subtracting 0.25% of one year’s amount from that amount, equivalent to multiplying by 99.75% (1 – 0.0025 = 0.9975) each year. But this didn’t fit with anything she had learned about. (In fact, it would correspond to an exponential decrease, not a linear function, but she hadn’t learned about that.)

In answer, I pointed out that the statement was ambiguous, and we need to interpret it in a way that is consistent with the mention of a linear function. What the problem should have said, for clarity, is

In 1950, Americans spent 22% of their budget on food. This has decreased at an average rate of approximately 0.25 percentage points per year since then.

That is, the number 22 is to be reduced by 0.25 each year; that will be the slope of the function. Both 22 and 0.25 are to be thought of as measured in the same unit, percentage points, rather than the latter being a percentage of the former.

## A fraction of a percent

A different sort of issue arises when we mix together a fraction and a percentage in one measurement. Here is a question from 2005:

How to Pronounce a Fraction of a Percentage

Is it correct to say "one tenth of one percent" as opposed to saying "one tenth percent" for 0.1%?  Why or why not?

It seems wrong to refer to a percentage as a fraction of a percentage, but news people and the financial industry do it all the time.  I can't seem to find out what the precedence is for this.

Both mean the same thing; neither is wrong.  The reason for the longer phrase is probably the usual reason for using a longer phrase: to avoid ambiguity or possible confusion.  Many people are not quite clear on what percentages mean, and might well take "one-tenth percent" as if it were just "one-tenth" (which, of course, is really 10%).  So people tend to expand it to make it clear that they are using BOTH a fraction AND a percent; that is,

0.1% = 1/10 * 1% = 1/10 * 1/100 = 1/1000

I suppose you could compare this to using "a quarter OF A dollar" or "twenty-five hundredths OF a dollar", rather than just reading "\$0.25" as "a quarter dollar" or "twenty-five hundredths dollars".

The latter just feels subtly wrong (despite the fact that American coins do say “quarter dollar”). I imagine it is not much more than habit, just as many other aspects of language feel right or wrong though we can’t point to a rule for it, or explain why it should be that way.

I was reminded, in writing this just now, of a fascinating conversation with a French reader in 2002 about a different issue that, we decided, depended heavily on what language we were using:

Use of Plural with Decimal Numbers

In part of my answer to this question about whether we write, for example, 1.5 degrees or 1.5 degree (we say the former in English, while they say the latter in French, for a very interesting reason), I said this:

My understanding is that we consider ONLY the number 1 as singular; in particular, zero is a plural: we say "0 degrees" or "0 (no) apples," not "0 degree" or "0 apple." We do not use fractions as adjectives at all, but say "half (of) an apple" or "two thirds of a degree" with the fraction standing alone as a noun phrase, so it would not be quite accurate to say that a proper fraction is singular. With a mixed number, we tend to use a plural: "one and a half apples."

This ties in to what I said above about “a quarter (of a) dollar”; possibly the real reason we say “a quarter of a percent” is exactly the same: The fraction is treated as a noun, not as an adjective or adverb; and “percent” is treated as a unit.

## Percent vs. percentage

A distantly related issue came up in the following question from 2008:

Difference between Percent and Percentage

What is the difference between percent and percentage are there any difference?  I think both are out of 100.

The only difference is in how they are used grammatically--and people differ even on that.  I take "percentage" to refer to the concept, and "percent" to be a unit, much like "voltage" vs. "volts" in electricity, or "mileage" vs. "miles" in distance.  The mileage you put on a car during a trip might be 40 miles; the percentage of people who expect gas prices to rise might be 40 percent (40%).

Here is one dictionary's take on it (www.merriam-webster.com):

Percentage: noun

1 a: a part of a whole expressed in hundredths
<a high percentage of students attended>
b: the result obtained by multiplying a number by a percent
<the percentage equals the rate times the base>

: in the hundred : of each hundred

The main difference is that they report "percentage" as a noun and "percent" as an adverb.  That fits my understanding.  Note that "percentage" is not used with a number, while "percent" is (and not without).

Looking at the current definitions, I see that I must have missed two of three entries:

: in the hundred : of each hundred

percent (noun)

1 plural percent

a : one part in a hundred

b : percentage
a large percent of their income

2 percents plural, British : securities bearing a specified rate of interest

1 : reckoned on the basis of a whole divided into 100 parts

2 : paying interest at a specified percent

Most uses are probably nouns, as in “20 percent of people”, though the adjective use is also common (“a 20 percent solution”). But note that they say “percent” is also used as a synonym for “percentage”, which I would take as a concession to common but mistaken usage.

I then referred to a previous question from 2002:

Percent vs. Percentage

There, Doctor Sarah quoted a dictionary and two usage books.

Percent or Percentage?

Would you please explain the exact difference between the words percent and percentage?  I have used textbooks that use them as if they are the same.  I have have always explained it as percent is a % and percentage is a number that is the same unit as the base number.

This questioner seems to take “percentage” in sense 1b from my dictionary reference above, as the amount itself rather than the number of hundredths, which I have seen used in some textbooks but don’t think I really use in practice. (They might say “percentage = percent times whole”; I’d rather say, “part = percentage times whole”.) I’m not sure either Doctor Sarah or I recognized this detail. Her response was:

You're on the right track, and even a regular dictionary can help with this.

percent -  one part in a hundred

percentage -  a part of a whole expressed in
hundredths; the result obtained by
multiplying a number by a percent

From the Guide to Grammar and Writing on the Web ("Notorious Confusables"):

http://ccc.commnet.edu/grammar/notorious2.htm

"We use the word percent as part of a numerical
expression (e.g., Only two percent of the
students failed.). We use the word percentage
to suggest a portion (e.g., The percentage of
students who fail has decreased.)."

Unfortunately, you will find percent and percentage incorrectly used everywhere on the Web and in textbooks.

Although the dictionary definition quoted can be taken as representing the part of the whole itself, not the fraction, the example from the grammar site doesn’t have that meaning, as it is not the actual number of students who fail, but the fraction, that has decreased.

The same page includes a 2003 question on the same topic:

Is there a difference between the meaning of these two words, or are they totally interchangeable?

I always thought the word percent required the correct notation using the symbol % and that percentage was referred to as AN AMOUNT BASED ON A GIVEN TOTAL, NOT NECESSARILY BASED ON 100.

For example: Given 4/16
The percentage is 4 out of a total of 16, the percent is 25%

I replied:

To answer your specific question, I would say "the fraction used is 4 out of 16; the percentage is 25% [read as 25 percent]." That is, both "percent" and "percentage" refer to an amount "out of 100" (since that is what "per cent" means), and the only difference is how they are used in a sentence. We (should) use "percent" only in phrases like "25 percent" where it can be directly replaced by the phrase "out of 100"; we use "percentage" as a name for the concept.

Although I don’t always defer to dictionaries in areas related to math, I do like to refer to them, and especially to what they say about common usage.

My American Heritage dictionary has this usage note:

_Percent_ and _percentage_ are both used to
express quantity with relation to a whole.
_Percent_ is employed only specifically and
always with a number or numeral.
_Percentage_ is never preceded by such a
figure, but should be qualified by a
general term to indicate size (since
_percentage_ does not necessarily imply
smallness). The number of the noun that
follows _percent_ or _percentage_, or is
understood to follow them, governs the
choice of the verb: _Forty percent of his
estate is in securities. A large
percentage of the patients are children._

Because they are probably looking largely at non-technical material, I think they have missed some usages of “percentage”, as all their examples of it tend to be, as they say, with “a general term”. It can also be specific: the percentage of patients who are children is  75%, or whatever.

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