Percentage nature

Question

In mathematical expressions, the internationally recognized symbol % (percent) may be used with the SI to represent the number 0.01. Thus, it can be used to express the values of dimensionless quantities. According to Le Système international d’unités/The International System of Units, (Brochure sur le SI/SI brochure), 2006.

In mathematics, a percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", or the abbreviations "pct.", "pct"; sometimes the abbreviation "pc" is also used.[1] A percentage is a dimensionless number (pure number). Based on Wikipedia, https://en.wikipedia.org/wiki/Percentage

Is percentage considered as a number?, e.i. can I add a number to percentage?

Is the following considered as a valid question?

Evaluate 30+10%

Answer: 30+10%=30.1

Answer 1

I agree with you that $30+10\%=30+10\cdot 0.01=30.1$. However, you asked "Is the following considered a valid question?" and I'm afraid that it would be very confusing for students. That is, if a teacher asked "Evaluate $30+10\%$" in an exam, many students (and their parents) would complain because they would find it "unclear." They are not accustomed to having pure numbers added with percentages.

Answer 2

Percentage is not a number: it's only a form or notation for a number, as well as, for example, scientific notation.

I write scientific notation not by accident. Consider, for any numer $c$, the following factorization, with $n$ arbitrary integer: $$c=c^* \times 10^n.$$ When $1 \leq c^*<10$, then the above is known as scientific notation; when $n\mod 3 = 0$ it is called engineering notation.

But not everyone realizes that even percentage falls into such case: the one with $n=-2$ and the symbol % replacing the multiplication by $10^-2$.

So, for example, $0.15=15 \times 10^{-2}=15 \frac1{100}=15\%$.

You ask if 15% is a number or a percentage. I answer: it's a number in percentage form.

It's not the form that makes a number, but its value: 0.15, $\frac3{20}$, $1.5 \times 10^{-1}$, 15% are only different ways to express the same number.

So, yes, you can calculate 30+10%, because you aren't adding a number to a percentage: you are simply playing with two numbers expressed in different forms.

Anyway, I sometimes fell discomfort when speaking about percentages with colleagues and others, when reading textbooks and online resources, because they seem to have a different (but inconsistent) definition of percentage, where a multiplication by 100 plays a very detrimental role.

Even if I don't consider a ratio a prerequisite to have a percentage, many sources explain percentages starting from a ratio.

For example, some textbooks (most about chemistry, as far as I known) multiply a ratio by 100 and then add the symbol % to the result, as if it was a unit of measure to be written just one time (see for example this instance reported on Mathematics Educators). But $$\frac{15}{100} \times 100 = 15\%$$ is not a valid equality, according to the definition of the symbol %! One should write "parts per hundred" in place of % next to the result or (SI considers "parts per hundred" a synonym of %) multiply by 100% instead than by 100: $$\frac{15}{100} \times 100\% = 15\%$$

Sadly, several Wikipedia pages contribute to such confusion.

For example, on the English Wikipedia page about percentage, we read:

The percent value is computed by multiplying the numeric value of the ratio by 100. For example, to find 50 apples as a percentage of 1250 apples, one first computes the ratio $\frac{50}{1250}=0.04$, and then multiplies by 100 to obtain 4%.

And then, paradoxically:

It is not correct to divide by 100 and use the percent sign at the same time

when the most frequent error is another: to multiply by 100 and put it equal to the result followed by the % sign.

Here is another example from the French Wikipedia page:

Exemple : 56 personnes parmi 400 (population de référence) ont une particularité P. Pour exprimer cette proportion sur une population de 100, il faut diviser 400 par 4, et faire de même avec 56 pour conserver la proportion. Or 56 / 4 = 14. Donc 14 % ont une particularité P.

Le calcul de ce pourcentage revient à trouver le numérateur d'une fraction dont le dénominateur serait 100 et qui serait égale à $\frac {56}{400}$. C'est ainsi que l'on confond souvent la fraction de dénominateur 100 avec le pourcentage et donc le pourcentage avec le nombre décimal 0,14.

Cette confusion, très pratique en mathématique, induit parfois des incompréhensions dans le domaine technique puisque l'on rencontre souvent l'indication de calcul suivante : pourcentage de personne ayant la particularité P : $\frac{56}{400}\times 100=14$