什么是百分数？

(Video) What are Percentages?

by Math Antics

Here's a video by mathantics that explains percents.

$\large{\text{Summary}}$

A percent is a fraction that always has $100$ as its denominator. $$\frac{\#}{100}$$ But instead of writing the number over $100$ every time, we use the percent symbol $\%$. $$\#\,\%$$

For example, if we saw $15\%$ we would read it as "fifteen percent". But it's the same as $\frac{15}{100}$. $$15\%=\frac{15}{100}$$

To go from the fraction to the percent, multiply by 100. Then add the percent sign at the end.

$100\%$

One-hundred percent, $100\%$, is just the fraction $\frac{100}{100}$, which is just $1$. $$100\%=\frac{100}{100}=1$$ $100\%$ is just all of something, or one whole. Thus it makes sense it equals $1$.

Over $100$

So what about percent over $100$? What is $126\%$? Well, it works the exact same way. Just place the number over $100$ in a fraction. $$126\%=\frac{126}{100} = 1 \frac{26}{100}$$ The $\frac{126}{100}$ is an improper fraction, and $1 \frac{26}{100}$ is a mixed number; they mean the same thing. Thus these fractions and the percent are all bigger than $1$.

A Percent Is a Fraction of $100$!

The word "percent" means "per hundred." So, whenever you see a number with a percent sign $(\%),$ you can imagine it as a fraction, with the number before the $\%$ as the numerator (the top number) and $100$ as the denominator (the bottom number).

For example, if you have a container that holds $100$ mL total, but there is only $60\text{ mL}$ of water in it, you could say that it was $60\%,$ or $\frac{60}{100}$, full.

We can also write percents as fractions or as decimals. Sometimes those are easier to work with.

If a percent is just a fraction, with $100$ as the denominator, why make a special thing of it? Why not just have fractions?

Fractions can be hard to compare, but percents make comparison at a glance much easier.

Suppose the pupils at two different schools voted on whether to abolish school holidays, and the votes in favor were like this:

At School A, $457$ pupils out of a total of $985$ voted to abolish holidays. At School B, $518$ out of $1137$ voted to abolish holidays.

Which school had the larger proportion of pupils voting against school holidays? And was there a majority against holidays in either school?

The problem here is that the denominators of the two fractions are different. It's not so easy to compare $\frac{457}{985}$ (School A) to $\frac{518}{1137}$ (School B).

But, when we translate those fractions into percentages—we see this:

At School A, $46.4\%$ voted to abolish holidays. At School B, $45.6\%$ voted to abolish holidays.

We have forced those two awkward fractions into forms which are easy to compare—and we can see that a slightly higher proportion of pupils at School A are against them, than at School B.

We were also asked if a majority of students voted to abolish holidays at either school. Both percentages are less than $50\%$. So, the answer to this question is no.

The price we pay for this convenience is a loss of information. These results are rounded, and we don't know how many students were in each school.

But it's much easier to compare the opinions of the two schools when we put things in percentages. If we were comparing more than two schools, we would appreciate the convenience of percentages even more.