Percent sign after a fraction? Convert to percent

Question

Isn't it wrong to write the following with only the percent sign? Instead of $100 \%$?

The change in height as a percentage is $$ \frac{a - b}{a} \% \tag 1 $$ where $a$ is the initial height and $b$ is the final height.

Because if $a=10$, $b=5$ we have $$ \frac{10-5}{10}\%=\frac{1}{2} \% = 0.5 \frac{1}{100} = 0.005 \quad \text{what?!} \tag 2 $$

If we convert a decimal number to percent we multiply it by $100$ and add the percent sign. We have $1\%=\frac{1}{100}$, so with $100 \%$ we multiply the number by $1$, i.e. \begin{align} \frac{10-5}{10} \cdot \color{blue}{1} &= \frac{10-5}{10} \cdot \color{blue}{100 \%} = \frac{5}{10} \cdot \color{blue}{100 \frac{1}{100}} =\frac{1}{2} \cdot \color{blue}{100 \frac{1}{100}} \tag 3 \\ &=0.5\cdot \color{blue}{100 \frac{1}{100}} =50 \color{blue}{\frac{1}{100}} = 50 \color{blue}{\%} \tag 4 \end{align} So, shouldn't we instead write $(1)$ as $$ \frac{a - b}{a} 100 \% \quad ? \tag 5 $$

Answer 1

In what concerns exclusively the use of the percentage sign $\%$, it is simply a shorthand notation for $1/100$, so e.g. five percent of a population of $n$ elements is equivalently expressed as the amount $$n \cdot 5\%=n\cdot \frac{5}{100} =\frac{n}{20}= 0.05 n$$

So it is true that $$\frac{10-5}{10}\%=\frac{1}{2}\%=0.005$$ although it certainly doesn't correspond to the percentage representation of the ratio $\frac{1}{2}$, which would of course be $50\%$.

Answer 2

"Isn't it wrong to write the following with only the percent sign? Instead of 100%?"

It depends on what you want to say.

There's nothing wrong with saying $a=10$ and $b=5$ and $\frac {a-b}{b}\% = \frac 12\% = 0.005$ if that is what you want to say.

But you are correct. That is not what the author wanted to say.

The author wants to say that if something was originally $10$ feet and it ended up as $5$ feet then it SHRUNK by $50\%$ or that it grew by negative $50\%$.

And the formula for that is:---- well lets think it out.

If $a$ is the starting and $b$ is the ending then the absolute change is $b-a$.

In comparison to what it was, the thing has grown by the following fraction $\frac {b-a}a$ from what it was.

And in terms of percentages that is $\frac {b-a}a\cdot 100\%$. In this case it grew by $-50\%$.

The formula $\frac {a-b}a\%$ would mean something else entirely. It'd be hard to put it in practical terms to come up with a question that that would be a solution for.

....

Note: If the question was what was the original value as a precent of the final value. Then if $a=$ the final value and $b=$ the original value then $\frac {a-b}a 100\%$ would be the correct formula. If it started at $5$ feet and grew to $10$ feet then it grew by what percentage of the final result. Well it grew $5$ feet which is $50\%$ of $10$.

Answer 3

Caution: some authors improperly write expressions like $\dfrac{a-b}a\%$ with the meaning that the value should be thought of as percents. In this case, the percent sign should not be taken for a factor. $\dfrac{a-b}a(\%)$ would be safer.

Leaving this on the side, IMO $$\dfrac{a-b}a\%$$ should rarely occur because there are no common reasons to multiply a fraction by one percent (if this really occurred, I would certainly write $0.01$ instead of $\%$). On the opposite, an evaluation like

$$\frac{a-b}a=5\%$$ makes perfect sense.

I also think that $$\dfrac{a-b}a100\%$$ is ugly/superfluous and would puzzle people.