How can something be greater than $100\%$?

Question

Apologies, my mathematics is poor but I saw an advert recently which said:

Up to $350\%$ more likely to quit

How can something be greater than $100\%$? It can't be can it?

Answer 1

Say 1 out of 10 people quit in year 1. Now if 4.5 out of 10 people quit in year 2 it is up 350%. Does this make sense? The percentages compare numbers relative to each other.

Answer 2

$100$% of something means all of it.

$200$% of the same thing means twice as much.

$350$% means $3.5$ times the original amount.

Answer 3

$\%$ (percent) literally means "over 100".

So when people say "$350\%$ more likely to quit", what they loosely mean is that, if $x$ is how many people (usually) quit, then $x + 3.5x$ represents how many people will quit in this new scenario.

Your question "how can it be bigger than 100%" suggests that you are thinking about probabilities which never go higher than 1. But that's because of its definition and the fact that people frequently use percentages to represent probabilities.

Answer 4

What you likely have in mind is the fact that the probability of something happening cannot be more than 100%, that is true. However, in this context, 350% more likely means that the current probability is 4.5 times that which it used to be, so it is still alright.

Answer 5

A percentage is just a fraction - say: $25\% = \frac{25}{100}=\frac{1}{4}$ - that is used as a multiplicative factor of some quantity.

Hence we speak of "$25\%$ of the people present" and we have 80 people, the we are speaking of $\frac{25}{100} \times 80=20$ people.

So, if a percentage is used to denote a part of some total, yes, the fraction must be between 0 and 1 (nothing and all), and the percentage must be between $0\%$ and $100\%$.

But we can also use the fraction to denote some arbitrary change or proportional difference. Suppose the probability of some events $A,B$ are respectively $p_A=0.1$, $p_B=0.3$. We see that the latter is three times more probable n the former: $p_B = 3 \times p_A$ ; hence we can say that the event $B$ is "$300\%$ more likely" than $A$.

Answer 6

How can something be greater than 100%?

Nothing can be greater than itself. However, a value can be greater than a reference value.

Consider the sentence

$x\%$ of something.

If something is an abstract value, then the sentence makes sense with $x$ greater than $100$. In your particular example, something refers to probability which is a abstract value. So, makes sense to take $x$ greater than $100$ and this corresponds to multiply by a number greater than one.

If something is a concrete object, then the sentence makes sense only with $x$ less than $100$, or equal $100$.

Examples:

• Makes sense: I will eat $50\%$ of this pizza.
• Makes no sense: I will eat $150\%$ of this pizza.

Answer 7

A percentage $x\%$ is literally nothing more than shorthand for $x/100$. It doesn't quite make sense to talk about, say, eating $150\%$ of a candy bar, but that's not a problem mathematically. I can't easily talk about drinking $-2$ or $\pi$ or $i$ candy bars, but those are all perfectly reasonable numbers.

In this particular case, the advertisement is saying that if (say) people had a $10\%$ chance of quitting without the product, then they'd have a $(3.5 + 1) * 10\% = 45\%$ chance of quitting with the product. It would be odd to say that $350\%$ of the specific people who quit without the product later quit with it, but the message is clear as it stands.