Misunderstanding of percentage increase [duplicate]
Misunderstanding of percentage increase [duplicate]
This question already has an answer here:
If something increase $50$ to $200$, I know that it is $400%$ increment using common sense.
I can get this using $dfrac20050times 100% = 400%$.
If something increase $50$ to $52$, I know that it is $4%$ increment using common sense.
But if I apply the same logic, $dfrac5250times 100% = 104%$.
What is the problem in my logic?
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@badjohn Yes, Even after doing maths for AL and even after getting "B" grade I am really ashamed of myself.
– I am the Most Stupid Person
Aug 28 at 8:22
Don't be too harsh on yourself, as I said, it is a common mistake. An interesting question is at what point people tend to switch. $100 %$ is usually correctly interpreted as double. By your example of $400 %$, the mistake is common. How is $200 %$ commonly perceived?
– badjohn
Aug 28 at 9:32
Note that the other words around the percent number itself make a big difference. $200$ is $400%$ of $50$, but $200$ is a $300%$ increase from $50$, or is $300%$ more than $50$. $52$ is $104%$ of $50$, but $52$ is a $4%$ increase from $50$, or is $4%$ more than $50$. "Percent more than" is equal to "percent of" minus $100$.
– Todd Wilcox
Aug 28 at 13:19
I have heard an anecdotal story on percentage increase: There was a debate on "Do law students have to study Mathematics?" The lawyer claimed it was not necessary for them. The mathematician asked him the question: "Last year there were $80$ crimes, this year there were $100$ crimes. How much percentage increase is this?" The lawyer quickly and with confidence replied "$20$ percent"... The moral of the story is the mathematics is necessary for each and every subject and profession.
– farruhota
Aug 28 at 14:31
6 Answers
6
Percentage increase is $$fractextnew number - old numbertextold numbertimes 100 %$$
The right comptuation should be $$frac200-5050 times 100 %=300%$$
If something increases from $50$ to $200$, it increases by $300%$ and has a new value that is $400%$ of the old value.
Similarly, if something increases from $50$ to $52$, it increases by $4%$ to a new value that is $104%$ of the old one.
The convention is that "percentage increase" is the number of percentage points that are added.
So it is assumed that you always start with $100%$ of a number and then add an $n%$ percent increase to that, so you end up with $(100 + n)%$ of the original number.
If you take the ratio of the starting and ending amounts and multiply by $100%,$
you end up with the figure $(100 + n)%.$ You then have to subtract $100%$ if what you want is the percentage increase.
Indeed $52$ is $104%$ of $50,$ but the added amount is only $2,$ which is $4%$ of $50.$
Likewise $200$ is $400%$ of $50,$ but the added amount is only $150,$ which is $300%$ of $50.$
You are making the classic mistake of confusing ratio with change.
$ratio = fracnew;valueold;value$
$percentage;ratio = fracnew;valueold;value times 100%$
$difference = new;value - old;value$
$percentage;change = fracdifferenceold;value times 100% = fracnew;value - old;valueold;value times 100%$
Change is more commonly known as growth or increase.
This is were the ratio makes more sense
That is when
50:400
are divided one both sides by 50 giving us
1:4
so my understanding is that it is four times more.
If you see your question, you'll see that you have answered it yourself. In the second statement, you said 50 to 52 increment means 4% which is equal to 100 subtracted from 104 which you have calculated. Similarly, if you subtract 100 from 400 you will get 300%.
It is worth mentioning that your mistake is very common. For small increases, the percentage is usually correct. As the increase gets larger, mistakes become frequent. It is common to confuse $4$ times bigger with $400 %$ more. Think about $100 %$ bigger which means double and not the same.
– badjohn
Aug 28 at 7:56