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If something increase $50$ to $200$, I know that it is $400\%$ increment using common sense.

I can get this using $\dfrac{200}{50}\times 100\% = 400\%$.

If something increase $50$ to $52$, I know that it is $4\%$ increment using common sense.

But if I apply the same logic, $\dfrac{52}{50}\times 100\% = 104\%$.

What is the problem in my logic?

gen-ℤ ready to perish
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I am the Most Stupid Person
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    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 '18 at 07:56
  • @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 '18 at 08:22
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    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 '18 at 09:32
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    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 '18 at 13:19
  • Related, possibly helpful: https://math.stackexchange.com/questions/2417020/what-is-the-difference-between-ratio-change-vs-percent-change-finance-questio/2417032#2417032 – Ethan Bolker Aug 28 '18 at 14:13
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    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 '18 at 14:31
  • If you'll notice, both of the answers that you got (400% and 104%, respectively) are 100 percentage points away from the actual answers (300% and 4%, respectively) when we're talking about percentage increase, which involves addition and subtraction. However, when we're talking about percentage of, that involves multiplication and division, and you're exactly correct on both counts: 50 * 400% = 50 * 4 = 200, and 50 * 104% = 50 * 1.04 = 52. So you had the right answers, but were calling them the wrong thing. – John Doe Aug 28 '18 at 15:30
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    @badjohn Arguably, "4 times bigger" should mean "400%". One should say instead "four times as big" to avoid ambiguity. – Acccumulation Aug 28 '18 at 22:30
  • @farruhota I don't understand the anecdote. Someone whose profession is P claims that subject S isn't required in profession P, and makes a mistake in S. How does it show that S actually is needed in P? – JiK Aug 29 '18 at 15:24
  • @JiK, the lawyer underestimated (answered $20%$ instead of $25%$) and showed his incompetence in math, consequently in law and lost the argument... – farruhota Aug 29 '18 at 15:54
  • @farruhota I don't see how being incompetent in subject S and losing an argument about subject S shows that subject S is important. – JiK Aug 29 '18 at 21:56
  • @JiK, do not worry, I myself do not understand some anecdotes. Cheers. – farruhota Aug 30 '18 at 04:18
  • Of course this has already been answered but maybe it’s intuitive to write it as 50+x50=200 => x=(200-50)/50=3 -> in % : 3100=300% .. What do u think? – SheppLogan Aug 12 '21 at 17:01

6 Answers6

82

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.

5xum
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73

Percentage increase is $$\frac{\text{new number - old number}}{\text{old number}}\times 100 \%$$

The right comptuation should be $$\frac{200-50}{50} \times 100 \%=300\%$$

Siong Thye Goh
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13

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 $50.$

David K
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You are making the classic mistake of confusing ratio with change.

$ratio = \frac{new\;value}{old\;value}$

$percentage\;ratio = \frac{new\;value}{old\;value} \times 100\%$

$difference = new\;value - old\;value$

$percentage\;change = \frac{difference}{old\;value} \times 100\% = \frac{new\;value - old\;value}{old\;value} \times 100\%$

Change is more commonly known as growth or increase.

Based
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This is were the ratio makes more sense

That is when

$50:400$

are divided one both sides by $50$ giving us

$1:8$

so my understanding is that it is eight times more.

Brahm Bothma
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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 $300$%.

Devesh
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