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If I usually drive at 100 km/h, but today I'm driving at 80km/h, I am driving 20% slower than usual.

But if I speed up to 100 km/h, I am driving 25% faster, aren't I?

25% seems like a lot and encourages me to speed, but 20% seems like not that bad and I'll slow down and relax.

Please help me to understand why the difference can be both 20 and 25, and preferably, can there be some unified number I can think about instead? I don't like this double-think.

amWhy
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Andrew
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    Nothing strange: $20$ is the $20 %$ of $100$ while it is the $25 %$ of $80$. – Mauro ALLEGRANZA Feb 21 '18 at 14:58
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    @MauroALLEGRANZA Use \% to make the $%$ appear. – Arthur Feb 21 '18 at 14:58
  • The difference $20$ is "absolute" while percentages are relative to something. A quarter of a mile is differnt from a quarter of a yard. – Mauro ALLEGRANZA Feb 21 '18 at 14:59
  • The answer is "both". 100 is $25/%$ bigger than $80$, and $80$ is $20%$ smaller than $100$. – 5xum Feb 21 '18 at 15:00
  • If you're looking for sth unified, you should compare all your values with respect to one value (Let's say everything respect to $100$) – Mehrdad Zandigohar Feb 21 '18 at 15:02
  • I don't like it. Maybe that absolute 20 km/h might be the unified number thing i need. Yes need something unified – Andrew Feb 21 '18 at 15:02
  • If you go $80$km / h from the typical speed of $100$, that is going $20%$ slower than the typical speed. Therefore, of course if I go back up my typical speed of $100$km / h, to cancel out how much slower I went, I go $20%$ faster. It is a little bit like doing $100 - 20 + 20$ but not completely. Don’t take that last sentence too seriously or it could prove to be misleading when dealing with percentages. – Mr Pie Feb 21 '18 at 15:06
  • if you take an absolute value of speed difference, it will be the same no matter how you arrange two speed values. Maybe that's why we call it an "absolute value". – Vasili Feb 21 '18 at 15:06
  • Suppose you invest 100 $ in a stock. Next day the price rises by x%, the following day the price falls by x%, then the scenario repeats itself. What would the price be after 2n days (n=1,2,...)? – Dr. Wolfgang Hintze Feb 21 '18 at 15:11
  • that's a very useful "incongruence" for .. politicians. – G Cab Feb 21 '18 at 15:11
  • Maybe the more helpful way of thinking about it would be related to time it takes to get to destination. But even that I think ends up with the same 25 / 20 conundrum. – Andrew Feb 21 '18 at 15:19

6 Answers6

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This seems like a paradox to people because we like to pretend that percentages are additive things when actually it really is more multiplicative.

So, reducing something by $20\%$ (alternatively, reducing something to $80\%$ of its original value) means multiplying by $0.8$. It is not a subtraction, but a multiplication by a number less than $1$. Most are taught to think of it as a subtraction because that's generally viewed as easier for elementary school students to cope with, but it is not what's really going on.

To reverse multiplying by $0.8$ and get back the original value, you have to divide by $0.8$. Dividing by $0.8$ is also known as multiplying by $1.25$. We are trained to think of multiplying by $1.25$ as increasing by $25\%$ (or increasing to $125\%$), but as we see, it can just as well be thought of as reversing a decrease by $20\%$.

This also means that if something first decreases by $20\%$, then increases by $20\%$, the end result is a multiplication by $0.8$, followed by a multiplication by $1.2$. Since multiplication is associative, this is the same ass a single multiplication by the number $0.8\cdot 1.2 = 0.96$, which means a net effect of a $4\%$ decrease.

Arthur
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  • Well multiplying and dividing by 0.8 now that is a unified number. Will need to take a little more time to think about this, my poor elementary mind has trouble coping :D – Andrew Feb 21 '18 at 15:07
  • Very good point! – user Feb 21 '18 at 15:21
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Let two real numbers $a>b$.

The percentage decrease from $a$ to $b$ is $\frac{\Delta}{a}$, and is surely smaller than the percentage increase from $b$ to $a$, $\frac{\Delta}{b}$.

If you want to make "sense" out of the two percentage changes, consider the harmonic mean of $a$ and $b$.

Consider a third real number $c$, such that $a>c>b$. You want the percentage decrease from $a$ to $c$ to be the same as the percentage increase from $b$ to $c$.

$$ \begin{align} \frac{a-c}{a} &= \frac{c-b}{b} \\ 1-\frac{c}{a} &= \frac{c}{b}-1\\ c&= \frac{2}{\frac{1}{a}+\frac{1}{b}} \end{align} $$

"Typically, it (harmonic mean) is appropriate for situations when the average of rates is desired."

In your case, the harmonic mean of your speed = $\frac{2}{1/80+1/100}=88.9$

Edit

I can ask a millionaire to give me 100 dollars. I can also give 100 dollars to a beggar with only 20 dollars. You see, although I had the same amount of money, what I asked of the millionaire is insignificant (small percentage decrease), whereas what I did to the beggar is quite the opposite (large percentage increase). This discrepancy in percentage change surely exists.

Now consider the arithmetic mean of the millionaire and beggar (1000020/2=500010 dollars). This number has virtually no meaning at all in reflecting the relative richness or poorness between the two individuals. In very crude terms this mean does not do justice in reflecting the participation of the beggar. The percentage increase of the beggar to this arithmetic mean is surely much larger (and meaningless) than the percentage decrease of the millionaire to this arithmetic mean.

Now try for yourself the harmonic mean: Find the two percentage differences from the harmonic mean to the millionaire and beggar respectively. and see what happens...

Perhaps think of harmonic mean as the useful one, or rather the "meaningful" one in this case

L Parker
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  • I suppose thinking about the average could work, but seems like some made up number because i dont intend on driving inbetween the two speeds. Also not sure how the mean of 80 and 100 isn't 90 :D. You guys are too smart. – Andrew Feb 21 '18 at 15:30
  • see my new edit – L Parker Feb 21 '18 at 16:06
  • ok, I believe you, seems like this could be a way of thinking about it, but the average would probably be just as useful and only 0.1 different – Andrew Feb 21 '18 at 16:18
  • Using arithmetic mean, the beggar increased by several thousands of folds, whereas the millionaire reduced by half. Using harmonic mean however makes the percentage changes of these two the same :} – L Parker Feb 21 '18 at 16:21
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Note that

  • 20% slower is referred to 100km/h that is $\frac{100-80}{100}=0.20=20\%$

and

  • 25% faster is referred to 80km/h that is$\frac{100-80}{80}=0.25=25\%$

Note also that as pointed out by Arthur since

$$ 80 km/h =0.80 \cdot 100 km/h \iff100 km/h=\frac1{0.80}80km/h$$

More in general if $r$ is the reducing factor from $v_2$ to $v_1$ defined by

$$r=\frac{v_1}{v_2}$$

the amplification factor $a$ is given by

$$a=\frac{v_2}{v_1}=\frac{1}{r}$$

user
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The only way you could express this as a single number is in the decrease/increase in speed. Your starting speed is different, so the ratio is different.

William Grannis
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If you slow down from $100$ km/h to $80$ km/h, it means you are going $20\%$ slower compare to your normal (regular, everyday) speed.

If you speed up from $80$ km/h to $100$ km/h, it means you are going $25\%$ faster compare to your current speed (for today only, $80$ km/h), but your velocity goes up by $20\%$ of your normal speed.

user061703
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  • You offered a unified way of thinking about it thanks, just always compare to the "normal" speed rather than flipping to current speed. – Andrew Feb 21 '18 at 15:28
  • Yeah, it's like when you lower the prices, then raise the prices of the products back to normal in a store. – user061703 Feb 22 '18 at 01:37
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Thinking of it in terms of time to arrival is the best way to unify the thought process. Because you do need to think of that "absolute" difference for there to be any single number that you can use that is easily comprehended in a way that could influence your driving. And since its the time it takes you get somewhere that matters to you, and not the speed you are going, that's what you should think about.

If you drive at 100 km/h, your time to drive 100 km will be 1 hour.

If you drive at 80 km/h, your time to drive 100 km will 1 hour 15 minutes.

So driving at 100, for every 100 km you drive, you will be saving 15 minutes of your life (as long as the speeding doesn't kill you).

And driving at 80, for every 100 km you drive, you will lose 15 minutes.

Andrew
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