Am I driving 20% slower or 25% faster?

Question

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.

Answer 1

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

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

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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

Answer 6

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.