Why is the sum of three different percent differences not the same as the total percent difference?

Question

To be more specific, I am looking at the numbers $1234, 1345, 1456,$ and $1567$.

The percent increase between each number and the number which follows ($1234$ to $1345$, $1345$ to $1456$...) are below:

$~9\%$, $8.25\%$, and $7.62\%$.

The sum of the differences $(9+8.25+7.62)$ is $24.87\%$.

The percent difference between the first and the last number ($1234$ and $1567$) is $26.99\%$.

I know that they are different numbers, but I don't quite understand WHY they are different.

Answer 1

Any time there is a percentage in the picture, there is multiplication going on. As an example, suppose you are told to increase the value 30 by 5%. $$30 + \dfrac{5}{100}\times 30 = 30 +1.5 = 31.5,$$ but you can also look at it like this $$30 + \dfrac{5}{100}\times 30 = 30 \left(1 + \dfrac{5}{100}\right).$$ Looking at it this way lets you see that if you apply, say, a 2% increase after a 5% increase you get$$30\left( 1+\dfrac{5}{100}\right)\left(1+\dfrac{2}{100}\right).$$ Now lets multiply out the two terms on the right $$= 30\left( 1+\dfrac{5}{100} + \dfrac{2}{100} + \dfrac{5}{100} \cdot \dfrac{2}{100}\right)$$ $$= 30\left( 1+\dfrac{5+2}{100} + \dfrac{5}{100} \cdot \dfrac{2}{100}\right).$$

Hopefully you can see the expected 7% increase in there but notice that there's also a "second order" term, that corresponds to $5\cdot2/100 \% = 0.1 \% $ additional increase. That term is always there any time you compose two percentage changes, but it's usually much smaller than the other terms so it's often ignored.

When you compound three percentage changes, like you are doing, there are actually 3 "second order" terms, and even one "third order" term, and that's enough to make the difference obvious. The rule to take away with you is that percentages actually don't add, unless you're okay with ignoring the second order terms. If you need to be exact, rewrite "+5%" as "multiply by 1.05".

(Another phenomenon that this explains is the fact that if you increase something by 5% and then decrease it by 5% you don't get your original value back. If you write it out you can see how the "first order" terms cancel each other out but there is still a "second order" term left over.)

Answer 2

So you have four numbers $a,b,c,d$. The differences from $a$ to $b$ is $(b-a)/a$.

Let's assume that you are right and that

$$ \frac{d-a}{a}=\frac{b-a}{a}+\frac{c-b}{b}+\frac{d-c}{c}.$$

Now notice that the left side does not change when you change $b,c$ but the right side does.