Number minus 17% then plus 21% is always higher than the original number but 18% is always less, why?

Question

10000 - 17% = 8300 + 21% = 10043 (more than orig)
10000 - 18% = 8200 + 21% = 9922 (less than orig)

The point where it turns is between 17.3 and 17.4 but I want to know why.

Answer 1

The two identities (which aren't identities because you haven't actually written what you mean) are because of the fact that $$(1-0.17)\times(1+0.21)=1.043\\(1-0.18)\times (1+0.21)=0.9922$$ The value of $x$ such that $(1-x)\times(1+0.21)=1$ is $$x=1-\frac{1}{1+0.21}=\frac{21}{121}=17.\overline{3553719008264462809917}\%$$

Answer 2

$x\%$ of a number is just the number, multiplied by $\frac{x}{100}$.

Using this knowledge, you can see that if number is $x$, then number minus $17\%$ is just $x-\frac{17}{100}x=0.83x$.

Also, if number is $y$, then number plus $21\%$ is just $y+\frac{21}{100}y=1.21y$.

So, if you start with $x$, and first decrease the number $x$ by $17\%$ to get $y$, and then increase $y$ by $21\%$ to get $z$, then

$$z=1.21\cdot y = 1.21\cdot (0.83\cdot x) = (1.21\cdot 0.83)\cdot x=1.043x>x$$

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Your last part of the question raises an interesting question: How much should you decrease the original number so that, once you increase the result by $21\%$, you get the original number again?

This can be a wonderful exercise you can try to do, and I strongly advise you try to do it. To do that, I recommend you try to not calculate the actual percentage $p$, but rather the value $1-\frac{p}{100}$, as that is the actual value being multiplied by $x$ in the equation above. Once you have that value, it is easy to calculate $p$.