What's the difference between $10\%$ of $10\text{ cm}$ and $1\text{ cm}$? I overhead a physics professor at my university on the phone: 

I interviewed that student you sent me, but he didn't know the difference between increasing the length of a $10\text{ cm}$ rod by $1\text{ cm}$ versus by $10\%$, so I didn't accept him into the program.

Is this some sort of trick question? What difference could he possibly be talking about? A philosophical one? 
 A: $10\%$ of $10\text{ cm}$ is $1\text{ cm}$. $10\text{ cm}\times\frac{10}{100}=1\text{ cm}$.
I really don't see how this is a 'nit-pick', or 'relative' vs 'absolute', as suggested in the comments.
Are you absolutely, $100\%$, or at least relatively, sure you got the quote right? (There are very close variations of it, where indeed it would be rather stupid not to know the difference.)
A: If the 10 cm was a measured value with a corresponding error estimate, then the results would be different. Suppose the bar was $10 \text{ cm} \pm 0.10 \text{ cm}$. Adding exactly $1 \text{ cm}$ to the length of the bar would yield $11 \text{ cm} \pm 0.10 \text{ cm}$, whereas increasing the length of the bar by 10% yields $11 \text{ cm} \pm 0.11 \text{ cm}$.
Nevertheless, this solution takes advantage of what is clearly a nitpicky difference, especially without being provided with a context of measurement error.
A: Just a guess, but perhaps he actually said (or meant) "1 cubic centimeter".
Increasing the rod length by 10% is not the same as adding 1 cubic centimeter (because the rod is circular, not square, in the other dimensions).
Also assuming the diameter of the rod is 1cm.
A: One thing would be that it's not practical to use '%' in that example. I can easily imagine metal worker going nuts if whole 'tech spec' is in ratios.
Aftervards you can't just decrease by 10% and get the same original length.
Or maybe because he didn't specify that it's 10% of original length. 
I have this 'dejavu' feeling that I have heard something similar and there's something more.
A: I think that there just is no difference. The professor would have wanted to hear
"There is no difference!"
Of course, most people being put in such a situation will assume the answer should be more elaborate and respond
"I don't know"
which seemed unsufficient from the professor's point of view.
