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I don't mean to waste your time with this simple thing but I came here because I haven't found a solution. The textbook is Mathematics All Around 5th ed. by Thomas Pirnot. The exercise says:
"Assume the board of trustees for your school is bound by state law to increase tuition by no more than 10% over the next three years. If tuition is raised by 2%, then 3%, and then 5%, has the board followed its mandate?"
So I figured from a previous related exercise that incrementing tuition by, say, 2% and then 3% is not the same as increasing it by 3% and then 2%.
So this is how I rephrased the question assuming that tuition is 1: Is 1.1 $\geq$ 1.05 (1.03 (1.02))?
No, because 1.1 $\lt$ 1.10313. So the board hasn't followed its mandate.
I think the way I calculated the increase is wrong because if incrementing by 2%, then 3% and finally 5% was calculated with plain multiplication then it doesn't matter in which order I multiply the 1.02, 1.03 and 1.05. But it does matter. What am I doing wrong here?

Update: I think I figured it out. I should have taken into account the "over 3 years" part. Assuming that each increase corresponds to years 1, 2 and 3:
Is 1.02 + 1.03(1.02) + 1.05(1.03(1.02)) $\leq$ 1.1 $\times$ 3?
The answer is yes, 3.17373 $\leq$ 3.3, so the board has followed its mandate. The related exercise that I mentioned was:
"The board of trustees is considering a tuition increase. Some want to raise it by 5% this year and 8% the next. Some want to raise it by 8% this year and 5% the next. Some say that it doesn't matter because you will end up paying the same. Does it make any difference?"
Assuming that tuition is 1, then 1.05 + 1.08(1.05) $\neq$ 1.08 + 1.05(1.08). So the order does matter when the increases have to be added like in this case.

David Vallejo
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    Why do you think it should matter? I agree with your answer and also with the observation that these increases are commutative (order doesn't matter). – Brian Tung Jul 09 '22 at 04:52
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    "figured from a previous related exercise that ..." You should expand on that, since it sounds like you misunderstood something that might be worth correcting. – dxiv Jul 09 '22 at 04:57
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    It is kind of surprising that the order doesn't matter. Maybe post another question showing the previous problem, how you got the result that the order mattered, and pointing to these comments, where three people are telling you that it shouldn't. – JonathanZ Jul 09 '22 at 05:03
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    "I figured from a previous related exercise that incrementing tuition by, say, 2% and then 3% is not the same as increasing it by 3% and then 2%." I suspect that you are misremembering that previous exercise's conclusion, which might have been something along the lines of "increasing by 2% then decreasing by 2% is not the same as doing nothing". – ryang Jul 09 '22 at 05:58
  • Re: your Update: (1) something done over the next 3 years or alternatively the next 5 years differs only in its duration, not in its multiplicity; "over each of the next 3 years" on the other hand does imply 3 increases. (2) I'm not sure I understand the LHS of the new inequality, and I disagree with the last two sentences. – ryang Jul 10 '22 at 17:28
  • To exapand on @ryang's comment the lesson to learn that $+2%$ and then $-2%$ is not the same as $0%$ is not that order matters (it doesn't) but that you percentages are not additive. If you do 2% then 3%, it is not the same as doing 5%. You don't add percentages. (You multiply them $2%$ then $3% =(1+2%)(1+3%)=1.02\times 1.03=1.0506=5.06%$) But order doesn't matter. – fleablood Jul 10 '22 at 17:34
  • Oh... I see the issue. In the "related puzzle" in both cases what the final tax is order does not matter. $1.05\times 1.08 = 1.08 \times 1.05$. What does matter is how much tax is paid in total along the way. If you raise taxes but 8 percent the first year every one pays 8 percent that year and then $1.08\times 1.05$ the next. But if you only raise it by 5 percent the every one pays only 5 percent that year* and then $1.05\times 1.08$ the next. In the end the final tax rate is the same but the taxes you pay along the way will be different. – fleablood Jul 10 '22 at 17:40
  • You answered the question correctly in that in the end the tuition is more than 10% what it was 3 years ago. But notice if the question had been. "There is a rule that students can not spend more than a total number of XXXX dollars in three years and in one case the tuition goes up 2 than 3 than 5 and in another it goes up by 5 then 3 then 2, then it's possible that the first is acceptable the other is not $1.02 + 1.02\times 1.03 + 1.02\times 1.03\times 1.05 < XXXX < 1.05 + 1.05\times 1.03 + 1.05\times 1.03\times 1.02$. – fleablood Jul 10 '22 at 17:47

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