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How can you easily compute in your head 84.4084% of 25?

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    $\begingroup$ I don't think either answer currently answers the question on it's own, but combined they do. Both leave out a leap in logic that the other fills. $\endgroup$ – computercarguy Feb 7 at 17:19
  • $\begingroup$ Agree. More explanation is needed, so I removed the tick for now. $\endgroup$ – Dmitry Kamenetsky Feb 7 at 23:38
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    $\begingroup$ There's not really a puzzle here, is there? $\endgroup$ – shoover Feb 8 at 5:03
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    $\begingroup$ @FIreCase you are almost there. You still haven't explained why division by 4 can be done easily. $\endgroup$ – Dmitry Kamenetsky Feb 11 at 6:36
  • $\begingroup$ But that's how we learned division in primary school, go from digit to digit...no remainders in this case... "easy to do it" Is that the "explanation" or am I missing something? $\endgroup$ – FIreCase Feb 11 at 15:54
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$21.1021$, because $84.4084\%$ of $25$ equals $25\%$ of $84.4084$.

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    $\begingroup$ It had never occurred to me that you can do that! $\endgroup$ – Aww_Geez Feb 7 at 17:45
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    $\begingroup$ Even having graduated as a math major, it still took me a while to figure out why this was true. It might help others to add in a quick explanation as to why this works, something like rot13(k creprag bs l vf k qvivqrq ol bar uhaqerq gvzrf l, juvpu vf rdhvinyrag gb k gvzrf l qvivqrq ol bar uhaqerq, juvpu vf l creprag bs k) (Obviously with numbers and symbols it would be more clear, but I didn't want to give away the solution in a non-spoiled comment) $\endgroup$ – Davy M Feb 7 at 18:52
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    $\begingroup$ Great and simple. Shame I was too late for it. $\endgroup$ – Nautilus Feb 11 at 11:19
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Notice that:

All the non-zero digits in 84.4084 are divisible by 4.

From there you can do the following steps in one go easily:

Divide 84.4084 by 100 to get the raw decimal form, then multiply by 25 to get the answer (i.e. 84.4084 * 1/100 * 25). But this is the same as multiplying 84.4084 by 25/100 (= 1/4), which is the same as dividing by 4. And since all the non-zero digits of 84.4084 are divisible by four, we can trivially compute the answer: 21.1021

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Well, not sure if this is easy enough, but

multiplying 84.4084 by 5 two times is definitely doable in the head (however, I only managed to do that from the second attempt). $84 \cdot 5 = 420$, $844 \cdot 5 = 4220 \implies 84.4084 \cdot 5 = 422.042$. In a similar way we get $422.042 \cdot 5 = 2110.21$. The answer is therefore $21.1021$.

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