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A grocer has a set of two-pan balance scales for weighing food. A customer weighs a bag of sugar on one of the pans, which registers at 45 pounds. Suspicious, she swaps the bag to the second pan of the same scale and registers it at 20 pounds.

How much does the bag actually weigh?

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  • $\begingroup$ I'm not really sure why people are downvoting this question... $\endgroup$ – Steven_BDawg Feb 24 '15 at 16:27
  • $\begingroup$ It may be that people think you created this puzzle yourself and are posing it as a challenge to the community, as opposed to genuinely asking for the solution. The former would make it highly unoriginal, whereas the latter makes it a legitimate inquiry. $\endgroup$ – Aza Feb 24 '15 at 17:23
  • $\begingroup$ Oh okay. I was genuinely asking this. One of the issues I was having, which ghosts_in_the_code helped explain, was how the 20 and 45 pounds were being determined. As most are aware of, there are different types of scales. $\endgroup$ – Steven_BDawg Feb 24 '15 at 17:34
  • $\begingroup$ @Emrakul A puzzle created by the poster is original by definition. How can it be unoriginal? Anyway, I don't see anything wrong with that. $\endgroup$ – Florian F Feb 25 '15 at 17:34
  • $\begingroup$ @Florian By unoriginal, I consider "You have sixty-three balls and a scale..." - which obviously hasn't been asked before, but there are many variants a simple search would turn up, and so it's not new. Once you get to the sixty-third question, you'd be pretty tired of answering with the same thing. $\endgroup$ – Aza Feb 25 '15 at 17:37
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It depends whether it is the pans that have different weights or the arms that have different lengths.

It would look highly suspicious if the pans don't balance when empty. So the second option is more likely.

The effect of different arm lengths is that the actual weight is multiplied or divided by a factor depending which way it is used (on which side are the reference weights). The actual weight is the geometrical mean of the two measured weights: $weight = \sqrt{20\times 45} = 30\ pounds$.

So, the bag weighs 30 pounds (or 13.6 kg).

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    $\begingroup$ Today, the desired answer was revealed. The guy said he in fact did intend for the scales to have different arm lengths, so the answer he was looking for was indeed 30 pounds. However, if you have the same amount of base weight at different arm lengths (IE the pans themselves), then you actually end up with the balance STILL not being balanced when empty. $\endgroup$ – Steven_BDawg Feb 25 '15 at 15:56
  • $\begingroup$ Indeed. The calculation assumes either that the pans have negligible weight, or that the pans have different weights that compensate for the lengths and make the pans balance when empty. $\endgroup$ – Florian F Feb 25 '15 at 17:25
  • $\begingroup$ I was thinking 30 pounds myself, but didn't know how to justify it. $\endgroup$ – Joe Z. Feb 25 '15 at 21:06
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The bag weighs $mean(20,45)=\frac{20+45}2=32.5$ kg

Since the difference between the pans is same, the weight of an object would be affected by the same value on both sides, one side positively, the other negative. So we just find the mean.

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  • $\begingroup$ I guess I'm having a hard time wrapping my head around the concept of the scale itself. If the scale reads 20, does that mean one side of the scale is 20 pounds heavier than the other? $\endgroup$ – Steven_BDawg Feb 24 '15 at 16:29
  • $\begingroup$ @Steven_BDawg The scale itself is unfair, with one pan weighing 12.5 kg more than the other. $\endgroup$ – ghosts_in_the_code Feb 24 '15 at 16:32
  • $\begingroup$ Right I figured there was something like that going on, but whether or not the scale itself is unfair, is that how a scale would read 20? If one side was 20 pounds heavier than the other? $\endgroup$ – Steven_BDawg Feb 24 '15 at 16:35
  • $\begingroup$ @Steven_BDawg Yes, a reading of 20 means pan 1 + bag = pan 2 + 20. $\endgroup$ – ghosts_in_the_code Feb 24 '15 at 16:41
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    $\begingroup$ I'm sure it was just an oversight, but your answer is about $2.2\times$ the correct answer. The bag weighs $32.5$ pounds, not kilograms. $\endgroup$ – KSmarts Feb 24 '15 at 17:27

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