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Hi guys sorry if there is a really easy way to do this...

Our flat got a powerbill and wants to break it up evenly..

The bill is over a 61 day period and the total amount is $343.31

There are six flat occupants.

Occupants A and B were there for 27 days out of the 61.

Occupant C was there for 47 days.

Occupant D was there for 13 days.

Occupant E was there for 45 days.

Occupant F was there for 44 days.

An answer would be awesome but an answer with working would be amazing!

Thanks heaps in advance!

Will
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  • What exactly do you mean by "evenly"? $343.31$ isn't divisible by $6$. – Abstraction Mar 03 '16 at 08:24
  • Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you write what your thoughts are on the problem and include your efforts (work in progress) in this and future posts – JKnecht Mar 03 '16 at 08:25
  • Cool thanks for that. To be honest I tried and ended up with a large jumble of numbers from not knowing where to start! – Will Mar 03 '16 at 08:29
  • You got three answers to your question. Is any of them what you want? If so, please accept the best answer and upvote all useful answers. It's how this site works. – 5xum Mar 03 '16 at 11:10

3 Answers3

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Basically, you want numbers $a,b,c,d,e,f$ such that:

  1. The sum of the numbers is $343.31$
  2. The ratio of the numbers is $a:b:c:d:e:f = 27:27:47:13:45:44$

The second equation tells you a lot about the numbers. Basically, it's equivalent to saying that there exists some number $x$ such that:

$$a=27x\\b=27x\\c=47x\\d=13x\\e=45x\\f=44x$$

Plugging this into the first point (with the sum) you get that

$$27x+27x+47x+13x+45x+44x=343.31$$

which means that $203x=343.31$ or $x=\frac{343.31}{203}$. You can now plug in $x$ to see what $a,b,c,d,e,f$ are.

5xum
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  • Ah that's exactly what I was trying to figure out what to ask! What is the formula used to determine x in that scenario? – Will Mar 03 '16 at 08:27
  • @Will I edited the answer. Is everything clear now? – 5xum Mar 03 '16 at 08:27
  • Yes awesome thank you so much for that! Much clearer after that very logical explanation! – Will Mar 03 '16 at 08:28
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    @Will There is an assumption that "evenly" means "assuming each person each day uses the same amount of electricity (worth $x$)". From that, you get $x=$1.69$, as shown by 5xum. Note that total length of perion (61 days) doesn't matter at all in this model. – Abstraction Mar 03 '16 at 08:29
  • @Abstraction Yes, I admit I did make that assumption. But I think it's a reasonable assumption to make, don't you? I mean, my flat mate and I split costs 50:50, because we both spend all our days in the flat. We don't really measure each kWh everyone spends. – 5xum Mar 03 '16 at 08:30
  • @5xum I don't say it's unreasonable, I just say it's an assumption absent in initial puzzle (and to me, a non-native speaker, "evenly" means that each pays equal sum which can't be the case). It's rather obvious that it's how the puzzle was meant to be solved (since $203$ divides $34331$). – Abstraction Mar 03 '16 at 08:34
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    @Abstraction I agree with everything you said, except the "$203$ divides $34331$" part... – 5xum Mar 03 '16 at 08:46
  • @5xum Ew-w-w... that's what I get for relying on calculator and not noticing it's in integer mode. Then who should pay 24 cents? – Abstraction Mar 03 '16 at 09:06
  • @Abstraction Well, it's impossible to nicely divide the costs when you only have two decimal places. The solution is to either keep track of who overpaid, or maybe just forget the 24 cents... – 5xum Mar 03 '16 at 09:14
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Forget the $61$ days.

A total of $27+27+47+13+45+44=203$ "human days" were spent.

This means that the bill per "human day" is $343.31\div203\approx1.69$ dollars.

Now, for each occupant, simply multiply this value by the number of days spent by that occupant:

  • A needs to pay $1.69\times27\approx45.66$ dollars
  • B needs to pay $1.69\times27\approx45.66$ dollars
  • C needs to pay $1.69\times47\approx79.48$ dollars
  • D needs to pay $1.69\times13\approx21.98$ dollars
  • E needs to pay $1.69\times45\approx76.10$ dollars
  • F needs to pay $1.69\times44\approx74.41$ dollars

In order to verify this, note that $45.66+45.66+79.48+21.98+76.10+74.41\approx343.31$.

barak manos
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  • If you round the amounts to paid to nearest cent, $C$, $D$ need to paid one cent more and the total amount become $343.31$... – achille hui Mar 03 '16 at 08:52
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    @achillehui: Yep, I've disregarded those "cent fragment" parts, assuming that the question was genuine, and that OP just needed a good enough approximation in order to split up the bill (I mean, you gotta have some real tightwad flatmates if you end up arguing with them over $1$ cent). – barak manos Mar 03 '16 at 08:57
  • To be fair I did say too hard basket and ask A, C and D to just pay $100 between them and I'd sort the rest but they felt hard done by.. So there's always a chance that the 1 cent might be a factor! – Will Mar 03 '16 at 10:57
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Here is an alternate idea. Suppose we want everyone to pay exactly "fair" sum. Suppose that "fair" means that every person pays $x$ cents for each day he/she is present and $y$ cents otherwise, with $y$ as low as possible. Then we gain (adding days): $$203x+163y=34331$$ This is called Diophantine equation. Its solutions are $$x = 137+163k \\ y = 40-203k$$ where $k$ is any integer. For $k=0$, $y$ is the lowest possible positive, so sums to pay would be: $$A : \$50.59 \,(27*137+34*40)\\ B : \$50.59 \,(27*137+34*40)\\ C : \$69.99 \,(47*137+14*40)\\ D : \$37.01 \,(13*137+48*40)\\ E : \$68.05 \,(45*137+16*40)\\ F : \$67.08 \,(44*137+17*40)\\ \text{Total}:\$343.31$$

Abstraction
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