I imagined splitting a powerbill was easy...

Question

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!

Answer 1

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.

Answer 2

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 $45.66+45.66+79.48+21.98+76.10+74.41\approx343.31$.

Answer 3

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: $$A : \$50.59 \,(27*137+34*40)\\ B : \$50.59 \,(27*137+34*40)\\ C : \$69.99 \,(47*137+14*40)\\ D : \$37.01 \,(13*137+48*40)\\ E : \$68.05 \,(45*137+16*40)\\ F : \$67.08 \,(44*137+17*40)\\ \text{Total}:\$343.31$$