How to divide bill of amount x between different people with usage of different days.

Question

For example: Total electricity bill is $19930 for 60 days (~i.e 2 months) for a particular house.

Now there are 7 people who lived in the home for different days.

• Alice: 60 days
• Bob: 60 days
• Catharine: 45 days
• Douglous: 45 days
• Eve: 45 days
• Frank: 15 days
• George: 15 days

How much amount each person have to pay for this total bill.

I tried dividing total bill of 19930/60 = 332.16 332.16 of single day. between 7 people. and

332.16/7 = 47.45 per person for single day.

Now George have to pay 47.45 * 15 (the days for which she lived in the house) = ~711

Same for Frank = ~711

For Catharine, Douglous and Eve the amount will be 47.45 * 45 = 2135.25

and for Alice and Bob it will be 47.45 * 60 = $ 2847

But when I add this amount of all persons 2847*2 + 2135 * 3 + 711*2 != ~19930

Why is this happening?

Answer 1

You could weight each person's contributions according to the length of their stay.

Find the total number $T$ of "people-days". For your example this is $60+60+...$.

Then a person who has lived in the house for $D$ days could be expected to pay $$\frac{Dx}{T}$$ out of a bill of amount $x$.

Answer 2

Let $p$ be the price for one person in one day .

Then we have

$$60p+60p+45p+45p+45p+15p+15p=19930 .$$

Hence $285 p =19930 .$

Now you can determine $p$.

For example: the amount of Alice is $60 \cdot \frac{19930}{285}.$

Answer 3

A lot of people are disliking this question but I couldn't help myself from arguing my way into the answer here. So let's see.

Suppose you have $N$ people, and that person number $i$ has lived in the house for $n_i$ days in in the house. To be fair, the amount $x_i$ a person has to pay should be proportional to this $n_i$, that is, $x_i = x \,n_i$ for some value of $x$, and thus the total amount of money to be paid $x_{tot}$ is given by the formula

$$x_{tot} = \sum_{i=1}^N x\, n_i = x \sum_{i=1}^N \, n_i.$$

We can find $x$ simply by rewriting

$$x = \frac{x_{tot}}{\sum_{i=1}^N n_i},$$

so that person number $i$ has to pay

$$x_i = x \,n_i = \frac{n_i}{\sum_{i=1}^N n_i} x_{tot}. $$

Now you fill in the numbers.