Not sure where to post, but I have a question about averages

[Legacystar]

Math is not my strong suit but I ran into a problem that I'm really interested in seeing if there's an equation for. Situation is I'm trying to split a trip cost between people who are staying varying length. The goal is to make everyone have the same cost per day

Total amount paid $1355 for 3 nights
Total of 9 guests, but 4 are coming in early, staying 3 days, and then being joined by the other 5 who are staying 2 days.

I want each person to have the same per night cost

My basic math skills came up with this, but I want to know if there is a better equation
(Rounding to make things simpler)
$1355/3= $451/day
(451*2)/9=100 per person for 2 days
451/4=112 per person for the people staying an extra day
100/2= 50 per night for the 5 staying 2 nights
(100+112)/3=70 per night for the 4 people staying 3 nights

I know the average is somewhere around 60-61, but I'd like to know how to accomplish this the right way. I appreciate anyone who can help a math noob out

 

[blamocur]

$1355/3= $451/day

Dividing the total sum by the number of days makes sense when the expenses are the same each day -- is this true in your case?

The goal is to make everyone have the same cost per day

You can count the total number of "person days", or you can make an equation: if the cost per person per night is [imath]x[/imath] what is the total amount expressed in terms of [imath]x[/imath]?

 

[Steven G]

There are 9 people staying for a total of 22-people days (4*3+5*2) for a total of $1355
$1355/22 = amount each person pays for each day they are there for.


Another way is even easier.
The cost per day is $451
Day-1: Each person there pays $451/4
Day-2: Each person there pays $451/9
Day-3: Each person there pays $451/9

 

[The Highlander]

There are 9 people staying for a total of 22-people days (4*3+5*2) for a total of $1355
$1355/22 = amount each person pays for each day they are there for.
Another way is even easier.
The cost per day is $451
Day-1: Each person there pays $451/4
Day-2: Each person there pays $451/9
Day-3: Each person there pays $451/9

Hmmm,
your first method seems absolutely the way to go (for me). I would call it the room rate (say, R) and 4 people staying 3 nights means a total of (4×3) ie 12R would be charged by the hotel then those 5 only staying 2 nights would incur a charge of (5×2) ie: 10R making a total of 22R charged for all the guests' time there. So 22×R=$1355, therefore, the Room Rate, R, is $1355/22 per night for each guest, just as you say at the outset.

However, your "easier" method seems somewhat problematic (unless I have misunderstood what you wrote?).

Another way is even easier.
The cost per day is $451
Day-1: Each person there pays $451/4 = $112.75
Day-2: Each person there pays $451/9 ≈ $50.11
Day-3: Each person there pays $451/9 ≈ $50.11

Using the amounts I have appended (in Red, above) the 4 guests who stay 3 nights pay $112.75 each for their first night then (rounding) $50.11 for their second & third nights, ie: a total of $212.97 or $70.99 per night.
Whereas those 5 who only stay 2 nights pay $50.11 for each of them so they pay $20.88 less per night than those who stay a day longer!

That method may appear "easier" but it doesn't seem to give the right answer!

 

[The Highlander]

Math is not my strong suit but I ran into a problem that I'm really interested in seeing if there's an equation for. Situation is I'm trying to split a trip cost between people who are staying varying length. The goal is to make everyone have the same cost per day

Total amount paid $1355 for 3 nights
Total of 9 guests, but 4 are coming in early, staying 3 days, and then being joined by the other 5 who are staying 2 days.

I want each person to have the same per night cost

My basic math skills came up with this, but I want to know if there is a better equation
(Rounding to make things simpler)
$1355/3= $451/day
(451*2)/9=100 per person for 2 days
451/4=112 per person for the people staying an extra day
100/2= 50 per night for the 5 staying 2 nights
(100+112)/3=70 per night for the 4 people staying 3 nights

I know the average is somewhere around 60-61, but I'd like to know how to accomplish this the right way. I appreciate anyone who can help a math noob out

Hi Legacystar,

The first thing to note is that this problem has nothing to do with "averages". It is a straightforward calculation of how much each person should pay (per day or per night) for their stay given that they should each be paying the same 'daily' rate.

You don't describe exactly what this "Situation" is so I have no idea whether your total cost ($1,355) is for a camping weekend or you just had folks staying over at your home or a guest house or some kind of student accommodation (hostel?) or the like and a "Bill" for $1.355 was presented at the end.

But one (useful, I think) way to consider your problem is to assume that all your "guests" are staying at an hotel but this is an unusual hotel in that it takes them a week to service their rooms so, if you stay for more than 1 day/night, then you move into a different room the following day, so every guest uses up one room every day that they stay.
However, the charge for a room is the same regardless of how long you stay! And this charge, the (daily) Room Rate, is $R.

Now, 4 of your guests stay 3 days, so how many rooms do they occupy? (4×3=12)

And 5 guests stay 2 days so how many rooms do they occupy? (5×2=10)

So, over the 3 days, how many different rooms have been used by the "guests"? (12+10=22)

That means 22 rooms were occupied, each costing $R.

And you know the total cost of this was $1,355, therefore, 22×$R=$1,355 \(\displaystyle \Rightarrow\) $R=$1,355÷22.

Simples.

The "method" you have tried to adopt is more akin to the "easier" method suggested by Steven G but this leads to differing rates for the the two groups (as I explained in my reply to his post, above) and you then appear to have been attempting to "average" out these differing rates for each guest.
I trust you can now see that this is an entirely erroneous approach and it's just a simple calculation that's required to get your desired answer and nothing at all to do with "averages".

 

[Steven G]

Hmmm,
your first method seems absolutely the way to go (for me). I would call it the room rate (say, R) and 4 people staying 3 nights means a total of (4×3) ie 12R would be charged by the hotel then those 5 only staying 2 nights would incur a charge of (5×2) ie: 10R making a total of 22R charged for all the guests' time there. So 22×R=$1355, therefore, the Room Rate, R, is $1355/22 per night for each guest, just as you say at the outset.

However, your "easier" method seems somewhat problematic (unless I have misunderstood what you wrote?).


Using the amounts I have appended (in Red, above) the 4 guests who stay 3 nights pay $112.75 each for their first night then (rounding) $50.11 for their second & third nights, ie: a total of $212.97 or $70.99 per night.
Whereas those 5 who only stay 2 nights pay $50.11 for each of them so they pay $20.88 less per night than those who stay a day longer!

That method may appear "easier" but it doesn't seem to give the right answer!

Another way is even easier.
The cost per day is $451
Day-1: Each person there pays $451/4
Day-2: Each person there pays $451/9
Day-3: Each person there pays $451/9

The fee for day 1. Since there are 4 people, they each pay $451/4. Since there are 4 people and they each pay $451/4 which totals $451 which is correct.
The fee for day 2. There are now 9 people, they each pay $451/9 for the 2nd day (some of these people were there for the 1st day as well)
The fee for day 3. There are now 9 people, they each pay $451/9 for the 3nd day (some of these people were there for the 1st day as well)
It seems correct for me.
Imagine that the guest check out each night, pay their bill-shared between the number of guests-and then check back in with another set of people.

 

[Dr.Peterson]

The fee for day 1. Since there are 4 people, they each pay $451/4. Since there are 4 people and they each pay $451/4 which totals $451 which is correct.
The fee for day 2. There are now 9 people, they each pay $451/9 for the 2nd day (some of these people were there for the 1st day as well)
The fee for day 3. There are now 9 people, they each pay $451/9 for the 3nd day (some of these people were there for the 1st day as well)
It seems correct for me.
Imagine that the guest check out each night, pay their bill-shared between the number of guests-and then check back in with another set of people.

That's what I would probably consider most fair; but the question specifies

The goal is to make everyone have the same cost per day

That suggests that their idea of fairness is not based on splitting up the cost on each individual day, taking into account the actual use of the place, but to share the extra cost incurred by some coming earlier among everyone.

Accepting that standard, we have

Total amount paid $1355 for 3 nights
Total of 9 guests, but 4 are coming in early, staying 3 days, and then being joined by the other 5 who are staying 2 days.

I want each person to have the same per night cost

Total person-nights = 4*3 + 5*2 = 22​

Cost per person-night = 1355/22 = $61.59​

​

Cost for each early guest: 3*61.59 = $184.77​

Cost for each late guest: 2*61.59 = $123.18​

​

Total paid: 4*184.77 + 5*123.18 = $1354.98​

which is as good as you'll get. This is Highlander's approach, doing just as requested.

By Steven's "easier" approach, we have

The cost per day is $451
Day-1: Each person there pays $451/4
Day-2: Each person there pays $451/9
Day-3: Each person there pays $451/9

Total cost per night: 1355/3 = $451.67​

Cost per person for first night: $451.67/4 = $112.92​

Cost per person for each other night: $451.67/9 = $50.19​

​

Cost for each early guest: 1*112.92 + 2*50.19 = $213.30​

Cost for each late guest: 2*50.19 = $100.38​

​

Total paid: 4*213.30 + 5*100.38 = $1355.10​

​

The early guests, of course, pay more this way.

 

[The Highlander]

The fee for day 1. Since there are 4 people, they each pay $451/4. Since there are 4 people and they each pay $451/4 which totals $451 which is correct.
The fee for day 2. There are now 9 people, they each pay $451/9 for the 2nd day (some of these people were there for the 1st day as well)
The fee for day 3. There are now 9 people, they each pay $451/9 for the 3nd day (some of these people were there for the 1st day as well)
It seems correct for me.
Imagine that the guest check out each night, pay their bill-shared between the number of guests-and then check back in with another set of people.

I understand your logic but the problem is that all the OP "accurately/evidently" provides is the total cost: $1,355 (for the whole "trip") that s/he wants to be "split" equally among the 9 persons so that each pays the same per diem. The fact that this total is the "trip cost" implies (to me, at least) that it may well include, travel, food and (possibly several) other 'incidentals' not just the cost of accommodation.
It is then the OP's (somewhat muddled) attempt to work this out that divides the total cost up into $415 per day, thus 'emulating' your "easier" method. It is nowhere suggested (in the OP's original presentation of the "Situation") that $415 did have to be paid out each day or the travellers would, indeed, then be liable to pay differing amounts each (in my estimation of what would be "fair")!

 

[The Highlander]

I ran out of time (by a few seconds) when trying to edit my previous post.
I just wanted to add this further comment, Re:-

Imagine that the guest check out each night, pay their bill-shared between the number of guests-and then check back in with another set of people.

If that were the case then why would the (hotel?) "bill" still be $415 (on the 2nd/3rd days) when there are now 9 people staying there if it was $415 too when there were only 4 staying (on the first day)?

So dividing the total cost up into three equal (daily) expenditures of $415 really makes no "sense" at all!

 

[Legacystar]

There are 9 people staying for a total of 22-people days (4*3+5*2) for a total of $1355
$1355/22 = amount each person pays for each day they are there for.


Another way is even easier.
The cost per day is $451
Day-1: Each person there pays $451/4
Day-2: Each person there pays $451/9
Day-3: Each person there pays $451/9

you're first method is exactly what i'm looking for and it make a ton a sense, i never thought to do it that way. the second method is problematic because it puts more of the cost on the people staying 3 days.

if you run it the second way, it seems very unfair that 4 people people pay $851 of the total $1355. mor than half the bill, to stay an extra day is crazy.

the per day room rate of option one makes way more sense.