Wikipedia:Reference desk/Archives/Mathematics/2022 June 5

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June 5

Looking for a formula of which finds the percentage in the following case

Looking for a formula that can help me to find how much money every floor needs to pay in the following scenario: there are 5 r0ws in the show, but the fifth raw people (which are closer to the air conditioner and other services) need to pay 5 times more than the 1st raw people, and only 20% more than the 4th, 40% more than the 3rd, 60% more than 2nd, and 80% more than the first. Now I am looking for a formula that I can change the numbers of rows and get the percentage of the money, each row needs to pay based on the mentioned principle (last raw pay always more than everyone, depends on the number of raws, the more rows the more the last raw pay relatively to the 1st).--ThePupil (talk) 20:09, 5 June 2022 (UTC)

Do you mean 'row' instead of 'raw'? Also you said the fifth r*w people pay 5 times more than the first r*w and 80% more than the first. 5 times is not the same as 80% more; it's 400% more. Maybe do an example with exact amounts instead of ratios. --RDBury (talk) 03:39, 6 June 2022 (UTC)

Yes, it was my mistake. I meant to say row. --ThePupil (talk) 05:22, 7 June 2022 (UTC)

To be very explicit, the information given contradicts itself. Assume, for concreteness, that the occupants of the first row need to pay 100 cents. Then "the fifth row people ... need to pay 5 times more than the 1st row people" means that they need to pay 500 cents. But "the fifth row people ... need to pay ... 80% more than the first" means that they need to pay 180 cents.  --Lambiam 11:20, 6 June 2022 (UTC)

What this seems to imply is:

1. Row 1 pays 1. Proportion 1/15 = 6.6%
2. Row 2 pays 2. Proportion 2/15 = 13.3%
3. Row 3 pays 3. Proportion 3/15 = 20.0%
4. Row 4 pays 4. Proportion 4/15 = 26.6%
5. Row 5 pays 5. Proportion 5/15 = 33.3%

Total paid 15

To generalise, each row n therefore pays n (cents/yen/pounds). The total paid will be (m²+m)/2, where m is the maximum row number, so the proportion paid by row n will be n/((m²+m)/2), which is 2n/(m(m+1)). Multiply by 100 to get a percentage.

Any misunderstanding is entirely mine. -- Verbarson  ^talk_edits 16:18, 6 June 2022 (UTC)

I don't see where the "80% more than the first" enters your considerations.  --Lambiam 16:29, 6 June 2022 (UTC)

But if the amount paid by row 5 is defined as the full amount, or 100%, whereas the others pay a discounted percentage, we do indeed get these ratios, and the differences in percentage points fit with the original question:

row 1 pays   20%,

row 2 pays   40%,

row 3 pays   60%,

row 4 pays   80%,

row 5 pays 100%.

It is easier to work with fractions of the full amount than with percentages. (See our article Percentage on how to translate percentages to fractions, and the other way around.) The table above then becomes

row 1 pays 1⁄5,

row 2 pays 2⁄5,

row 3 pays 3⁄5,

row 4 pays 4⁄5,

row 5 pays 5⁄5.

It should be clear how to generalize this. If there are m rows in total, we get for the n-th row:

row n pays n⁄m.

Although this looks differently, this is essentially the same answer as given by Verbarson above, except that the fractions in their answer have been scaled so as to add up to 1. But the ratios are the same.  --Lambiam 23:30, 6 June 2022 (UTC)

Two points:

• From the point of view of row 5, row 1 only pays 20% of what they pay, so they are paying "80% more" than row 1. Colloquial English is not designed to convey accurate statistics, but that is how I understood it.
• I also understood the original question to be about what percentage of the whole amount paid was due from each row, hence my scaling the total amount to 100%.

-- Verbarson  ^talk_edits 07:55, 7 June 2022 (UTC)

Agreed, once I saw that the differences given as a percentage were meant to stand for differences in percentage points, everything fell in place; the "80% more" is the difference between 100% and 20%. You may be right that the original question asked to split the whole amount due by the rows combined into percentages for each row, adding up to 100%. Practically, it remains easier to skip that step and keep working with fractions:

row n pays a fraction of 2n⁄m(m+1) times the total combined dues.

This is easier in general, and particularly so for someone not accustomed to dealing with percentages; it is easy to get confused.  --Lambiam 19:44, 7 June 2022 (UTC)