How to describe and solve this problem mathematically

[sxtja]

I appreciate any hints on how to solve this problem mathematically. I'm also not sure how it should be solved, if it should be calculus then I will move the post to that forum. I tried uploading a spreadsheet but the site would not allow it so I will post the data below, which you can copy to a spreadsheet. The left column is the item #, which can be ignored, and the next column is a set of observations for each item, I'm calling this variable z.

1334	5333.2
1089	4644.4
173	4405
1855	2420.8
768	1958.2
1269	1668.4
785	1553.2
560	1414.4
298	1391.6
1404	1088.2
783	1039.6
1637	780
746	723.6
1562	711.6
1271	650
306	630.2
1816	582.8
994	533.2
1649	491.2
1779	440.6
95	414.4
302	393
1000	390.2
65	388.6
193	383
1059	336
1255	334.4
1329	313.8
1689	310.4
2177	282.8
805	270.2
464	253.8
1201	252
1273	242.4
1378	242.2
1653	241.6
1106	192.6
512	192.4
2094	178.4
167	137.2
1621	136.6
2148	134
1546	131.8
401	131
967	101.2
1596	99.4
1557	75.8
809	73.8
1045	73
2082	66
161	65.8
521	60
1029	57.6
1202	50.6
1042	46.8
1992	46.4
933	41.4
1655	39.8
1850	39.6
862	35
192	33.6
1940	31.6
1506	30.2
2080	30
911	28
1725	27.6
2018	26.8
130	25
1569	24.4
1730	22.8
225	22.8
196	22
1994	20.8
1220	18.2
938	16.6
1376	16.2
1193	14
251	13.4
1250	12.4
727	10.4
2093	9.2
1380	8
2013	6.6
1398	6.6
272	3.8
1346	3
1091	2.6
750	2.6
1543	2.4
178	1.8
887	1.4
1504	1
1141	1
2186	0.8
652	0.8
1118	0.6
1040	0.6
842	0.4
492	0.4
393	0.2
1050	0.2
565	0
1610	0
179	0
1203	0
1559	0
2202	0
1741	0
1449	0

There are 109 items. I sorted the items by descending value of z. I need to find the solution such that 5000 divided by (number of items) will be greater than the minimum value of z (I can delete items/observations). As an example, if I take 5000/ 109, it is 45.9, which is greater than 0. By removing the bottom items below that value, I can take 5000/ 56, which still does not work. By trial and error I then used 46 and 44, which is the solution, as 5000/44 = 113.6 and the 44th item has a value of 131, which is greater than 113.6. I know there must be some way to express this mathematically but I have not figured it out. Thanks for any help!

 

[Ishuda]

I appreciate any hints on how to solve this problem mathematically. I'm also not sure how it should be solved, if it should be calculus then I will move the post to that forum. I tried uploading a spreadsheet but the site would not allow it so I will post the data below, which you can copy to a spreadsheet. The left column is the item #, which can be ignored, and the next column is a set of observations for each item, I'm calling this variable z.
...
There are 109 items. I sorted the items by descending value of z. I need to find the solution such that 5000 divided by (number of items) will be greater than the minimum value of z (I can delete items/observations). As an example, if I take 5000/ 109, it is 45.9, which is greater than 0. By removing the bottom items below that value, I can take 5000/ 56, which still does not work. By trial and error I then used 46 and 44, which is the solution, as 5000/44 = 113.6 and the 44th item has a value of 131, which is greater than 113.6. I know there must be some way to express this mathematically but I have not figured it out. Thanks for any help!

1334	5333.2		1000	390.2		967	101.2		2018	26.8		1543	2.4
1089	4644.4		65	388.6		1596	99.4		130	25		178	1.8
173	4405		193	383		1557	75.8		1569	24.4		887	1.4
1855	2420.8		1059	336		809	73.8		1730	22.8		1504	1
768	1958.2		1255	334.4		1045	73		225	22.8		1141	1
1269	1668.4		1329	313.8		2082	66		196	22		2186	0.8
785	1553.2		1689	310.4		161	65.8		1994	20.8		652	0.8
560	1414.4		2177	282.8		521	60		1220	18.2		1118	0.6
298	1391.6		805	270.2		1029	57.6		938	16.6		1040	0.6
1404	1088.2		464	253.8		1202	50.6		1376	16.2		842	0.4
783	1039.6		1201	252		1042	46.8		1193	14		492	0.4
1637	780		1273	242.4		1992	46.4		251	13.4		393	0.2
746	723.6		1378	242.2		933	41.4		1250	12.4		1050	0.2
1562	711.6		1653	241.6		1655	39.8		727	10.4		565	0
1271	650		1106	192.6		1850	39.6		2093	9.2		1610	0
306	630.2		512	192.4		862	35		1380	8		179	0
1816	582.8		2094	178.4		192	33.6		2013	6.6		1203	0
994	533.2		167	137.2		1940	31.6		1398	6.6		1559	0
1649	491.2		1621	136.6		1506	30.2		272	3.8		2202	0
1779	440.6		2148	134		2080	30		1346	3		1741	0
95	414.4		1546	131.8		911	28		1091	2.6		1449	0
302	393		401	131		1725	27.6		750	2.6

Are you talking about a way to express this mathematically or a way to compute what you want automatically? Expressing something mathematically is not always a simple formula. In your case I think one might write
Given the real numbers a_i, i= 1, 2, 3,, ...., n with a_j < a_k if j > k, find a m such that 5000/m > a_m
There are other mathematical symbols which could make the statement even more concise but none of those tell you how to solve the problem.

The way I would go about solving the problem with a spread sheet is add two new columns. For ease of discussion, let column A be the items number, column B be the item value (your z), column C be the index value, and column D a computed value discussed after describing column C. Column C comes about after you have sorted columns A and B with column B (your z) in descending order. Then add the index in column C: 1 would go in the row next to 5333.2, 2 in the row next to 4644.4, 3 in the row next to 4405, etc.[put a 1 in for the first item, then the next is just the last plus 1 which can be copied and pasted for all but the first item] Now the formula for column D would be 5000 divided by the appropriate index value from column C minus the appropriate value in column B. If you format this column so that negative numbers are red [maybe easiest done by formatting as currency], then the change from black to red would give you a solution.

As an example using your numbers, all item numbers from 1 to 44 are red and all those following are black.

 

[sxtja]

This is magnificent! Yes, you are correct; I realized what I want was a way to solve it automatically. I am amazed by how concise and elegant your solution is. It didn't occur to me at all.