Real life problem: projecting sales for shrimp

Aqua

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Hello,

I run a family shrimp aquaculture business. I need to come up with reasonable sales projections for the next six months, and though I've figured out most of the overall problem, the middle part is giving me fits. I'm the most gifted in math in my family, but you'll likely see that I'm still rather feeble and really need the help.

We take in a batch of 30k shrimp each month. I figured out that we average selling 23% of each batch as small size (about 35 pieces per pound), 42% as medium (~27 pieces/lb), 23% as large (~22 pieces/lb), and 12% as jumbo (~16/lb). Each batch is sold over the length of three months, and naturally the longer a shrimp lingers, the bigger it gets. Here's where I get lost.

I need to figure out how many pieces out of 30k shrimp makes up 23% of sales in small pounds, 42% in medium pounds, 23% in large pounds, and 12% in jumbo pounds. I know enough that I don't multiply 30k by 0.23 because that will only show me whatever 23% of 30k is, not how many small pieces make up the 23% of the weight of the entire batch, which includes other larger sizes.

If anyone reading this needs some clarifications, I will do my best to do so. Thank you for your time.
 
Hello,

I run a family shrimp aquaculture business. I need to come up with reasonable sales projections for the next six months, and though I've figured out most of the overall problem, the middle part is giving me fits. I'm the most gifted in math in my family, but you'll likely see that I'm still rather feeble and really need the help.

We take in a batch of 30k shrimp each month. I figured out that we average selling 23% of each batch as small size (about 35 pieces per pound), 42% as medium (~27 pieces/lb), 23% as large (~22 pieces/lb), and 12% as jumbo (~16/lb). Each batch is sold over the length of three months, and naturally the longer a shrimp lingers, the bigger it gets. Here's where I get lost.

I need to figure out how many pieces out of 30k shrimp makes up 23% of sales in small pounds, 42% in medium pounds, 23% in large pounds, and 12% in jumbo pounds. I know enough that I don't multiply 30k by 0.23 because that will only show me whatever 23% of 30k is, not how many small pieces make up the 23% of the weight of the entire batch, which includes other larger sizes.

If anyone reading this needs some clarifications, I will do my best to do so. Thank you for your time.
As I see the problem:

Your total sales is ~30,000 (=3*10^4) pieces of shrimps.

If you declare that 23% of your sales is small → then 0.23*3*10^4 pieces of shrimps are small. That converts to ~6900 pieces of small shrimp, 12,600 pieces of medium, 6900 pieces of large ans 3600 pieces of jumbo, totaling6900+12600+6900+3600 =) 30000 pieces in total sales.
 
As I see the problem:

Your total sales is ~30,000 (=3*10^4) pieces of shrimps.

If you declare that 23% of your sales is small → then 0.23*3*10^4 pieces of shrimps are small. That converts to ~6900 pieces of small shrimp, 12,600 pieces of medium, 6900 pieces of large ans 3600 pieces of jumbo, totaling6900+12600+6900+3600 =) 30000 pieces in total sales.

This is true if I know that 23% of sales are small shrimp pieces. The nuance here is that I know 23% of pounds (not pieces) are small. A pound of small averages 35 pieces. By comparison, 12% of sales are jumbo pounds, each pound having 16 pieces on average. The stats on the other sized pounds are above.

If I do things out fully starting with the assertion that "23% of total pounds are small" equates to "23% of the total pieces are small" (etc.), you'll see that the percentages at the end diverge from the already known values.

So--

0.23 * 30,000 total shrimp = 6900 small shrimp
6900 small shrimp * (1 small lb/35 small shrimp) = 197.14 small pounds

0.42 * 30,000 total shrimp = 12,600 medium shrimp
12,900 medium shrimp * (1 medium lb/27 medium shrimp) = 477.78 medium pounds

0.23 * 30,000 total shrimp = 6900 large shrimp
6900 large shrimp * (1 large lb/22 large shrimp) = 313.64 large pounds

0.12 * 30,000 total shrimp = 3600 jumbo shrimp
3600 jumbo shrimp * (1 jumbo lb/16 jumbo shrimp) = 225.0 jumbo pounds

That makes 1213.56 total lb.

Now, in the projection I'm trying to make based on factual past performance, 23% of pounds should be small, 42% of pounds should be medium, 23% of pounds should be large, and 12% of pounds should be jumbo. However, when you look at the calculations and totals above, you'll see that (197.14 lb/1213.56 lb =) 16% of the pounds are small, (477.78 lb/1213.56 lb =) 39% of the pounds are medium, (313.64 lb/1213.56 lb =) 26% of the pounds are large, and (225 lb/1213.56 lb =) 19% of the pounds are jumbo. That means the method above is the wrong one for what I'm trying to accomplish.

The problem would be much easier if I knew the total pounds that will be made out of the 30k shrimp, then I could just take that pound total, apply the percentages, apply the pieces/lb to each size, and get the right answers. However, that bit of info is missing.
 
I'm getting a headache...does that enter the "picture":confused:

Sure, why not? Still, sorry if I confused you. To better explain, the longer a shrimp stays in the tank, the more time it has to grow and the larger it gets. That means when we start selling shrimp out of a batch (around 3 months), they're around 1/35 a pound each. When we sell the last of the shrimp in the same batch (about 3 months later), those final shrimp are about 1/16 a pound each and much bigger than they were three months ago. That's all I was getting at there.

I see it like this:
23% : 35 : .35 * 30000 = 10,500 = shrimps that weigh 1/35 of pound..
42% : 27 : .27 * 30000 = 8,100
23% : 22 : .22 * 30000 = 6,600
12% : 16 : .16 * 30000 = 4,800
=====================
100%:100 : 1 * 30000 = 30000

I think (hope) you're on the right track, but I don't think you have the right answer.

10,500 small shrimp / (35 small shrimp/lb) = 300lb of small shrimp
8,100 medium shrimp / (27 medium shrimp/lb) = 300lb of medium shrimp
6,600 large shrimp / (22 large shrimp/lb) = 300lb of large shrimp
4,800 large shrimp / (16 jumbo shrimp/lb) = 300lb of jumbo shrimp

So the final portions by pound are 25% each, which doesn't match the 23%, 42%, etc.

Your work did clue me in to a way to simplify the problem a little bit, and I received a clue the other day from a customer/math teacher ("Try chi squared and goodness of fit"). So it looks like I should show this problem to the statistics forum.

Thank you both for trying :)
 
A real life business problem with chi square and/or goodness of fit

I first posted this in the business forum, but a customer/math teacher clued me in that this could be a chi square or goodness of fit problem. I was horrible at statistics and don't remember either approach. Therefore, I was hoping someone(s) could help me out here. A quick background to the problem: I run an indoor shrimp growing company. Shrimp come to us at miniscule size and we grow them to sellable sizes. I need to come up with a projection of sales for the next six months for one batch of shrimp. I don't need a perfect answer, so I've simplified the problem below as much as I can.


The batch in question will have 30 thousand shrimp. Presume none of them die during production. Presume that they start at exactly the same mass and grow at exactly the same rate. Presume that the batch will sell out entirely in the end. Furthermore, presume that I will only sell shrimp from the batch at four exact mass milestones: 1/35 lb each (size "small") ; 1/27 lb each (size "medium") ; 1/22 lb each (size "large") ; 1/16 lb each (size "jumbo").


Here's where things get too tricky for me. Using actual past sales figures, I can reasonably project that 23% of the pounds (not pieces of shrimp) sold will be "small" pounds ; 42% will be "medium" pounds ; 23% will be "large" pounds ; and 12% will be "jumbo" pounds. I somehow need to use this percentage information to figure out the actual number of pounds of small I will sell, the actual number of pounds of medium, the actual number of pounds of large, and the actual number of pounds of jumbo.

If it's important in thinking through the problem, keep in mind that the four parts of the batch will be sold off at four different times. At the small size milestone, I'll sell from the batch until there are no more takers. The population of the batch will go down, and on the way to the next milestone, the remaining individuals in the batch will gain mass. This process repeats at the medium, large, and jumbo milestones.


This would be a simple problem if I knew the total poundage of all the shrimp sold in the batch, but I'm lacking that important bit of info.
 
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