Sets and Grouping, Part IV

For the past few days I have been demonstrating sets and groupings using t-shirts as my sample material. Today I'll do some math for you. Here's the basic question:

If Duluth Trading stocked one of each combination of color and size of each variety, how many men's t-shirts would Duluth have?

(Use the 4 tables from Part I of this blog for your source of data. If you go to their website you might find that some shirts are out of stock, and some sizes discontinued, etc. but we will work from data in the catalog I used for the tables.)

I'd expect our typical student to do it this way:

Look at each line (lines are what I'll call a shirt model) in the 4 tables. Multiply the number of colors times the number of sizes to get the possible variations. Let's call these variation sums V. That's

 C x S = V

Add the products of our multiplication for all the lines (models) in that table to get a sub-total:

V1 + V2 + V3 ...  = ST

Take the sub-totals from each table and add them together for the grand total. This is simple math.

ST1 + ST2 + ST3 + ST4 = T

I typed the data into a spreadsheet (from my Duluth catalog - I don't work for them),  so I can create formulas to do the multiplication and addition work for me. Otherwise I would use paper to make notes as it's hard to hold the sum of a string of 13 numbers (table 1) in my head while I do tables 2, 3 and 4. So here are my sums:

ST1 = 572, ST2 = 110, ST3 = 22 and ST4 = 90.

We can add these in our heads. I'd do it this way 90 + 110 = 200 then 200 + 22 = 222 and 572 + 222 = 794. I did this out of order because I could see the subtotals were going to be easier.

Duluth would need to have 794 shirts in stock just to have one of each variation.

Now imagine we will buy these 794 shirts and give them to the local Rescue Mission so they can put a shirt on the back of every needy guy in town. Assuming we couldn't get any discounts, what would it cost to buy all these shirts

Could we come up with an average price per table by adding the prices in each table and dividing by the number of shirt models? Let's try that. There are 13 shirts in table 1, 5 in table 2, 3 in table 3 and 3 in table 4. I get table average prices of: 

ST1 = $17.12, ST2 = $21.30, ST3 = $21.83 and ST4 = $21.50. Add the sub-totals and divide by 4 = $20.44. This is our average price per shirt. Multiply by 794 to get $16,229.36.

We'll have to ante up more than 16 thousand dollars for the shirts using this approach.

Will it be different if I added all the numbers and prices first, and divided by the total shirts instead of using the 4 separate tables? Let's see ... the sum of shirt prices is $459 divided by 24 models = $19.13. This is our new average price per shirt. Multiply by 794 to get $15,189.22.

Wow! I can get the same shirts for $1000 less.

Will it be different if I do the math on each model (multiplying colors x sizes x prices) and then add those products? Indeed it will.

This approach results in a total of just $14,347.00 for all the shirts.

Notice the total is an even number of dollars (no cents). Because all the shirt prices are even or multiples of 50¢ and 794 is an even number, I think this total is more accurate than the ones with 22¢ or 36¢ at the end of the number. Why?

Later we can divide the total price by 794 to find an average price per shirt of $18.07.

All of these calculations were done properly by the spreadsheet. I checked with a calculator. But the totals on the first two approaches were wrong! Why?

These different yet inaccurate approaches are taken by hopeful students (and business people) all the time. If you were the generous t-shirt donor, and it was your $15,000, would you be interested in who was doing the math with your donations? I would be. Let's hope it's a former Excel Math student!