**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!