Today's class was our "pre-test" for our Matrices unit. This pre-test prepares us for our up comming unit test which will take place on Friday.

We were given about half the time in class to write the pre-test. After that, Mr. K split us up into groups and following that we were to compare our answers with our group members. After comparing our answers we were to hand in one pre-test for our whole group which will be worth marks. The paper each group chose to hand in was the modified test paper (meaning the answers were looked over, and corrected by what the entire group felt was the right answer within reason.)

After that, Mr. K went over every single question and explained each question step by step.

Here is an example of a similar problem which was on the pre-test.

At a local grocery store, there are 2 available sizes in which milk can be purchased in. There is the 2 litre carton, and there is a 4 litre jug. Research shows that 23% of the 2 litre carton buyers switch to the 4 litre jug on their next purchase. And 41% of the 4 litre jug buyers switch the the 2 litre carton on their next purchase. The original market share is 40% of the 2 litre carton and 60% of the 4 litre jug.

a.) What is the market share for each the 2 litre and the 4 litre in the next round of purchases?

Well in order to find the next market share for the next round of purchases, you must write the initial state matrix, and you must also write the transition matix. After that, you must multiply the two together inorder to firgure out the market share for the next round of purchases.

The inital state matrix is the original percentage market share, which is 40% for the 2 litre and 60% for the 4 litre. Also, you must remember that the percentages in a state matrix should always be expressed as a decimal.

Let's call the initial state matrix " I "


Now we must figure out the transition matrix.

The information given states that 23% of the 2 litre carton buyers switch to the 4 litre jug on their next purchase. And 41% of the 4 litre jug buyers switch the the 2 litre carton on their next purchase.

So since 23% of the 2 litre buyers switch to the 4 litre, 77% remain as a 2 litre buyer. And since 41% of the 4 litre buyers switch the the 2 litre, 59% remain as a 4 litre buyer.

Lets call our transition matrix " T "



Now we must multiply the two matrices together (I*T) in order to get the market share for the next round of purchases. Lets call the resultant matrix "R"


Now the resultant is..

Meaning the market share between the two is 55% for the 2 litre carton and 45 % for the 4 litre jug.

b.) In the long term, what would the market share be for both sizes?

Well, in order to find the long term market share you must chose a large amount of years. Lets say 50. And then you must multiply I by T to the power of 50.

So it should look like this.
L is the resultant.


Now, we must put up the power by 1 making it 51 to clarify that the percentages have stabilized.

So we now find that the matrice has stabilized. Making the final market share 64% for the 2 litre carton and 36% for the 4 litre jug.



*JUST A REMINDER! *



The final unit matrices test is on Friday, Febuary 20. Remember to study and to be prepared!



I choose K_Hannah to scribe next:)!