I hope you had a good summer break and are ready to start school!
This month, I want to talk about the last two questions you’ll see on the SAT. As a reminder, there are two math sections on the new SAT (Section 3 and Section 4). Section 3 is “NO CALCULATOR” allowed, and Section 4 requires a calculator. Both sections have multiple choice and free response questions, and both sections progress from easier to harder questions. This progression happens for each question type, multiple choice and free response, independently. The last two questions you will see on your exam, then, will be math, calculator required, free response, and difficult, as determined by The College Board.
These two problems are always grouped together and based on the same information. Most of the time, these problems will ask you to model a situation in some way. Here is an example of the information that could be given before these two problems:
“If shoppers enter a store at an average rate of r shoppers per minute and each stays in the store for an average time of T minutes, the average number of shoppers in the store, N, at any one time is given by the formula N = rT. This relationship is known as Little’s law.
The owner of the Good Deals Store estimates that during business hours, an average of 3 shoppers per minute enter the store and that each of them stays an average of 15 minutes. The store owner uses Little’s law to estimate that there are 45 shoppers in the store at any time.”
The key to understanding these problems is having a firm idea about what the variables represent. The variable r, for example, has units of shoppers per minute. This means you will need to divide the number of shoppers by total minutes to calculate variable r.
“Little’s law can be applied to any part of the store, such as a particular department or the checkout lines. The store owner determines that, during business hours, approximately 84 shoppers per hour make a purchase and each of these shoppers spend an average of 5 minutes in the checkout line. At any time during business hours, about how many shoppers, on average, are waiting in the checkout line to make a purchase at the Good Deals Store?”
Let’s match units to variables:
- “…shoppers spend an average of 5 minutes…” This sounds like the description of variable T. We now know that T = 5 in our problem.
- “…approximately 84 shoppers per hour…” is close to the units needed for r; however, r is in shoppers per minute, not hour. To fix this, we convert 84 shoppers / 60 minutes = 1.4. We know that r = 1.4.
- We want to know, “…about how many shoppers, on average…,” which are the units for N.
This paragraph was a long-winded way of asking you to solve for N! The actual math involved in this problem looks like the following:
N = rt
N = (1.4)(5)
N = 7
Now, try the next problem in the set:
“The owner of the Good Deals Store opens a new store across town. For the new store, the owner estimates that, during business hours, an average of 90 shoppers per hour enter the store and each of them stays an average of 12 minutes. The average number of shoppers in the new store at any time is what percent less than the average number of shoppers in the original store at any times?