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The SAT loves to ask a particular kind of question about averages that can pretty confusing without a nice, easy way to organize your information. Enter The Average Table. KNEEL WHEN IT ENTERS THE ROOM, KNAVE! Seriously, this thing kicks ass.

To build it, just remember what you have known for a long time about averages: how to calculate them. If I gave you a set of 5 test scores and asked you to average them, what would you do? You’d add them up, and then divide the total by 5. That’s because…

That’s just how averages work. But what if we multiplied both sides of that equation by [Number of Values]?

You can use this to set up a very handy little table, which will help you solve even the hairiest looking average questions. I’m going to use colors to help you see how! Aww yiss.

Let’s illustrate with a problem that looks like it sucks:

  1. A delivery truck is loaded with seven packages weighing an average of 30 pounds. At his first stop, the delivery man drops off three packages weighing a total of 60 pounds. He also picks up one package weighing 15 pounds. He makes one more stop to deliver two more packages, which weigh 42 and 48 pounds. What is the average weight, in pounds, of the packages that remain on the truck?
     
    (A) 15
    (B) 17
    (C) 19
    (D) 25
    (E) 30

OK. So let’s set up the average table, using the colors in the problem above to show what came from where (if you’re colorblind I’m so sorry):

7 packages
30 pounds
-3 packages
-60 pounds
+1 package
+15 pounds
-2 packages
-90 pounds

See where everything’s coming from? When we have an average (30 pounds for the initial 7 packages) we put it in the average column. When we have a sum, we put it in the sum column. We keep track of whether the packages are being delivered or picked up with + and – signs. Now let’s fill in the rest of the table just to see how everything works together (calculated values are in bold type…make sure you understand where they come from):

7 packages
30 pounds
 210 pounds
-3 packages
 20 pounds
-60 pounds
+1 package
 15 pounds
+15 pounds
-2 packages
 45 pounds
-90 pounds

We filled everything in just for practice, but for the next step we’re only going to need the values in the outer columns. So we know we started with a total weight in the truck of 210 pounds. We dropped off 3 packages weighing 60 pounds, picked up 1 package weighing 15 pounds, and dropped off 2 more weighing 90 pounds. Using our table, we can easily see that the number of packages left on the truck is 7-3+1-2 = 3, and the total weight on the truck is 210-60+15-90 = 75. So the average weight of the 3 packages left on the truck is 25 pounds! That’s choice (D)!

7 packages
30 pounds
 210 pounds
-3 packages
 20 pounds
-60 pounds
+1 package
 15 pounds
+15 pounds
-2 packages
 45 pounds
-90 pounds
= 3 packages 25 pounds = 75 pounds

Some last notes about the average table before I give you a few more practice problems:

  • You can only add or subtract up and down the outer columns. Try adding and subtracting averages and you’ll get all screwed up. You can only use the middle column for
    • calculating the sum by multiplying the number by the average, or
    • calculating the average by dividing the sum by the number.
  • This will work with questions that have variables instead of numbers, as long as you follow the rules (but it’s a good idea to substitute real numbers to make your life easier whenever possible).
Try these two for practice:

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Comments (13)

The question says the SUM of the 3 numbers is g, so you don’t need to multiply g by anything.

Also, I would be remiss if I didn’t take this opportunity to say that plug-in can really clarify a question like this. Say f = 10 and g = 20. If the average of 5 numbers is 10, then the sum of them is 50. If the sum of 3 of them is 20, then the sum of the other 2 must be 30. So the average of those 2 would be 15. Which answer choice gives you 15 when f = 10 and g = 20?

Sure, although it’s maybe even easier to understand in table form, so I’ve included a screenshot of the solution in my book below. The basic idea, as in most average questions, is that you need to work with the sums. If 12 students had an average score of 74, then the sum of their scores is 12×74=888. The 4 kids who averaged 96 contributed 4×96=384 to that total.

If you want to know the average of the 8 other kids, you first find the sum of their scores. 888-384=504. Now, to get the average, you divide that by the number of kids. 504/8=63.

To find the average of the remaining two numbers, find their sum! You know that all five numbers have an average of f, so their sum must be 5f. Then you know three of the numbers have a sum of g. So the sum of the remaining two numbers must be 5f – g. Since there are two numbers, you divide that by 2 to get their average.

You might find it helpful to plug in here, as well. I talked through a solution that uses that technique in this comment.

Can all average problems be solved with this table? What about this one:

A baseball team averages 5 runs scored per game over a span of 6 games. The team scored one run in the first game, two runs in the second game, three runs in the third game and four runs in the fourth game. What is the average of the runs they scored in games 5 and 6?

Sure, that can be done. I’ve attached an image below. Look at it, and follow along. Each game is one game, so you keep putting 1 in the number of numbers column for each of the first four games, where the team scored 1, 2, 3, and 4 games, respectively. Then you put a 2 for number of games, and an x for their sum total of runs.

You know the number of numbers times the average equals the sum, so if the team averages 5 runs a game for 6 games, you know they score a total of 30 runs.

1 + 2 + 3 + 4 + x = 30
x = 20

So the team scored 20 runs in its last two games—that’s an average of 10 runs per game for games 5 and 6.

Can you explain the question on Averages in your Diagnostic Drill #1? I think it’s question 6.
In a class of 6 students, the average height is 5 feet and 8 inches. If a student joins the class and causes the average height of the class to increase by 1 inch, what is the height of the new student? (1 foot = 12 inches)

Can it be solved with the Average Table?

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