4.10 SAT Math - Averages

Rule for Averages: Every time you see the word 'average' in a word problem, write down this formula!  This is the number of separate equations you will need to solve.

Take a look at this example:

The average distance traveled to school by 7 girls is 5 miles.  If the average distance traveled to school by three of the girls is 7 miles, what is the average distance traveled by the other four girls?

This kind of question can read a little confusingly, but remember the Average Rule.  How many average equations should you have written down right now?  You should have three.  That's how many equations you need.  From here, simply plug the information that relates into the formula:

The average distance traveled to school by 7 girls is 5 miles.   
What do we know in our formula ?  We know the Average, and we know the number of things, so plug those in: .  We can do the math to find that Total: 35.

If the average distance traveled to school by three of the girls is 7 miles,
What do we know here?  Same thing, plug those numbers in to your second formula: , and solve for T, giving you 21.

what is the average distance traveled by the other four girls?

Well, now we only know the number of things, four girls, right?  But can we find their Total distance traveled?  Sure, because we have the total all the girls traveled and the total the other 3 girls traveled.  So the total this group traveled must be All-3 girls, or 35-21.  So that number is 14.  So now we have numbers for our third  formula: .  Reduce that fraction and you get 3.5.  The average distance traveled by the other 4 girls is 3.5 miles.

-SAT Math Hint: Whatever information you're given, simply plug that in to the formula where it goes and solve for what's missing.  If you're given the average and the total, solve for the number of things.  If you're given the total and the number of things, solve for the average.  For the most part, these aren't going to be the simple averages you do in school.

Starting SAT Score 550 or Above:

Weighted Averages

Weighted averages are averages in which one or more groups contribute more to the outcome than others.  We saw an example of a weighted average at the beginning of this program under the discussion of Overly Obvious Answers.  We'll use that same example here:

17.  Susan goes to work at an average speed of 60 mph.  Assuming she takes the same route and makes no stops, if she averages 40 mph on the way home, what is her average speed in mph for the entire trip?
(A) 40
(B) 48
(C) 50
(D) 52
(E) 60

In this case, Susan spends more time on the road on the way home because she is driving slower.  So her way home takes longer, requiring her to spend more time going at the slower speed, and therefore weighs more into the equation.

When you're dealing with weighted averages that have to do with distances, speeds and the like, the simplest way to solve them is to simply pick a number for her distance.  Pick an easy number that her speeds divide into easily, let's say 120.  It doesn't matter if 120 miles seems a really far way to drive to work, it works nicely so we'll use it.  If work is 120 miles away and she drives at 60 mph, how long will it take her to get there?  2 hours.  Going home, she'll be going 120 miles at 40 mph, so it will take her 3 hours to get home.  How many miles in Total will she have driven?  240.  How many hours will it have taken her? 5.  So into our  we can now plug numbers in: and get an answer of 48.  She averaged 48 mph for the trip.

When dealing with a weighted average in which they tell you the contribution of each group, you can simply make sure that group is accurately represented.  Example:

14.  There have been 7 tests so far this semester.  Sam scored an 85 on three of them and a 90 on all the rest.  What grade must he get on the last test to achieve an average of 89 for the semester?

Here, they specifically tell you the "weight" of each group: there were 7 tests total, consisting of 3 85s and (once you subtract to find that out) 4 90s.  Let's plug those into our formula: .  See what happened?  We multiplied each group by its weight.  You could write out 85+85+85+90+90+90+90, but that's a lot of unnecessary work. 
Now what?  The problem is, we don't care what his average for the year is now, we need his average for the year after he takes another test.  So what do we do when we have an unknown?  We plug in a variable.  So the numerator should now read: 3•85 + 4•90 + x.  Has the denominator changed?  Definitely, because he's taking another test.  So the new formula all together should read: .  So how do we solve?  Well, we know our Average, too, don't we?  He wants an 89 for the semester.  So now we have: .  From here, solve for x.  You should end up with x=97.