Arithmetic Mean with Variables

Question

What is the arithmetic mean of a set one numbers with 1 variable? (i.e. - 10, 10, 10, 10, 8, 8, 8, 8, 12, 12, 11, y) ?

 

The question asks for us to find the arithmetic mean (average) of the ages of 12 houses on a city block. The ages are listed in the data set. I assume the equation would look something like this:

 

(107 + y) / 12 

 

The correct answer was 10 but I have no idea how they came to that conclusion. 

Answer 1

You are on the right track!

Yes,   (107 +y)/12  is the correct expression (so far) for finding the arithmetic mean (average).

Now, somewhere in the question there must be a clue about the variable y.  For example,

 - are all the values whole numbers?

 - was construction done in a specific pattern (e.g., outside in; note: the age must be 13)?

If the value of y is 13, the expression (107 + 13)/12 yields an average of 10 (whole number).

Now, the interesting question:  If the answer is to be rounded to the nearest whole year, what values (low and high) would make the mean NOT 10?

For values of y=7 up to y=18 (seems quite likely from the data), the arithmetic mean would be rounded to be 10.  If the question is advanced or is a reasoning question, then this explains it.  Otherwise, look for a clue in the wording of the question.

Answer 2

We need the average of 12 items and we only know 11 of them. Let's add them:

 

10 + 10 + 10 + 10 + 8 + 8 + 8+ 8 + 12 + 12 + 11 + y =  sum

 

107 is the sum of 11 items, and the average is 9.7. Let's guess y = 9.7 and add 12 terms.

 

116.7 is the sum of 12 items, and the average is 9.725. That's a little more than our our guess, but very close.

 

Average is just another way to say arithmetic mean, and if we round up to the nearest integer, the average is 10 years.  

 

 

 

Answer 3

Your answer is correct.  

 

There's no way to get a numeric answer (like 10) from an undefined variable (like y) so unless there's something else in the statement of the problem to suggest that y is equal 13 then I would say it's probably a misprint and they should have listed 13 instead of y.