Unit 30: Missing Average

There is a specific type of problem that appears frequently that we call “missing average”.  You will be presented with a question like the following:

A teacher said the average for a test was 86.  There are six students in the class, and five of the students shared their scores with each other.  They had scored 80, 82, 90, 94, and 91.  They were then able to calculate the sixth student’s score.  What was it?

A) 63
B) 79
C) 86
D) 88

How to identify it’s a missing average problem:

The problem will discuss the average of some group, and how that average will change when members of the group are added or taken away.

Strategy:

  1. Draw out your first missing average circle and fill it in.
  2. If necessary, draw your second missing average circle and fill in what you can.
  3. Always be looking for the Total Points. These are the numbers needed to solve missing average problems.

For answering these problems, we will use the average pie method. In the average pie method, draw yourself a pie, representing the idea that the total points = the # of items times the average of the items. 

$$Total\;points=(\#\;of\;items)(Average\;of\;the\;items)$$

For our initial example, we must first figure out the total points.  If six students averaged 86, then the total number of points must have been 86 x 6 = 516 total points.  

The five scores we were given add up to 437:

$$80+82+90+94+91=437$$

Since the total of the 6 scores must be 516, according to our circle, we’ve now revealed the missing number:

$$437+x=516$$

We subtract 437 from both sides, and the missing score is B) 79.

Another example:

A company wants to average 4.5 on a survey where 5 is the best possible score.  They give the survey to their best customers, and receive scores of 4.2, 4.7, 5, 4.9, and 4.8.  One more customer still needs to return their survey.  What is the lowest possible rating their next customer can give to keep at least a 4.5 average?

A) 4.5
B) 5
C) 2.7
D) 3.4

Again, we need to figure out the total points.  We see that six total customers are taking the survey, and they want to average 4.5, so we multiply \(4.5\times6=27\).

Then we find out how many points the company already has by taking: 

$$4.2+4.7+5+4.9+4.8=23.6\\23.6+x=27\\x=3.4$$

The lowest possible score they can receive while keeping a 4.5 average is 3.4.