Grade and Test Averages

Date: 12/02/2001 at 16:35:47
From: adam
Subject: Algebra

1. In  a class of 21 students the average score on a math test was 
   77%. If Suzie Smart got 95% and Jack Average got 78%, find the 
   average grade of the other 19 students.

2. Andy had an average of 87 on 4 test. What does he have to get on 
   his next test to get an average of 90?

Date: 12/03/2001 at 19:33:23
From: Doctor Greenie
Subject: Re: Algebra

Hi, Adam -

Here are descriptions of two ways to solve each of these problems.  
The first way for each problem is the way which is traditionally 
taught, and it is the way which is easiest to understand. The second 
way is not as easy to understand, but it is often much faster. (In 
fact, in both of these problems, the second method is much faster, 
because the numbers you have to work with are much smaller, and so the 
arithmetic is easier.)

I would certainly encourage you to use the second method if you 
understand how and why it works; if you aren't sure you understand it, 
the first method is always a good one.

First (traditional) method...

(1) 21 students averaged 77% on a test.  To simplify things, let's 
    just say there were 100 points and they averaged 77 points.  Then 
    the total number of points scored was 21*77 = ?. 

    Suzie and Jack together scored 95+78 = ? points. Figure out how 
    many points were scored altogether by the other 19 students and 
    divide that number by 19 to get the average of those other 19 
    students.

(2) On 4 tests, Andy averaged 87 points; the total number of points   
    on those 4 tests is 4*87 = ?. In order to average 90 points on 
    5 tests, the total number of points on the 5 tests must be 
    5*90 = ?.  

    The difference between those two totals is how many points he 
    needs to score on the 5th test (I hope there is some extra credit 
    available...!)

Second (alternative) method...

(1) The average of all 21 students was 77 points. Suzie's score of
    95 was 18 points above the average, and Jack's score of 78 was 
    1 point above the average. Together, those two scores were a total 
    of 18+1 = 19 points above the average. That means the other 19 
    scores must be a total of 19 points below the average; that means 
    that the average of those 19 scores must be 1 point below the 
    class average, or 76 points.

(2) Andy's average on the first 4 tests was 87 points; the desired 
    average for the 5 tests is 90 points. Each of the first 4 tests is 
    3 points below the desired average, so in all Andy is 4*3 = 12 
    points short on his first 4 tests. That means he must be 12 points 
    above his desired average on the 5th test, so his score on the 
    last test must be 90+12 = 102 points.

If you work both problems by the first method as outlined above, you 
should get the same answers I got using the alternative method - but 
you will see in both cases that you are working with bigger numbers, 
so the arithmetic is a bit harder.

I hope this helps.  Write back if you have any further questions on 
this type of problem.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/