Making the Grade

Date: 04/13/2001 at 00:41:53
From: LaChelle
Subject: Math word problem on average word problems

I am having difficulty with a certain kind of math word problem. My 
dad has tried to explain it to me, but I do not get it. Can you please 
explain to me why you do each step and how to this word problem?

After three tests, Amanda's average score is 88. What grade does she 
need on her next test to score a four-test average of 90?

Thank you so much.

Date: 04/13/2001 at 12:17:47
From: Doctor TWE
Subject: Re: Math word problem on average word problems

Hi - thanks for writing to Dr. Math.

I'll do a similar problem in two different ways. Maybe one of them 
will click for you, and you can use it to solve your problem.

Suppose Joe had an average of 77 after two tests, and he wanted to get 
a three-test average of 80. What would he have to score on the next 
test?

Method 1: Using Algebra

The average of two tests is defined as:

     Avg = (S1 + S2) / 2

where S1 and S2 are the scores on each of the two tests. Rearranging 
this, we can find that the sum of the scores of the tests thus far 
must be:

     (S1 + S2) = 2 * Avg
               = 2 * 77
               = 154

Now to get the average of three tests, we must take:

     Avg = (S1 + S2 + S3) / 3

We can substitute 154 for S1 + S2, since we found that value in the 
previous step. We also know that Joe wants the average after three 
tests to be 80, so we can replace Avg in the equation by 80. Then we 
have:

     80 = (154 + S3) / 3

Now all we have to do is solve for S3:

         80 = (154 + S3) / 3
     3 * 80 = 154 + S3
        240 = 154 + S3
         S3 = 240 - 154 = 86

So Joe must get an 86 on the next test. Can you think of a way to 
check this answer?


Method 2: "Find the differences"

One way to average 80 over three tests would be to get exactly 80 on 
each test. Likewise, one way to average 77 over two tests would be to 
get exactly 77 on each test.

So Joe's current average is the equivalent of getting exactly 77 on 
each of the first two tests. (Maybe he actually did, maybe not. But 
it's the equivalent of getting two 77's.) But to average 80, he needs 
to get the equivalent of an 80 on each of the three tests. How far 
short of the mark did he fall so far?

                 Test 1   Test 2
                 ------   ------
     Needed:       80       80
     Scored:       77       77
                  ----     ----
     Difference:   -3       -3

So far, he's 6 points short of his goal. Now, how much will he need to 
score on the next test to make up those points?

                 Test 1   Test 2   Test 3
                 ------   ------   ------
     Scored:       77       77       ??
     Needed:       80       80       80
                  ----     ----     ----
     Difference:   -3       -3       +6

To make up those 6 points, he'll need to score 80+6 = 86 points on the 
next test. Can you think of a way to check this answer? Of course, 
since we got the same answer using both methods, it looks pretty good, 
but it always helps if we can check our final answer.  ;-)

I hope this helps. If you have any more questions, write back.

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