Word problem

Question

To earn an A in a course, a student must get an average score of at least 90 on three tests and a final exam. His final exam score counts as double, and his lowest score is removed if the final score is higher than the lowest score. If his first three tests scores are 86, 79, and 96, what is the lowest score he needs on the final exam to obtain a 90 average?

Answer 1

Katie:

 

Good question.  Tell me if I what I understand is correct. I assume that the scores on all tests and the final are out of 100.  Let x be his score on the final.

 

Case 1:  If the final score is less than or equal to 79, the highest average he can get is (86+79+96+2*79)/5, which is 419 / 5 = 83.8, This case does not obtain a  90 average.

 

Case 2: If the final score x is greater than 79 (80 or more), the score 79 is dropped.  The average score is then calculated as (86+96+2x) /4.  For this to be at least 90, we have to have 2x greater than or equal to 4*90 - (86+96) = 178. Hence x must be at least 178/2 = 89.  Note that this computed x is larger than 79, so the first case will not apply.

 

Answer: The lowest score he needs is 89 to get an average of at least 90.

 

Please confirm via posting a comment whether or not you agree.

 

Dr. G.

 

 

Answer 2

All right, Katie--let's rock this one.

 

The key is to think of the "double-counted" final as two single scores.

 

You have 86, 79, 96 so far, plus a final that will count double, AND you can drop the 79 if your score on the final is better than 79.

 

Let's assume that your score on the final WILL be better than 79, so drop the 79 off the list--buh-bye!

 

So now you have (so far) 2 single scores: 86 and 96.

 

And since the final counts as double, let's count it as 2 singles, plus the 2 singles you already have. That makes 4 singles, whose average has to be at least 90.

 

So you need a big number that, when you divide it by 4 (the number of tests), will give you 90 (the average). So an average score of 90, times 4 separate test scores, is 360 total points, right? Right. :-)

 

So to get 360 points, your final, which counts double, has to be a double 90, a.k.a. a 180, IF the other 2 add up to 180 as well.

 

BUT... they don't. They are an 86 + a 96, which adds up to 182. AWESOME--because now you don't even need a "double 90 = 180" on the final--you only need 178, because you have 2 extra points from your quizzes. So if your "doubled" score on the final needs to be 178, your regular score on the final needs to be ... (you've got this... say it out loud!... 89 ! BOOM!)

 

BUT I may be kind of dumb, so we better double check: if you get an 89 on the final, are you allowed to drop the low score? (Yes.) Does the 89 on the final count double? (Yes, so it's worth 178, right? Right.) And do your 182 more points add up to 360, so that your average of the 4 scores = 90?)

 

Yup. NOW before you're REALLY done, remember that you have to "answer the question that was asked." The question was,"what is the lowest score that the dude needs on the final in order to get a 90 average for the course?"

 

So your answer CAN'T just be a number, or your teacher will be all, "Katherine, you didn't answer the question," and we hate it when teachers do that. So make sure you state that "The dude needs to get at least 89 on the final to have an average of 90 or better for the course, Miss Dorkenberg..."

 

Hope this helps!

 

Matt in NY

Answer 3

Let's assume we do well on the final, represented by f. The lowest score, 79, would be dropped. So we are averaging 86, 96, f, and f since the final counts twice.

 

(86 + 96+ f +f)/ 4   >= 90

182 + 2f >= 360                simplify and cross multiply

2t    >=  178

t     >= 89

 

The problem with this is that the question is worded funny. This average is based on 2 tests and the final since we were allowed to drop one of the tests. The first line says I'm calculating the average by 3 tests and the final. I was expecting 4 test scores to be given so I could drop one and have 3 test and the final which would count twice.